# The Weyl tube theorem for Kähler manifolds

Andreas Bernig<EMAIL_ADDRESS>Institut für Mathematik, Goethe-
Universität Frankfurt, Robert-Mayer-Str. 10, 60054 Frankfurt, Germany  ,
Joseph H.G. Fu<EMAIL_ADDRESS>Department of Mathematics, University of
Georgia, Athens, GA 30602, USA  ,  Gil Solanes<EMAIL_ADDRESS>Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193
Bellaterra, Spain, and Centre de Recerca Matemàtica, Campus de Bellaterra,
08193 Bellaterra, Spain  and  Thomas Wannerer<EMAIL_ADDRESS>Friedrich-Schiller-Universität Jena, Fakultät für Mathematik & Informatik,
Institut für Mathematik, Ernst-Abbe-Platz 2, 07743 Jena, Germany

###### Abstract.

As sharpened in terms of Alesker’s theory of valuations on manifolds, a
classic theorem of Weyl asserts that the coefficients of the tube polynomial
of an isometrically embedded riemannian manifold
$M\hookrightarrow\mathbb{R}^{n}$ constitute a canonical finite dimensional
subalgebra $\mathcal{LK}(M)$ of the algebra $\mathcal{V}(M)$ of all smooth
valuations on $M$, isomorphic to the algebra of valuations on Euclidean space
that are invariant under rigid motions. We construct an analogous, larger,
canonical subalgebra $\mathcal{KLK}(M)\subset\mathcal{V}(M)$ for Kähler
manifolds $M$: i) if $\dim M=n$, then
$\mathcal{KLK}(M)\simeq\mathrm{Val}^{\mathrm{U}(n)}$, the algebra of
valuations on $\mathbb{C}^{n}$ invariant under the holomorphic isometry group,
and ii) if $M\hookrightarrow\tilde{M}$ is a Kähler embedding, then the
restriction map $\mathcal{V}(\tilde{M})\to\mathcal{V}(M)$ induces a surjection
$\mathcal{KLK}(\tilde{M})\to\mathcal{KLK}(M)$. This answers a question posed
by Alesker in 2010 and gives a structural explanation for some previously
known, but mysterious phenomena in hermitian integral geometry.

###### 2010 Mathematics Subject Classification:

32Q15 53A07 53A55 53C65

A.B. was supported by DFG grant BE 2484/5-2. J.H.G.F. was supported by a
Simons Collaboration Grant. G.S. was supported by the Serra Húnter Programme,
MICIU/AEI/FEDER grant PGC2018-095998-B-I00, and MICINN/AEI María de Maeztu
grant CEX2020-001084-M. T.W. was supported by DFG grant WA 3510/3-1.

###### Contents

  1. 1 Introduction
    1. 1.1 Lipschitz-Killing curvatures
    2. 1.2 The Kähler case
    3. 1.3 Local setting
    4. 1.4 Our approach
  2. 2 Background on valuations and curvature measures
    1. 2.1 Valuations on manifolds
      1. 2.1.1 Translation invariant valuations
    2. 2.2 Complex space forms
    3. 2.3 Riemannian valuations and curvature measures
  3. 3 Double forms
    1. 3.1 Double forms associated with a smooth map
      1. 3.1.1 Double exterior algebras
      2. 3.1.2 Double forms
    2. 3.2 Double forms on a riemannian manifold
    3. 3.3 Contractions
    4. 3.4 Exterior derivatives of riemannian double forms
    5. 3.5 Lipschitz-Killing curvature measures
  4. 4 Hermitian double forms
    1. 4.1 Algebraic curvature tensors
    2. 4.2 Double forms on Kähler manifolds
    3. 4.3 More relations among hermitian double forms
    4. 4.4 The Rumin differential
    5. 4.5 Restrictions
    6. 4.6 Some identities for pullbacks of double forms
      1. 4.6.1 The Weyl lemma
      2. 4.6.2 Proof of (61)
      3. 4.6.3 Proof of (62)
    7. 4.7 Embedded Kähler curvature tensors
    8. 4.8 Angularity of the $\Gamma_{k,p}^{M}$
  5. 5 The Gray algebra
    1. 5.1 Canonical elements of $\mathcal{G}(n)$
    2. 5.2 Structure of the algebra $\mathcal{G}(n)$
    3. 5.3 Proof of Theorem 5.5
  6. 6 The Kähler-Lipschitz-Killing algebra
    1. 6.1  The Kähler-Lipschitz-Killing curvature measures and the hermitian intrinsic volumes
    2. 6.2 Basic properties of the Kähler-Lipschitz-Killing curvature measures
    3. 6.3 Structure of $\mathcal{K}\mathcal{L}\mathcal{K}(M)$
    4. 6.4 Structure of $\widetilde{\mathcal{KLK}}$ as a $\mathcal{K}\mathcal{L}\mathcal{K}$-module
  7. 7 Integral geometry of complex space forms
    1. 7.1 The isomorphisms $r_{\lambda}$ and $R_{\lambda}$
    2. 7.2 Isomorphisms intertwining the kinematic operators
  8. 8 Open problems

##  1\. Introduction

###  1.1. Lipschitz-Killing curvatures

Given a smooth compact submanifold $M\subset\mathbb{R}^{n}$, Weyl [weyl_tubes]
expressed the volume of a tubular neighborhood about $M$ of small radius $r$
as a polynomial of degree $n$ in $r$. Weyl’s deeper point, however, was that
the coefficients are given by integrals of invariants of the curvature tensor
of $M$ with respect to the induced metric. These integrals, now known as total
Lipschitz-Killing curvatures, are extensions of the intrinsic volumes of
convex geometry, the same quantities that arise in the kinematic formulas of
Blaschke, Chern, and Santaló [santalo76]. Examples of total Lipschitz-Killing
curvatures are the Euler characteristic, the total scalar curvature and the
volume of $M$.

The (total) Lipschitz-Killing curvatures of riemannian manifolds appear in
various contexts. Let us mention just some of them. Cheeger-Müller-Schrader
proved that the Lipschitz-Killing curvatures of a riemannian manifold can be
obtained as limits of the Lipschitz-Killing curvatures of a sequence of
piecewise linear manifolds converging to the manifold in an appropriate sense.
The Lipschitz-Killing curvatures of piecewise linear manifolds are of a more
combinatorial nature, and have been used by physicists in Regge calculus, an
attempt to discretize important quantities in general relativity such as the
total scalar curvature.

Weyl’s tube formula was the main ingredient in the proof by Allendoerfer and
Weil of the Chern-Gauss-Bonnet theorem. The heat equation approach to the
Chern-Gauss-Bonnet theorem by Atiyah, Patodi, and Singer is also related to
Lipschitz-Killing curvatures. Recall that the short time asymptotics of the
heat kernel associated to the Laplacian acting on differential forms is given
by complicated polynomials in the curvature tensor and its derivatives. McKean
and Singer conjectured that some particular linear combination of these
polynomials equals the integrand in the Chern-Gauss-Bonnet theorem. In
particular, this would imply a ”fantastic cancellation” of all terms involving
derivatives. The conjecture was shown by Patodi and led to a heat equation
proof of the Chern-Gauss-Bonnet theorem and of the more general Atiyah-Singer
index theorem [atiyah_bott_patodi]. Some years later, Donnelly [donnelly]
observed that there are other linear combinations of the heat kernel
coefficients that do not contain derivatives of the curvature tensor; and
these combinations give precisely the Lipschitz-Killing curvatures. In
particular, the total Lipschitz-Killing curvatures are spectral invariants.

More recently, the theory of valuations, introduced by Alesker in the early
2000s, has enriched the context for these classical facts: the functionals
$\mu_{i}^{M}$ that yield the tube coefficients may be viewed as smooth
valuations—finitely additive measures—canonically associated to the riemannian
structure of $M$. They are called the intrinsic volumes on $M$. Remarkably,
Chern’s most general version of the Gauss-Bonnet theorem [chern45] shows that
$\mu_{0}^{M}=\chi$ is a smooth valuation (without using this terminology of
course).

A new and fascinating development is the extension of Lipschitz-Killing
curvatures to Alexandrov spaces. Alesker [alesker_alexandrov_spaces] stated
precise and far-reaching conjectures about the limits of the Lipschitz-Killing
curvatures of a sequence of manifolds with a uniform lower bound on the
sectional curvature converging to an Alexandrov space. Some special cases of
the conjecture are known (roughly for volume and total scalar curvature). For
recent results in this direction we refer to [alesker_riemannian_submersions,
alesker_katz_prosanov].

The main feature of the space $\mathcal{V}(M)$ of smooth valuations on $M$ is
the existence of a natural commutative product. From the perspective of
Alesker’s theory the intrinsic volumes are distinguished by two properties:
invariance under (i) isometric embeddings and (ii) the formation of Alesker
products. The resulting subalgebra $\mathcal{LK}(M)\subset\mathcal{V}(M)$ is
canonically isomorphic to $\mathbb{R}[t]/(t^{n+1})$, where $n=\dim M$, and is
called the Lipschitz-Killing algebra. Up to scale, the powers $t^{i}$ coincide
with the intrinsic volumes $\mu_{i}^{M}$. In the case $M=\mathbb{R}^{n}$, this
subalgebra coincides with the algebra $\operatorname{Val}^{\mathrm{O}(n)}$ of
valuations invariant under the euclidean group. In this context a natural
grading exists, with $\deg t=1$.

The existence of the product structure is related to the existence of
kinematic formulas on isotropic spaces. The algebras of isometry invariant
valuations on euclidean space, the euclidean sphere and hyperbolic space
coincide with the Lipschitz-Killing algebra and are thus mutually isomorphic.
This can be translated into the fact that the kinematic formulas on the real
space forms are isomorphic.

Bernig, Faifman, and Solanes [bernig_faifman_solanes,
bernig_faifman_solanes_part2, bernig_faifman_solanes_part3] have recently
constructed intrinsic volumes with analogous properties for pseudo-riemannian
manifolds.

###  1.2. The Kähler case

A natural question is to what extent the above can be generalized in the
complex setting. To be more precise, at Oberwolfach in 2010, Alesker asked
whether the classical picture admits a Kähler analogue: is there a canonical
subalgebra of valuations for Kähler manifolds $M$, analogous to the Lipschitz-
Killing algebra for the riemannian case?

Although tubes around complex submanifolds of flat hermitian space
$\mathbb{C}^{n}$ or of complex projective space
$\operatorname{\mathbb{C}P}^{n}$ do contain information about Chern classes
[shifrin81, bernig_fu_solanes], it is no surprise that they do not reflect the
complex geometry. We are therefore forced to look at this problem from the
valuations’ point of view.

In the flat hermitian case $\mathbb{C}^{n}$, we have a complete understanding
of the space of invariant valuations. The entry point was the determination
[alesker03_un, fu06] of the structure of the algebra
$\operatorname{Val}^{\operatorname{U}(n)}$ of smooth valuations on
$\mathbb{C}^{n}$ invariant under the group of holomorphic isometries: in this
case

(1)
$\operatorname{Val}^{\operatorname{U}(n)}\simeq\mathbb{R}[s,t]/(f_{n+1},f_{n+2}),$

where $\deg s=2$ and the relations are homogeneous of degrees $n+1,n+2$. Thus
the monomials $s^{i}t^{j}$ furnish a natural spanning set for
$\operatorname{Val}^{\operatorname{U}(n)}$. Up to a factor, $t$ corresponds to
the first intrinsic volume, which can be defined by averaging the Euler
characteristic of intersections with real hyperplanes. The other generator,
$s$, is defined by averaging intersections with complex hyperplanes.

Another useful basis, more natural from a geometric viewpoint, are the
hermitian intrinsic volumes $\mu_{k,p}$ from [bernig_fu_hig]. The isomorphism
(1) can be translated into explicit kinematic formulas for the pair
$(\mathbb{C}^{n},\operatorname{U}(n)\ltimes\mathbb{C}^{n})$, see
[bernig_fu_hig].

On complex projective space $\operatorname{\mathbb{C}P}^{n}$, $t$ may be
defined as the first intrinsic volume (with respect to the riemannian
structure provided by the Fubini-Study metric), while $s$ can be defined using
intersections with complex hyperplanes inside
$\operatorname{\mathbb{C}P}^{n}$. The algebra generated by these two
valuations equals the algebra of isometry invariant valuations on
$\operatorname{\mathbb{C}P}^{n}$. Surprisingly, and opposed to what happens in
the real case, the map $t\mapsto t,s\mapsto s$ from
$\operatorname{Val}^{\operatorname{U}(n)}$ to
$\mathcal{V}(\operatorname{\mathbb{C}P}^{n})^{\mathrm{Isom}}$ is not an
isomorphism. Even more surprisingly, as observed in [bernig_fu_solanes], there
are nevertheless some isomorphisms between
$\operatorname{Val}^{\operatorname{U}(n)}$ and
$\mathcal{V}(\operatorname{\mathbb{C}P}^{n})^{\mathrm{Isom}}$. In particular,
the kinematic formulas are isomorphic.

Previously, there was no satisfactory explanation of this phenomenon. Inspired
by Weyl’s tube approach and Alesker’s refinement in the riemannian setting, we
clarify the situation by providing a vast generalization of the notion of
hermitian intrinsic volumes to Kähler manifolds.

Our main result gives a positive answer to Alesker’s question.

######  Theorem A.

For any Kähler manifold $M$ of complex dimension $n$, there is a canonical
subalgebra $\mathcal{K}\mathcal{L}\mathcal{K}(M)\subset\mathcal{V}(M)$,
isomorphic to $\operatorname{Val}^{\operatorname{U}(n)}$, with the property
that if $M^{\prime}\hookrightarrow M$ is a Kähler embedding then the natural
restriction map $\mathcal{V}(M)\to\mathcal{V}(M^{\prime})$ restricts to a
natural surjection
$\mathcal{K}\mathcal{L}\mathcal{K}(M)\to\mathcal{K}\mathcal{L}\mathcal{K}(M^{\prime})$.

In geometric terms, the isomorphism with
$\operatorname{Val}^{\operatorname{U}(n)}$ maps the hermitian intrinsic volume
$\mu_{k,p}$ to the hermitian intrinsic volume $\mu_{k,p}^{M}$ that we will
construct.

Since the isometries of complex projective (or complex hyperbolic) space are
Kähler embeddings, the elements of
$\mathcal{KLK}(\operatorname{\mathbb{C}P}^{n})$ are isometry invariant. Our
main theorem thus provides a new perspective on invariant valuations and
integral geometry of $\operatorname{\mathbb{C}P}^{n}$.

######  Corollary 1.1.

The algebras of isometry invariant smooth valuations on $n$-dimensional
complex projective and on $n$-dimensional complex hyperbolic spaces are
canonically isomorphic to $\operatorname{Val}^{\operatorname{U}(n)}$.

Several isomorphisms between these algebras, algebraically simple but
geometrically unmotivated, were previously discovered in [bernig_fu_solanes].
The isomorphism constructed in the present paper, however, is canonical in the
sense that it arises from the Kähler structures on the underlying spaces. In
terms of the valuations $t$ and $s$ described above, the canonical isomorphism
is given by $t\mapsto t$ and $s\mapsto s/(1-\lambda s)$ (see Prop. 7.2). This
particular isomorphism was a key ingredient in the recent paper
[bernig_fu_conjecture].

###  1.3. Local setting

The kinematic formulas due to Chern, Blaschke, and Santaló were generalized by
Federer in two directions. First, instead of taking compact convex sets, he
considered sets of positive reach. Note that compact convex bodies as well as
compact submanifolds, possibly with boundary, are sets of positive reach.
Secondly, he localized the kinematic formulas by looking at so called
curvature measures, which are localizations of the intrinsic volumes. In the
euclidean case, there is a one-one correspondence between intrinsic volumes
and curvature measures. Consequently, Federer’s local kinematic formulas look
formally the same as the previously known global kinematic formulas. The same
happens on other real space forms, where each invariant valuation admits a
unique localization. We thus find that on all real space forms of the same
dimension, the local kinematic formulas are isomorphic.

In [fu_wannerer], a sequence of curvature measures naturally associated to
riemannian manifolds was constructed. The main observation was that the space
of such curvature measures is a universal module over the Lipschitz-Killing
algebra, i.e. the structure coefficients do not depend on the riemannian
manifold. In the case of real space forms, the module structure is related to
a version of kinematic formulas called semi-local kinematic formulas. Hence
such formulas on all real space forms of the same dimension are isomorphic.

In the complex case, it follows easily from Howard’s transfer principle (which
applies more generally to homogeneous manifolds [howard93]) that local
kinematic formulas on complex space forms of the same dimension are formally
identical. To write them down explicitly is highly non trivial. The reason is
that, in contrast to the real case, the globalization map is not injective, so
the local kinematic formulas cannot be derived from the global ones. The
problem was solved in [bernig_fu_solanes], see also
[bernig_fu_solanes_proceedings] for a shorter way of describing such formulas.

In view of our first main theorem A and of the results concerning riemannian
curvature measures mentioned above it is natural to ask for a Kähler version.
This is our second main theorem.

######  Theorem B.

For any Kähler manifold $M$ of complex dimension $n$, there is a canonical
$\mathcal{K}\mathcal{L}\mathcal{K}(M)$-module
$\widetilde{\mathcal{K}\mathcal{L}\mathcal{K}}(M)\subset\mathcal{C}(M)$,
isomorphic to the $\operatorname{Val}^{\operatorname{U}(n)}$-module
$\operatorname{Curv}^{\operatorname{U}(n)}$, with the property that if
$M^{\prime}\hookrightarrow M$ is a Kähler embedding then the natural
restriction map $\mathcal{C}(M)\to\mathcal{C}(M^{\prime})$ restricts to a
natural surjection
$\widetilde{\mathcal{K}\mathcal{L}\mathcal{K}}(M)\to\widetilde{\mathcal{K}\mathcal{L}\mathcal{K}}(M^{\prime})$.

Applied to complex projective and complex hyperbolic space the theorem yields

######  Corollary 1.2.

There is a canonical module isomorphism between the modules of invariant
curvature measures on complex space forms of the same dimension, compatible
with the canonical isomorphism from Corollary 1.1.

This is a new insight in hermitian integral geometry and a nice extra property
of the canonical valuation isomorphism. It follows from these results that,
quite unexpectedly, there are isomorphisms for all integral geometric
structures on complex space forms: global, semi-local and local kinematic
formulas; algebra structure on valuations, module structure on curvature
measures.

###  1.4. Our approach

Our strategy for proving Theorems A and B is a hybrid of two different
approaches to the construction of the Lipschitz-Killing algebra in the
riemannian case. In the first, the Nash theorem is used to embed a given
riemannian manifold $M$ in some $\mathbb{R}^{N}$. The valuations giving the
tube coefficients are then restrictions of the intrinsic volumes $\mu_{i}$ of
$\mathbb{R}^{N}$, which by Weyl’s theorem depend only on the riemannian
structure of $M$. The resulting restrictions $\mu_{i}^{M}\in\mathcal{V}(M)$
then span $\mathcal{LK}(M)$: fundamental results of Alesker show that this
span is closed under Alesker multiplication.

That approach has the advantage of being simple and direct. On the other hand,
it fails to show how the $\mu_{i}^{M}$ and their algebraic properties arise
from the riemannian structure of $M$. A second approach, remedying this
defect, was given in [fu_wannerer], in which the $\mu_{i}^{M}$ are constructed
from canonical differential forms associated to the connection and the
curvature tensor of $M$. Certain combinations of these forms were shown to be
subject to an algebraic model of the fundamental formula of [alesker_bernig]
for the Alesker product. Using invariant theory, the possible outcomes were
shown to be highly constrained, so that the actual values could be computed
from a few simple templates.

Attempting to extend these approaches to the Kähler case we consider here, we
meet two obstacles: i) there can be no Kähler analogue of the Nash theorem, so
the first approach is impossible; and ii) the relevant space of invariant
differential forms is large and unwieldy, to the point where we were unable to
mimic the second approach above in the new setting. Happily, however, we are
able to combine weak versions of these two devices to bring the proof to
fruition. First, via the formalism of double forms we construct algebraic
models for the differential forms underlying the $\mu_{k,p}^{M}$. Although our
understanding of the invariant theory is insufficient to conclude that the
Alesker product of two such valuations lies in their span, we are nonetheless
able to deduce that the underlying differential form of such a product depends
polynomially on the curvature tensor $\operatorname{\mathsf{R}}$ of $M$
(Proposition 4.7). Second, we point out (Theorem 4.12) that we do have an
extremely weak Kähler isometric embedding theorem: for
$\operatorname{\mathsf{R}}$ lying in some open subset of the space of
algebraic curvature tensors at a single point $x\in M$, there is a local
embedding of $M$ into some $\mathbb{C}^{N}$ for which the curvature tensor at
$x$ of the Kähler metric induced on $M$ by that of $\mathbb{C}^{N}$ is equal
to $\operatorname{\mathsf{R}}$. Since the $\mu_{k,p}^{M}$ behave well under
restrictions (this will be shown in Subsection 4.5), the polynomial dependence
on $\operatorname{\mathsf{R}}$ of the Alesker product of hermitian intrinsic
volumes implies that the multiplicative structure is as claimed.

To conclude this introduction, let us point out the importance of the
formalism of double forms— introduced originally by de Rham [deRham] and
subsequently used to brilliant effect in Gray’s work on integral geometry
[gray69], [gray_book]— in this paper. Section 3 clarifies the construction of
the canonical curvature measures of [fu_wannerer] in these terms. In fact they
are polynomials in the canonical $(1,1)$ forms $\operatorname{\mathsf{g}}$
(the metric) and $\bm{\omega}$ (the connection), as well as the $(2,2)$-form
$\operatorname{\mathsf{R}}$ representing curvature. Section 4 then constructs
similar objects in the Kähler case, where the resulting algebraic structure is
much richer, due to the appearance of the canonical (2,2)-form
$\operatorname{\mathsf{G}}$ and the additional symmetries enjoyed by Kähler
curvature tensors $\operatorname{\mathsf{R}}$. As in [fu_wannerer], the fact
(Proposition 4.6) that the Rumin differential of these elements is again
expressible in this language is key.

##  2\. Background on valuations and curvature measures

###  2.1. Valuations on manifolds

Let us begin with a brief account of the theory of _valuations on smooth
manifolds_ $M$, introduced by Alesker [alesker_val_man1, alesker_val_man2,
alesker_val_man4, alesker_val_man3], in the case (sufficient to the purposes
of the present paper) where $M$ is an oriented riemannian manifold.

Given an oriented riemannian manifold $M$ of dimension $n$, we let $SM$ be its
sphere bundle. The contact form on $SM$ is denoted by $\alpha$. A sufficiently
regular set $X\subset M$ (e.g. a compact differentiable polyhedron
[alesker_intgeo], or a compact subset with positive reach [federer59], or a
compact subanalytic set [fu94], or a compact WDC set [pokorny_rataj,
fu_pokorny_rataj]) determines a normal cycle $\operatorname{nc}(X)$, which is
a current in $SM$. The normal cycle is finitely additive for such subsets
$X,Y\subset M$ in general position:

(2) $\operatorname{nc}(X\cup
Y)=\operatorname{nc}(X)+\operatorname{nc}(Y)-\operatorname{nc}(X\cap Y).$

A _smooth valuation on $M$_ is a functional of the form

(3) $\mu(X)=\int_{X}\phi+\int_{\operatorname{nc}(X)}\omega,$

where $\phi\in\Omega^{n}(M)$ is the inner term and $\omega\in\Omega^{n-1}(SM)$
is the boundary term. We denote it by $\mu=[[\phi,\omega]]$. A _smooth
curvature measure on $M$_ is a functional of the form

(4) $\Phi(X,B)=\int_{X\cap
B}\phi+\int_{\operatorname{nc}(X)\cap\pi^{-1}B}\omega,$

for $B\subset M$ a Borel subset. Such a curvature measure is written as
$[\phi,\omega]$. These objects inherit finite additivity from (2).

The space of smooth valuations is denoted by $\mathcal{V}(M)$, the space of
smooth curvature measures by $\mathcal{C}(M)$. The _globalization map_
$\operatorname{glob}\colon\mathcal{C}(M)\to\mathcal{V}(M)$ is the map given by
$\operatorname{glob}(\Phi)=[\Phi]:=\Phi(\bullet,M)$. The space
$\mathcal{V}(M)$ admits the structure of an algebra [alesker_bernig,
alesker_val_man3], while $\mathcal{C}(M)$ is a module over $\mathcal{V}(M)$
[bernig_fu_solanes]. If $e:M\subset N$ is an embedding of riemannian
manifolds, there are obvious restriction maps
$e_{\mathcal{V}}^{*}:\mathcal{V}(N)\to\mathcal{V}(M),e_{\mathcal{C}}^{*}:\mathcal{C}(N)\to\mathcal{C}(M)$
(cf. [alesker_intgeo, bernig_fu_solanes]). These maps respect the
multiplicative structure:

$e_{\mathcal{V}}^{*}(\mu\cdot\nu)=e_{\mathcal{V}}^{*}\mu\cdot
e_{\mathcal{V}}^{*}\nu,\quad e_{\mathcal{V}}^{*}\mu\cdot
e_{\mathcal{C}}^{*}\Phi=e_{\mathcal{C}}^{*}(\mu\cdot\Phi).$

The kernel of the map $(\phi,\omega)\mapsto[[\phi,\omega]]$ was described in
[bernig_broecker07] as the space of all pairs $(\phi,\omega)$ such that
$D\omega+\pi^{*}\phi=0,\pi_{*}\omega=0$. Here $D$ is the Rumin operator
[rumin94] and $\pi^{*},\pi_{*}$ are pull-back and push-forward of differential
forms with respect to the projection $\pi:SM\to M$. The kernel of the map
$(\phi,\omega)\mapsto[\phi,\omega]$ is given by the pairs $(0,\omega)$ where
$\omega$ belongs to the ideal generated by $\alpha$ and $d\alpha$, see
[bernig_faifman_solanes_part2, Proposition 2.4]. Set

(5) $\zeta:=\pi_{*}\omega\in
C^{\infty}(M),\quad\tau:=D\omega+\pi^{*}\phi\in\Omega^{n}(SM).$

It is easy to check that $\alpha\wedge\tau=0,d\tau=0,\pi_{*}\tau=d\zeta$.
Conversely, if $(\zeta,\tau)$ is a pair with $\zeta\in
C^{\infty}(M),\tau\in\Omega^{n}(SM)$ such that
$\alpha\wedge\tau=0,d\tau=0,\pi_{*}\tau=d\zeta$, then there exist
$\phi,\omega$ satisfying (5). We may therefore describe smooth valuations on
$M$ uniquely by such pairs $(\zeta,\tau)$. In this case we write

(6)
$\mu=\\{\\{\zeta,\tau\\}\\},\quad\mbox{and}\quad\zeta=\zeta(\mu),\quad\tau=\tau(\mu).$

The product of smooth valuations admits a description in these terms
[alesker_bernig]*Eqs. (48),(49), which for our purposes may be distilled as
follows. Fix $x\in M$. Then for each $\xi\in S_{x}M$, the Levi-Civita
connection induces a decomposition

(7) $T_{\xi}SM=H_{\xi}\oplus T_{\xi}(S_{x}M)$

into horizontal and vertical subspaces, where the derivative of the projection
map $\pi:SM\to M$ restricts to an isomorphism $H_{\xi}\simeq T_{x}M$. These
decompositions in turn induce an isomorphism

(8)
$\left.\Omega^{\bullet}(SM)\right|_{S_{x}M}\simeq{\bigwedge}^{\bullet}T_{x}^{*}M\otimes\Omega^{\bullet}(S_{x}M).$

Similarly, for a fixed $\xi\in S_{x}M$,

(9)
${\bigwedge}^{\bullet}T_{\xi}^{*}SM\simeq{\bigwedge}^{\bullet}(T_{x}M\oplus\xi^{\perp})^{*},$

where $T_{\xi}(S_{x}M)\simeq\xi^{\perp}\subset T_{x}M$. Thus, distinguishing a
unit vector $e_{0}\in\mathbb{R}^{n}$, and an isometry
$\sigma:\mathbb{R}^{n}\to T_{x}M$ with $\sigma e_{0}=\xi$ yields isomorphisms

(10) $\displaystyle\bar{\sigma}^{*}$
$\displaystyle\colon\left.\Omega^{\bullet}(SM)\right|_{S_{x}M}\to{\bigwedge}^{\bullet}(\mathbb{R}^{n})^{*}\otimes\Omega^{\bullet}(S^{n-1})$
(11) $\displaystyle\bar{\sigma}_{\xi}^{*}$
$\displaystyle\colon{\bigwedge}^{\bullet}T_{\xi}^{*}SM\to{\bigwedge}^{\bullet}(\mathbb{R}^{n}\oplus
e_{0}^{\perp})^{*}.$

######  Proposition 2.1.

For any isometry $\sigma\colon\mathbb{R}^{n}\to T_{x}M$ with $\sigma
e_{0}=\xi$ as above, there exist bilinear maps

$\Gamma:\left[{\bigwedge}^{\bullet}(\mathbb{R}^{n})^{*}\otimes\Omega^{\bullet}(S^{n-1})\right]^{2}\to{\bigwedge}^{\bullet}(\mathbb{R}^{n}\oplus
e_{0}^{\perp})^{*}$

and

$\mathrm{GT}\colon\Omega^{\bullet}(SM)^{2}\to\Omega^{\bullet}(SM),$

making the diagram

${\left[\left.\Omega^{\bullet}(SM)\right|_{S_{x}M}\right]^{2}}$${\left.\Omega^{\bullet}(SM)\right|_{S_{x}M}}$${{\bigwedge}^{\bullet}T^{*}_{\xi}SM}$${\left[{\bigwedge}^{\bullet}(\mathbb{R}^{n})^{*}\otimes\Omega^{\bullet}(S^{n-1})\right]^{2}}$${{\bigwedge}^{\bullet}(\mathbb{R}^{n}\oplus
e_{0}^{\perp})^{*}}$$\bar{\sigma}^{*}\oplus\bar{\sigma}^{*}$$\mathrm{GT}$$|_{\xi}$$\Gamma$$\bar{\sigma}_{\xi}^{*}$

commutative and such that the following hold:

  1. (1)

If $\mu_{i}=\\{\\{\zeta_{i},\tau_{i}\\}\\},i=1,2$ are smooth valuations, then
$\mu_{1}\cdot\mu_{2}=\\{\\{\zeta,\tau\\}\\}$ with

(12) $\displaystyle\zeta$ $\displaystyle=\zeta_{1}\cdot\zeta_{2},$ (13)
$\displaystyle\tau$
$\displaystyle=\mathrm{GT}(\tau_{1},\tau_{2})+\pi^{*}\zeta_{1}\cdot\tau_{2}+\pi^{*}\zeta_{2}\cdot\tau_{1}.$

  2. (2)

If $\mu=\\{\\{\zeta,\tau\\}\\}$ is a smooth valuation and $[\phi,\omega]$ is a
curvature measure, then
$\mu\cdot[\phi,\omega]=[\phi^{\prime},\omega^{\prime}]$ with

$\displaystyle\phi^{\prime}$ $\displaystyle=\pi_{*}(\omega\wedge
s^{*}\tau)+\zeta\cdot\phi,$ $\displaystyle\omega^{\prime}$
$\displaystyle=\mathrm{GT}(\omega,\tau)+\pi^{*}\zeta\cdot\omega,$

where $s\colon SM\to SM$ denotes the antipodal map.

The notation refers to the fact that $\mathrm{GT}$ is a Gelfand transform with
respect to a certain double fibration, described in the proof below.

###### Proof.

We use the description of the Alesker-Bernig formula given in
[fu_alesker_product]*Theorem 5.2 together with the notation introduced there.
In particular, as in [fu_alesker_product]*Section 5 consider the fiber bundle
$\Sigma\to M$ with the projections $\xi,\eta,\zeta\colon\Sigma\to SM$, given
by

$\Sigma:=\\{(\xi,\eta,\zeta)\in
SM^{3}:\pi\xi=\pi\eta=\pi\zeta,\xi\neq\pm\eta,\zeta\text{ lies in the open
geodesic segment }\overline{\xi\eta}\\}.$

At each point $(\xi,\eta,\zeta)$ of $\Sigma$ the Levi-Civita connection of $M$
defines a splitting

(14) $T_{\xi,\eta,\zeta}\Sigma=H^{\Sigma}\oplus V^{\Sigma}$

into horizontal and vertical tangent vectors, where $H^{\Sigma}\simeq T_{x}M$
and $x\in M$ is the common projection of $\xi,\eta,\zeta$. The vertical
subspace splits further as follows. First note that the riemannian metric of
$M$ induces an inner product on $V^{\Sigma}$. Let
$\Sigma_{\zeta}\subset\Sigma_{x}$ denote a fiber of $\zeta\colon\Sigma\to SM$.
Thus $V^{\Sigma}=V^{\prime}\oplus V^{\prime\prime}$, where $V^{\prime\prime}$
is the tangent space to the fiber $\Sigma_{\zeta}$ and $V^{\prime}\simeq
T_{\zeta}(S_{x}M)$ is its orthogonal complement inside $V^{\Sigma}$.

Recall that also the tangent spaces to $SM$ split into horizontal and vertical
vectors,

$T_{\zeta}SM=H\oplus V.$

Since the differential $\zeta_{*}\colon V^{\Sigma}\to V$ is onto with kernel
$V^{\prime\prime}$, $\zeta_{*}:V^{\prime}\to V$ is an isomorphism.

Consider the model fiber

$\Sigma_{0}=\\{(\xi_{0},\eta_{0},\zeta_{0})\in(S^{n-1})^{3}\colon\xi_{0}\neq\pm\eta_{0}\text{
and }\zeta_{0}\text{ lies in the open geodesic segment
}\overline{\xi_{0}\eta_{0}}\\}$

with the projections $\xi_{0},\eta_{0}\colon\Sigma_{0}\to S^{n-1}$. Fixing
$\zeta_{0}=e_{0}\in\mathbb{R}^{n}$, the tangent space to $\Sigma_{0}$ splits
as $T_{\xi_{0},\eta_{0},e_{0}}\Sigma_{0}\simeq V^{\prime}_{0}\oplus
V^{\prime\prime}_{0}$, where $V^{\prime}_{0}\simeq
e_{0}^{\perp}=T_{e_{0}}S^{n-1}$ and $V_{0}^{\prime\prime}\simeq
T_{\xi_{0},\eta_{0}}\Sigma_{0,e_{0}}$, mirroring the splitting above. For any
isometry $\sigma\colon\mathbb{R}^{n}\to T_{x}M$ with $\sigma e_{0}=\zeta$ , we
obtain from (14) induced isomorphisms

$\displaystyle T_{\xi,\eta,\zeta}\Sigma\simeq\mathbb{R}^{n}\oplus
T_{\sigma^{-1}\xi,\sigma^{-1}\eta,e_{0}}\Sigma_{0}\simeq\mathbb{R}^{n}\oplus
e_{0}^{\perp}\oplus V^{\prime\prime}_{0}$ $\displaystyle
T_{\xi}SM\simeq\mathbb{R}^{n}\oplus T_{\sigma^{-1}\xi}S^{n-1},$

both of which we denote by $\sigma_{*}$. Consequently, the following diagram
commutes

${T_{\xi,\eta,\zeta}\Sigma}$${\mathbb{R}^{n}\oplus
T_{\sigma^{-1}\xi,\sigma^{-1}\eta,e_{0}}\Sigma_{0}}$${T_{\xi}SM}$${\mathbb{R}^{n}\otimes
T_{\sigma^{-1}\xi}S^{n-1}}$$\xi_{*}$$\sigma_{*}$$\operatorname{id}_{\mathbb{R}^{n}}\oplus\xi_{0*}$$\sigma_{*}$

yielding a simple description of the differential $\xi_{*}$ in terms of the
model space. Likewise, the projection to top-degree forms on the fiber
$\Sigma_{\zeta}$ admits a description in terms of the model space

${{\bigwedge}^{\bullet}T^{*}_{\xi,\eta,\zeta}\Sigma}$${{\bigwedge}^{\bullet}(\mathbb{R}^{n}\oplus
T_{\sigma^{-1}\xi,\sigma^{-1}\eta,e_{0}}\Sigma_{0})^{*}}$${{\bigwedge}^{\bullet}(H\oplus
V)^{*}\otimes\bigwedge^{n}(V^{\prime\prime})^{*}}$${{\bigwedge}^{\bullet}(\mathbb{R}^{n}\oplus
e_{0}^{\perp})^{*}\otimes\bigwedge^{n}T^{*}_{\sigma^{-1}\xi,\sigma^{-1}\eta}\Sigma_{0,e_{0}}}$$\sigma_{*}$$\sigma_{*}$

Combining these two operations with integration over the fiber we obtain the
commutative diagram

${(\left.\Omega^{\bullet}(SM)\right|_{S_{x}M})^{\otimes
2}}$${({\bigwedge}^{\bullet}(\mathbb{R}^{n})^{*}\otimes\Omega^{\bullet}(S^{n-1}))^{\otimes
2}}$${\left.\Omega^{\bullet}(\Sigma)\right|_{\Sigma_{\zeta}}}$${{\bigwedge}^{\bullet}(\mathbb{R}^{n})^{*}\otimes\left.\Omega^{\bullet}(\Sigma_{0})\right|_{\Sigma_{0,e_{0}}}}$${{\bigwedge}^{\bullet}T^{*}_{\zeta}SM\otimes\Omega^{n}(\Sigma_{\zeta})}$${{\bigwedge}^{\bullet}(\mathbb{R}^{n}\oplus
e_{0}^{\perp})^{*}\otimes\Omega^{n}(\Sigma_{0,e_{0}})}$${{\bigwedge}^{\bullet}T^{*}_{\zeta}SM}$${{\bigwedge}^{\bullet}(\mathbb{R}^{n}\oplus
e_{0}^{\perp})^{*}}$$\xi^{*}\wedge\eta^{*}$$(\operatorname{id}_{\mathbb{R}^{n}}\times\xi_{0})^{*}\wedge(\operatorname{id}_{\mathbb{R}^{n}}\times\eta_{0})^{*}$$\int_{\Sigma_{0,e_{0}}}$$\int_{\Sigma_{\zeta}}$

where the horizontal arrows are isomorphisms induced by $\sigma$. Composed
with $\operatorname{id}\otimes s^{*}$, where $s$ is the antipodal map, the
left column defines $\mathrm{GT}$ and the right column defines $\Gamma$. ∎

####  2.1.1. Translation invariant valuations

If $M$ is a finite dimensional real vector space $V$, then any finite union of
compact convex bodies admits a normal cycle, and any two such finite unions
$X,Y$ satisfy (2). We denote the space of translation invariant smooth
valuations on $V$ by
$\operatorname{Val}=\operatorname{Val}(V)\subset\mathcal{V}(V)$. In fact it is
a graded algebra, with multiplication inherited from $\mathcal{V}(V)$ and
grading

$\operatorname{Val}=\bigoplus_{\begin{subarray}{c}k=0,\ldots,n\\\
\epsilon=\pm\end{subarray}}\operatorname{Val}_{k}^{\epsilon},$

where $n=\dim V$ and where $\operatorname{Val}_{k}^{\epsilon}$ is the subspace
of $k$-homogeneous valuations of parity $\epsilon$. (Note that there exists a
more primitive, and historically prior, notion of valuation in this context,
in which the smoothness condition is relaxed to continuity.)

By $\operatorname{Curv}\subset\mathcal{C}(V)$ we denote the space of smooth
and translation invariant curvature measures (meaning that
$\Phi(K+v,B+v)=\Phi(K,B)$ for all $v\in V$). It admits a grading by degree

$\operatorname{Curv}=\bigoplus_{k=0,\ldots,n}\operatorname{Curv}_{k},$

where $\deg\Phi=k$ iff $\Phi(tK,tB)=t^{k}\Phi(K,B),t>0$. $\operatorname{Curv}$
is a module over $\operatorname{Val}$.

A famous theorem of Hadwiger implies that the vector space
$\operatorname{Val}^{\mathrm{SO}(n)}$ of $\mathrm{SO}(n)$-invariant elements
in $\operatorname{Val}$ is spanned by the intrinsic volumes
$\mu_{0},\ldots,\mu_{n}$ (in fact Hadwiger’s theorem yields the same
conclusion even in the broader context mentioned above, where smoothness is
replaced by continuity). More generally, if $G\subset\mathrm{O}(n)$ is a
subgroup that acts transitively on the unit sphere $S^{n-1}$, then the
subspace of $G$-invariant elements $\operatorname{Val}^{G}$ is a finite
dimensional graded subalgebra of $\operatorname{Val}$.

In the present paper, we are particularly interested in the case
$G=\operatorname{U}(m)\subset\mathrm{O}(2m)$. In this case [alesker03_un,
fu06]

$\operatorname{Val}^{\operatorname{U}(m)}\cong\mathbb{R}[t,s]/(f_{m+1},f_{m+2}),$

where $\log(1+tx+sx^{2})=\sum_{k=1}^{\infty}f_{k}(t,s)x^{k}$. The valuation
$t$ is (up to scale) the mean width, or the first intrinsic volume, while
$s\in\operatorname{Val}_{2}$ is (up to scale) the average area of the
projections to complex lines. In particular,

$\dim\operatorname{Val}_{k}^{\mathrm{U}(m)}=1+\min\left(\left\lfloor\frac{k}{2}\right\rfloor,\left\lfloor\frac{2m-k}{2}\right\rfloor\right).$

The space of $\operatorname{U}(m)$-invariant and translation invariant
curvature measures was studied in [park02], [bernig_fu_hig] and
[bernig_fu_solanes]. It admits a natural basis $\Delta_{k,q},N_{k,q}$ such
that $[\Delta_{k,q}]=\mu_{k,q},[N_{k,q}]=0$; and another natural basis
$B_{k,q},\Gamma_{k,q}$ with $[B_{k,q}]=[\Gamma_{k,q}]=\mu_{k,q}$. The
dimensions are given by

$\dim\operatorname{Curv}_{k}^{\mathrm{U}(m)}=\min(k,2m-k{-1})+1.$

###  2.2. Complex space forms

Let $(\operatorname{\mathbb{C}P}^{n}_{\lambda},G_{\lambda})$ denote the
complex space form of complex dimension $n$ and holomorphic curvature
$4\lambda$ (depending on the sign of $\lambda$ this will be a rescaling of
complex hyperbolic space, hermitian space, or complex projective space); the
group $G_{\lambda}$ is the isometry group if $\lambda\neq 0$, or the unitary
isometry group $\mathrm{U}(n)\ltimes\mathbb{C}^{n}$ if $\lambda=0$. The paper
[bernig_fu_solanes] displayed several non-canonical isomorphisms

$\mathcal{V}(\operatorname{\mathbb{C}P}^{n}_{\lambda})^{G_{\lambda}}\cong\operatorname{Val}^{\operatorname{U}(n)}$

for all $\lambda$. The present paper gives a canonical one.

Relevant bases for
$\mathcal{V}(\operatorname{\mathbb{C}P}^{n}_{\lambda})^{G_{\lambda}}$ are the
monomials in certain canonical generators $t,s$; the valuations
$\mu_{k,q}^{\lambda}$; and the valuations $\tau_{k,q}^{\lambda}$. We refer to
[bernig_fu_solanes] for the definitions.

Given a point $x$ in a smooth manifold $M$, we can view (8) as an isomorphism

(15)
$\bar{\tau}_{x}\colon\left.\Omega^{\bullet}(SM)\right|_{S_{x}M}\to{\bigwedge}^{\bullet}T_{x}^{*}M\otimes\Omega^{\bullet}(ST_{x}M)\simeq\Omega^{\bullet}(ST_{x}M)^{T_{x}M}$

into the space of translation invariant forms of the sphere bundle $ST_{x}M$.
This induces the transfer map (cf. [bernig_fu_solanes]*Prop. 2.5)

(16) $\tau_{x}\colon\mathcal{C}(M)\to\operatorname{Curv}(T_{x}M),$

given by
$\tau_{x}([\phi,\omega])=[\phi_{x},\bar{\tau}_{x}(\omega|_{S_{x}M})]$.

It follows [bernig_fu_solanes]*Prop. 2.22 that there is a canonical
identification of $G_{\lambda}$-invariant smooth curvature measures on
$\operatorname{\mathbb{C}P}^{n}_{\lambda}$ and
$\mathrm{Curv}^{\operatorname{U}(n)}$. In particular, the curvature measures
$\Delta_{k,q},N_{k,q},B_{k,q},\Gamma_{k,q}$ from the flat case induce
curvature measures in the curved case, denoted by the same letters. For future
reference we note from [bernig_fu_solanes]

(17)
$[B_{k,p}]=\mu_{k,p}^{\lambda},\qquad[\Delta_{k,q}]=\mu_{k,q}^{\lambda}-\lambda\frac{q+1}{\pi}\mu_{k+2,q+1}^{\lambda}\\\
$

as well as

(18) $\displaystyle\tau_{k,q}^{\lambda}$
$\displaystyle=\sum_{i=q}^{\left\lfloor\frac{k}{2}\right\rfloor}\binom{i}{q}\mu_{k,i}^{\lambda}$
(19) $\displaystyle\tau_{k,q}^{\lambda}$ $\displaystyle=(1-\lambda
s)\frac{\pi^{k}}{\omega_{k}(k-2q)!(2q)!}v^{\frac{k}{2}-q}u^{q},$

where $v:=t^{2}(1-\lambda s),u:=4s-v$, and

$\omega_{n}:=\frac{\pi^{n/2}}{\Gamma\left(\frac{n}{2}+1\right)}$

is the volume of the $n$-dimensional unit ball. We will also use the notation
$s_{n}=(n+1)\omega_{n+1}$ for the volume of the $n$-dimensional unit sphere
and the formulas

(20)
$\displaystyle\omega_{2n}=\frac{\pi^{n}}{n!},\qquad\omega_{2n+1}=\frac{2^{2n+1}\pi^{n}n!}{(2n+1)!}.$

Recall also the recursion relation

(21) $\omega_{n+2l}=\frac{(2\pi)^{l}}{(n+2l)(n+2l-2)\cdots(n+2)}\omega_{n}.$

###  2.3. Riemannian valuations and curvature measures

Let us now summarize some results from [fu_wannerer].

On a riemannian manifold $M$ of dimension $n$, certain natural smooth
curvature measures $C_{k,p},0\leq 2p\leq k\leq n$ are constructed within
Cartan’s calculus, i.e. using solder, connection and curvature forms on the
frame bundle. The _Lipschitz-Killing curvature measures_ $\Lambda_{k},0\leq
k\leq n$ are given as special linear combinations of the $C_{k,p}$. We revisit
these expressions below, using Gray’s notion of double forms. The
globalizations of the Lipschitz-Killing curvature measures yield the
_intrinsic volumes_ $\mu_{k}=\frac{\pi^{k}}{\omega_{k}k!}[\Lambda_{k}]$, whose
linear span constitutes an algebra under the Alesker product, called the
_Lipschitz-Killing algebra_ $\mathcal{LK}(M)\cong\mathbb{R}[t]/(t^{n+1})$. The
main point of this construction is invariance: the action of the Lipschitz-
Killing algebra on the curvature measures $C_{k,p}$ is the same for every
riemannian manifold. Thus this action may be computed from easy manifolds
(like round spheres).

The Lipschitz-Killing curvature measures are characterized among all linear
combinations of the $C_{k,p}$ by the property that for an isometric immersion
$e:M\to N$ of riemannian manifolds we have
$e_{\mathcal{C}}^{*}\Lambda_{k}^{N}=\Lambda_{k}^{M}$.

##  3\. Double forms

###  3.1. Double forms associated with a smooth map

Several of the constructions in this section are taken from [gray69].

####  3.1.1. Double exterior algebras

Given finite dimensional real vector spaces $V,W$, we consider the bigraded
double exterior algebra

$\bigwedge\nolimits^{\bullet,\bullet}(W,V):=\bigwedge\nolimits^{\bullet}W^{*}\otimes\bigwedge\nolimits^{\bullet}V^{*}=\bigoplus_{p,q}\left(\bigwedge\nolimits^{p}W^{*}\right)\otimes\left(\bigwedge\nolimits^{q}V^{*}\right)$

with respect to the product

$(\theta_{1}\otimes\psi_{1})\wedge(\theta_{2}\otimes\psi_{2}):=(\theta_{1}\wedge\theta_{2})\otimes(\psi_{1}\wedge\psi_{2}).$

The elements (double forms) of the summands are said to be of type $(p,q)$. If
$\omega_{1}$ is of type $(p_{1},q_{1})$ and $\bm{\omega}_{2}$ of type
$(p_{2},q_{2})$, then clearly

$\bm{\omega}_{1}\wedge\bm{\omega}_{2}=(-1)^{p_{1}p_{2}+q_{1}q_{2}}\bm{\omega}_{2}\wedge\bm{\omega}_{1}.$

If $\bm{\omega}$ is of type $(p,q)$ and $X_{1},\dots,X_{p}\in W,\ Y_{1},\dots
Y_{q}\in V$ then we denote the evaluation

$\bm{\omega}(X_{1},\dots,X_{p};\ Y_{1},\dots
Y_{q}):=\bm{\omega}(X_{1},\dots,X_{p})(Y_{1},\dots Y_{q})\in\mathbb{R}.$

Given linear maps $V\to V^{\prime},W\to W^{\prime}$, there is a pullback map

${\bigwedge}^{\bullet,\bullet}(W^{\prime},V^{\prime})\to{\bigwedge}^{\bullet,\bullet}(W,V).$

Given linear maps $l\colon V\to W$ and $m\colon W\to V$ there is an operation
$\bm{\omega}\mapsto\bm{\omega}^{\prime}$ of degree $(1,-1)$

$\displaystyle{\bigwedge}^{p,q}(V,W)$
$\displaystyle\to{\bigwedge}^{p+1,q-1}(V,W),$
$\displaystyle\bm{\omega}^{\prime}(X_{1},\ldots,X_{p+1};Y_{2},\ldots,Y_{q})$
$\displaystyle:=\sum_{j=1}^{p+1}(-1)^{j+1}\bm{\omega}(X_{1},\ldots,\widehat{X_{j}},\ldots,X_{p+1};{l}X_{j},Y_{2},\ldots,Y_{q})$

and an operation $\bm{\omega}\mapsto\bm{\omega}^{\vee}$ from forms of type
$(p,q)$ to forms of type $(q,p)$:

$\displaystyle{\bigwedge}^{p,q}(V,W)$
$\displaystyle\to{\bigwedge}^{q,p}(V,W),$
$\displaystyle\bm{\omega}^{\vee}(X_{1},\dots,X_{q};Y_{1},\dots,Y_{p})$
$\displaystyle:=\bm{\omega}({m}Y_{1},\dots,{m}Y_{p};{l}X_{1},\dots,{l}X_{q}).$

Clearly

$(\bm{\omega}\wedge\bm{\phi})^{\prime}=\bm{\omega}^{\prime}\wedge\bm{\phi}+(-1)^{p+q}\bm{\omega}\wedge\bm{\phi}^{\prime}$

and

$(\bm{\omega}\wedge\bm{\phi})^{\vee}=\bm{\omega}^{\vee}\wedge\bm{\phi}^{\vee}.$

####  3.1.2. Double forms

Let $M_{1},M_{2}$ be smooth manifolds, and $f:M_{1}\to M_{2}$ a smooth map.
Define the algebra $\Omega^{\bullet,\bullet}(f)$ to be the space of smooth
sections of the bundle $\bigwedge\nolimits^{\bullet}T^{*}M_{1}\otimes
f^{*}(\bigwedge\nolimits^{\bullet}T^{*}M_{2})$ over $M_{1}$, i.e. with fibers

$\bigwedge\nolimits^{\bullet,\bullet}(T_{x}M_{1},T_{f(x)}M_{2}),\quad x\in
M_{1}.$

Given another smooth manifold $M_{3}$ and a smooth map $g:M_{3}\to M_{1}$,
there is a natural pullback map

$g^{*}:\Omega^{\bullet,\bullet}(f)\to\Omega^{\bullet,\bullet}(f\circ g)$

given explicitly by

$(g^{*}\bm{\omega})_{z}(Z_{1},\dots,Z_{k};Y_{1},\dots,Y_{l}):=\omega_{g(z)}(dg(Z_{1}),\dots,dg(Z_{k});Y_{1},\dots,Y_{l})$

for $\bm{\omega}\in\Omega^{k,l}(f),\ Z_{1},\dots,Z_{k}\in T_{z}M_{3},\
Y_{1},\dots,Y_{l}\in T_{f\circ g(z)}M_{2}$.

Suppose now that the bundle $TM_{2}$ is endowed with a connection $\nabla$.
Consider the pullback bundle $f^{*}(TM_{2})$ over $M_{1}$, together with the
pullback connection $f^{*}\nabla$, characterized by

(22) $(f^{*}\nabla)_{X}f^{*}Y=f^{*}(\nabla_{df(X)}Y),\qquad X\in T_{x}M_{1}$

for any vector field $Y$, considered as a section of $TM_{2}$. Consider also
the canonical extension of $f^{*}\nabla$ to
$f^{*}({\bigwedge}^{\bullet}T^{*}M_{2})$, given by

(23)
$((f^{*}\nabla)_{X}s)(Z_{1},\dots,Z_{k})=X(s(Z_{1},\dots,Z_{k}))-\sum_{j=1}^{k}s(Z_{1},\dots,(f^{*}\nabla)_{X}Z_{j},\dots,Z_{k}),$

where $s\in\Omega^{0,k}(f)$ (i.e. $s$ is a section of
$f^{*}{\bigwedge}^{k}T^{*}M_{2}$), and the $Z_{j}$ are smooth sections of
$f^{*}TM_{2}$.

Thus we may consider

$d_{\nabla}s:=(f^{*}\nabla)s,\qquad s\in\Omega^{0,k}(f)$

to be an element of $\Omega^{1,k}(f)$. Now define the exterior covariant
derivative $d_{\nabla}$ as the linear operator
$\Omega^{\bullet,\bullet}(f)\to\Omega^{\bullet,\bullet}(f)$ of degree $(1,0)$
given by

(24) $d_{\nabla}(\omega\otimes s)=d\omega\otimes
s+(-1)^{p}\omega\wedge(f^{*}\nabla)s,$

where $\omega\in\Omega^{p}(M_{1})$ and $s$ is a section of
$f^{*}{\bigwedge}^{\bullet}T^{*}M_{2}$. Explicitly, given
$\bm{\eta}\in\Omega^{p,q}(f)$, and smooth vector fields $X_{0},\dots,X_{p}$ on
$M_{1}$,

$\displaystyle(d_{\nabla}\bm{\eta})(X_{0},\ldots,X_{p})=\sum_{i=0}^{p}$
$\displaystyle(-1)^{i}(f^{*}\nabla)_{X_{i}}(\bm{\eta}(X_{0},\ldots,\widehat{X_{i}},\ldots,X_{p}))$
$\displaystyle+\sum_{i<j}(-1)^{i+j}\bm{\eta}([X_{i},X_{j}],X_{0},\ldots,\widehat{X_{i}},\ldots,\widehat{X_{j}},\ldots,X_{p}),$

where we regard the various $\bm{\eta}(X_{i_{1}},\dots,X_{i_{p}})$ as smooth
sections of $f^{*}{\bigwedge}^{q}T^{*}M_{2}$, and the terms of the first sum
may be unpacked as in (23).

######  Lemma 3.1.

  1. (1)

For double forms
$\bm{\varphi}\in\Omega^{p,q}(f),\bm{\theta}\in\Omega^{r,s}(f)$:

$d_{\nabla}(\bm{\varphi}\wedge\bm{\theta})=d_{\nabla}\bm{\varphi}\wedge\bm{\theta}+(-1)^{p}\bm{\varphi}\wedge
d_{\nabla}\bm{\theta}.$

  2. (2)

If $g:M_{3}\to M_{1}$ as above, then
$d_{\nabla}g^{*}\bm{\eta}=g^{*}d_{\nabla}\bm{\eta}$ with respect to the
pullback $g^{*}:\Omega^{\bullet,\bullet}(f)\to\Omega^{\bullet,\bullet}(g\circ
f)$.

  3. (3)

If $M_{2}$ is oriented riemannian and $\nabla$ is the Levi-Civita connection,
then

$d_{\nabla}(\eta\otimes\operatorname{vol})=d\eta\otimes\operatorname{vol}$

for any $\eta\in\Omega^{\bullet}(M_{1})$.

###### Proof.

(1) is straightforward using

$(f^{*}\nabla)(s_{1}\wedge s_{2})=(f^{*}\nabla s_{1})\wedge
s_{2}+s_{1}\wedge(f^{*}\nabla s_{2}).$

(2) follows at once from the definitions, and the corresponding property of
the usual exterior derivative.

(3) is immediate since the volume form
$\operatorname{vol}=\operatorname{vol}_{M_{2}}$ is parallel. ∎

Remark. The derivative $d_{\nabla}$ is denoted by $D$ in [gray69], a symbol we
reserve for the Rumin differential.

###  3.2. Double forms on a riemannian manifold

We specialize to the case where $M_{2}=M$ is a smooth oriented riemannian
manifold of dimension $n$ with Levi-Civita connection $\nabla$, $M_{1}:=SM$ is
its unit tangent bundle, and $f$ is the projection $\pi:SM\to M$. A spherical
double form  on $M$ is an element of

$\Omega^{\bullet,\bullet}(SM,M):=\Omega^{\bullet,\bullet}(\pi).$

Define similarly the space

$\Omega^{\bullet,\bullet}(M):=\Omega^{\bullet,\bullet}(\operatorname{id}_{M})$

of double forms on $M$. The pullback $\pi^{*}$ embeds the latter space into
the former. Using the maps $l=d\pi\colon T_{\xi}SM\to T_{x}M$ and the
horizontal lift $m=(d\pi|_{H_{\xi}})^{-1}\colon
T_{x}M{\longrightarrow}T_{\xi}SM$, we define the operations ′ and ∨ on
$\Omega^{\bullet,\bullet}(SM,M)$.

The riemannian metric of $M$ is a double form
$\operatorname{\mathsf{g}}\in\Omega^{1,1}(M)$, with
$\operatorname{\mathsf{g}}^{\prime}=0$. The curvature tensor
$\operatorname{\mathsf{R}}=\operatorname{\mathsf{R}}^{M}\in\Omega^{2,2}(M)$,
where as usual

$\operatorname{\mathsf{R}}(W,X;Y,Z)=\operatorname{\mathsf{g}}(\nabla_{[W,X]}Y-\nabla_{W}\nabla_{X}Y+\nabla_{X}\nabla_{W}Y;Z).$

The first Bianchi identity may be stated as

$\operatorname{\mathsf{R}}^{\prime}=0.$

Given $X\in T_{\xi}SM$, let

$X=X^{H}+X^{V}$

be its decomposition into horizontal and vertical components with respect to
the Levi-Civita connection. Identifying $X^{V}\in T_{\xi}(S_{\pi\xi}M)$ with
the corresponding vector in $T_{\pi\xi}M$, we define the connection double
form $\bm{\omega}\in\Omega^{1,1}(SM,M)$ by

$\bm{\omega}(X;Y)=\operatorname{\mathsf{g}}(X^{V};Y)=\operatorname{\mathsf{g}}((\pi^{*}\nabla)_{X}\xi;Y),$

where in the last expression $\xi$ denotes the tautological section of the
bundle $\pi^{*}TM$.

Similarly, the canonical contact form

$\bm{\alpha}(X)=\operatorname{\mathsf{g}}(X^{H};\xi)$

defines an element in $\Omega^{1,0}(SM,M)$.

We also define spherical double forms
$\operatorname{\mathsf{R}}_{0},\widehat{\operatorname{\mathsf{R}}}$ of
respective types $(2,1),(2,2)$ by

(25)
$(\operatorname{\mathsf{R}}_{0})_{\xi}(X_{1},X_{2};Y)=\operatorname{\mathsf{R}}_{x}({d\pi}X_{1},{d\pi}X_{2};Y,\xi),\qquad\widehat{\operatorname{\mathsf{R}}}=\operatorname{\mathsf{R}}+\bm{\alpha}^{\vee}\wedge\operatorname{\mathsf{R}}_{0}.$

In particular
$\widehat{\operatorname{\mathsf{R}}}_{\xi}(\cdot,\cdot;\cdot,\xi)=0$. It is
easily checked that

(26)
$\quad(\bm{\alpha}^{\vee})^{\prime}=\bm{\alpha},\quad\operatorname{\mathsf{R}}_{0}^{\prime}=0,\quad\widehat{\operatorname{\mathsf{R}}}^{\prime}{=\bm{\alpha}\wedge\operatorname{\mathsf{R}}_{0}},$

and

(27)
$(\operatorname{\mathsf{R}}_{0}^{\vee})^{\prime}=\operatorname{\mathsf{R}}_{0}.$

We will make frequent use of the natural identification

$\Omega^{\bullet}(SM)\simeq\Omega^{\bullet,0}(SM,M)\quad{\rm
via}\quad\phi\leftrightarrow\phi\otimes 1.$

Prominently, $\bm{\alpha}$ may be thought of as an element of either space.

It will be useful also to identify $\Omega^{\bullet,n}(SM,M)$ with the space
of differential forms on $SM$ with values in the orientation line bundle
$\Omega^{n}(M)$. In the main thrust of the present paper, $M$ will be a Kähler
manifold, hence endowed with a canonical volume form, so that

(28) $\Omega^{\bullet}(SM)\simeq\Omega^{\bullet,n}(SM)\quad{\rm
via}\quad\phi\leftrightarrow\phi\otimes\operatorname{vol}_{M}.$

Typically we will make this identification only after the formal calculations
in the algebra of double forms have taken place, and we will alert the reader
on these occasions in order to minimize confusion. We remark, however, that
integration over the normal cycle $\operatorname{nc}(A)$ of a subset $A\subset
M$, which will be a central construction for us, has the following property:
if $M$ is riemannian, but not oriented (or even orientable), one may pair
$\operatorname{nc}(A)$ with elements of $\Omega^{n-1,n}(SM)$ but not with
elements of $\Omega^{n-1}(SM)$.

Of course all of the above canonical double forms may be expressed in terms of
the classical language of moving frames. For $\eta$ lying in an open set
$U\subset SM$, let $e_{0}=\eta,\dots,e_{n-1}$ be a orthonormal frame for
$T_{\pi\eta}M$. In particular, each $e_{j}$ defines a section of $\pi^{*}TM$,
which we suppose smooth. Define $\theta_{i},\omega_{i,j}\in\Omega^{1}(U)$, and
$\Omega_{i,j}\in\Omega^{2}(U)$, by

(29) $\displaystyle\theta_{i}(X)$
$\displaystyle:=\operatorname{\mathsf{g}}(e_{i};d\pi X),$ (30)
$\displaystyle\omega_{i,j}(X)$
$\displaystyle:=\operatorname{\mathsf{g}}(e_{i};(\pi^{*}\nabla)_{X}e_{j}),$
(31) $\displaystyle\Omega_{i,j}(X,Y)$
$\displaystyle:=\operatorname{\mathsf{R}}(e_{i},e_{j};d\pi X,d\pi Y),$

for $X,Y\in T_{\eta}SM$. The corresponding structure equations are then

(32) $\displaystyle d\theta_{i}$
$\displaystyle=-\sum_{j=0}^{n-1}\omega_{i,j}\wedge\theta_{j},\quad\omega_{j,i}=-\omega_{i,j},$
(33) $\displaystyle d\omega_{i,j}$
$\displaystyle=-\sum_{k=0}^{n-1}\omega_{i,k}\wedge\omega_{k,j}+\Omega_{i,j},\quad\Omega_{j,i}=-\Omega_{i,j}$
(34) $\displaystyle d\Omega_{i,j}$
$\displaystyle=\sum_{k=0}^{n-1}(\Omega_{i,k}\wedge\omega_{k,j}-\omega_{i,k}\wedge\Omega_{k,j}).$

Note that the $\theta_{i}$ may also be viewed as local sections of
$\pi^{*}T^{*}M$. In these terms,

$\displaystyle\operatorname{\mathsf{g}}$
$\displaystyle=\sum_{i=0}^{n-1}\theta_{i}\otimes\theta_{i},$
$\displaystyle\operatorname{\mathsf{R}}=\frac{1}{2}\sum_{i,j=0}^{n-1}\Omega_{i,j}\otimes(\theta_{i}\wedge\theta_{j}),$
$\displaystyle\bm{\omega}$
$\displaystyle=\sum_{i=0}^{n-1}\omega_{i,0}\otimes\theta_{i},$
$\displaystyle\operatorname{\mathsf{R}}_{0}=\sum_{i=0}^{n-1}\Omega_{i,0}\otimes\theta_{i},$
$\displaystyle\bm{\alpha}$ $\displaystyle=\theta_{0}\otimes 1,$
$\displaystyle\widehat{\operatorname{\mathsf{R}}}=\frac{1}{2}\sum_{i,j=1}^{n-1}\Omega_{i,j}\otimes(\theta_{i}\wedge\theta_{j}).$

###  3.3. Contractions

Gray [gray_book] introduced a contraction operator $C$ on the space of double
forms,

$C(\bm{\eta})(X_{1},\ldots,X_{p-1};Y_{1},\ldots,Y_{q-1})=\sum_{i=1}^{n}\bm{\eta}(X_{1},\ldots,X_{p-1},e_{i};Y_{1},\ldots,Y_{q-1},e_{i})$

for $\bm{\eta}$ of type $(p,q)$ on a euclidean space $V$, where
$e_{1},\ldots,e_{n}$ is any orthonormal basis of $V$. We need the following
simple lemma.

######  Lemma 3.2.

If $\bm{\eta}$ is a double form of type $(p,p)$ on a euclidean space $V$ of
dimension $n$, then

$\frac{(n-p)!}{p!}C^{p}(\bm{\eta})=(\bm{\eta}\wedge\operatorname{\mathsf{g}}^{n-p})(e_{1},\ldots,e_{n};e_{1},\ldots,e_{n})$

for any orthonormal basis $e_{1},\ldots,e_{n}$ of $V$.

###### Proof.

From the definition of the wedge product of double forms it follows that

$\displaystyle(\bm{\eta}$
$\displaystyle\wedge\operatorname{\mathsf{g}}^{n-p})(e_{1},\ldots,e_{n};e_{1},\ldots,e_{n})$
$\displaystyle=\sum_{\rho,\sigma\in\mathcal{S}_{p,n-p}}\varepsilon_{\rho}\varepsilon_{\sigma}\bm{\eta}(e_{\rho_{1}},\ldots,e_{\rho_{p}};e_{\sigma_{1}},\ldots,e_{\sigma_{p}})\operatorname{\mathsf{g}}^{n-p}(e_{\rho_{p+1}},\ldots,e_{\rho_{n}};e_{\sigma_{p+1}},\ldots,e_{\sigma_{n}}),$

where $\epsilon_{\rho}$ is the sign of the permutation $\rho$ and
$\mathcal{S}_{p,n-p}$ denotes the set of $(p,n-p)$ shuffles, i.e. permutations
of $\\{1,\dots,n\\}$ that are monotonically increasing on $\\{1,\dots,p\\}$
and on $\\{p+1,\dots,n\\}$. Now

$\operatorname{\mathsf{g}}^{n-p}(e_{\rho_{p+1}},\ldots,e_{\rho_{n}};e_{\sigma_{p+1}},\ldots,e_{\sigma_{n}})=\begin{cases}(n-p)!&\text{if
}\\{\rho_{p+1},\ldots,\rho_{n}\\}=\\{\sigma_{p+1},\ldots,\sigma_{n}\\};\\\
0&\text{otherwise}.\end{cases}$

Hence

$\displaystyle(\bm{\eta}\wedge\operatorname{\mathsf{g}}^{n-p})(e_{1},\ldots,e_{n};e_{1},\ldots,e_{n})$
$\displaystyle=(n-p)!\sum_{1\leq i_{1}<\cdots<i_{p}\leq
n}\bm{\eta}(e_{i_{1}},\ldots,e_{i_{p}};e_{i_{1}},\ldots,e_{i_{p}})$
$\displaystyle=\frac{(n-p)!}{p!}\sum_{i_{1},\ldots,i_{p}=1}^{n}\bm{\eta}(e_{i_{1}},\ldots,e_{i_{p}};e_{i_{1}},\ldots,e_{i_{p}})$
$\displaystyle=\frac{(n-p)!}{p!}C^{p}(\bm{\eta}).$

∎

###  3.4. Exterior derivatives of riemannian double forms

######  Lemma 3.3.

Let $\theta_{i},\omega_{i,j}$ be defined by (29) and (30). Then

(35)
$d_{\nabla}(1\otimes\theta_{i})=-\sum_{j=0}^{n-1}\omega_{i,j}\otimes\theta_{j}.$

###### Proof.

From the defining relations (24), (23), (30)

$\displaystyle d_{\nabla}(1\otimes\theta_{i})(X;Y)$
$\displaystyle=(\pi^{*}\nabla)_{X}\theta_{i}(Y)$
$\displaystyle=X(\theta_{i}(Y))-\theta_{i}((\pi^{*}\nabla)_{X}Y)$
$\displaystyle=X(\theta_{i}(Y))-\sum_{j}\theta_{i}\big{(}X(\theta_{j}(Y))e_{j}+\theta_{j}(Y)(\pi^{*}\nabla)_{X}e_{j}\big{)}$
$\displaystyle=-\sum_{j}\theta_{j}(Y)\omega_{i,j}(X),$

as stated.∎

######  Proposition 3.4.

The exterior derivatives of
$\operatorname{\mathsf{g}},\operatorname{\mathsf{R}}\in\Omega^{\bullet,\bullet}(M)$
and $\bm{\alpha}^{\vee},\bm{\omega}\in\Omega^{\bullet,\bullet}(SM,M)$ are

(36) $d_{\nabla}\operatorname{\mathsf{g}}=0,\quad
d_{\nabla}\operatorname{\mathsf{R}}=0,\quad
d_{\nabla}\bm{\alpha}=\bm{\omega}^{\prime},\quad
d_{\nabla}\bm{\alpha}^{\vee}=\bm{\omega},\quad
d_{\nabla}\bm{\omega}=\operatorname{\mathsf{R}}_{0}.$

###### Proof.

Straightforward. Let us prove for instance the last one:

$\displaystyle d_{\nabla}\bm{\omega}$
$\displaystyle=\sum_{i=0}^{n-1}d\omega_{i,0}\otimes\theta_{i}-\omega_{i,0}\wedge
d_{\nabla}(1\otimes\theta_{i})$
$\displaystyle=-\sum_{i,j=0}^{n-1}\omega_{i,j}\wedge\omega_{j,0}\otimes\theta_{i}+\sum_{i=0}^{n-1}\Omega_{i,0}\otimes\theta_{i}+\sum_{i,j=0}^{n-1}(\omega_{i,0}\wedge\omega_{i,j})\otimes\theta_{i}$
$\displaystyle=\sum_{i=0}^{n-1}\Omega_{i,0}\otimes\theta_{i}$
$\displaystyle=\operatorname{\mathsf{R}}_{0}.$

∎

###  3.5. Lipschitz-Killing curvature measures

Let us now express in terms of double forms the Lipschitz-Killing curvature
measures $\Lambda_{k}$ defined in [fu_wannerer] and [bernig_fu_solanes].
Recall that these objects associate elements
$\Lambda_{k}^{M}\in\mathcal{C}(M)$ to any smooth riemannian manifold $M$, with
the property that if $N\subset M$ is a smooth submanifold of dimension $k$
then $\Lambda_{k}^{M}(N,\cdot)=$ the restriction of $k$-dimensional volume to
$N$.

Given a riemannian manifold $M^{n}$, define elements of
$\Omega^{n-k-1,n-k-1}(SM,M)$ by

$\displaystyle\bm{\psi}_{-1}:=$ $\displaystyle 0=:\tilde{\bm{\psi}}_{-1},$
$\displaystyle\bm{\psi}_{n-k-1}$
$\displaystyle=\sum_{l=0}^{\lfloor\frac{n-k-1}{2}\rfloor}d_{n,k,l}\operatorname{\mathsf{R}}^{l}\wedge\bm{\omega}^{n-2l-k-1}$
$\displaystyle\tilde{\bm{\psi}}_{n-k-1}$
$\displaystyle=\sum_{l=0}^{\lfloor\frac{n-k-1}{2}\rfloor}(n-2l-k)d_{n,k,l}\operatorname{\mathsf{R}}^{l}\wedge\bm{\omega}^{n-2l-k-1},\quad
k=0,\ldots,n-1,$

where

(37) $\displaystyle d_{n,k,l}$
$\displaystyle=\frac{\pi^{k}}{(k+1)!\omega_{k}}\frac{1}{2^{l}}\binom{\frac{k}{2}+l}{l}\frac{\omega_{k+2l}}{\pi^{k+2l}(n-2l-k)!\omega_{n-2l-k}}$
$\displaystyle=\left[(2\pi)^{l}l!(k+1)!(n-2l-k)!\omega_{n-2l-k}\right]^{-1}.$

These constants will be taken to be zero whenever any index $n,k,l$ is a half-
integer.

The spherical double form $\tilde{\bm{\psi}}_{n-k-1}$ will be used later in
Section 4. For later use we record the relation

(38) $(n-2l-k-1)d_{n,k,l}=(l+1)d_{n,k,l+1}$

and its immediate consequence

$d_{n,k,l}=\frac{(n-k-1)!!}{l!(n-2l-k-1)!!}d_{n,k,0},\quad
d_{n,k,0}=((k+1)!(n-k)!\omega_{n-k})^{-1},$

where as usual $n!!=n(n-2)!!$, ${(-1)}!!=1$, and $0!!=1$.

Also for $m<n$

(39)
$d_{2n,k,l}\frac{1}{(2\pi)^{e}e!}\frac{\omega_{2n-2l-k}(2n-2l-k)!}{\omega_{2m-2l-2e-k}(2m-2l-2e-k)!}=d_{2m,k,l+e}\binom{l+e}{e}$

and

(40)
$d_{2n,k,l}\frac{(2n-2l-k)!}{(2m-2l-k+1)!}\frac{2^{m-l-\frac{k}{2}+1}\omega_{2n-2l-k}}{\omega_{2m-2l-k+1}}=\frac{\pi^{k}}{(k+1)!\omega_{k}}\frac{1}{\left(\frac{k}{2}\right)!\left(m-\frac{k}{2}\right)!}\frac{1}{2^{m-\frac{k}{2}}\pi^{m}}\binom{m-\frac{k}{2}}{l}.$

Let

$\bm{\kappa}_{k}=\begin{cases}\frac{1}{k!\left(\frac{n-k}{2}\right)!(2\pi)^{\frac{n-k}{2}}}\operatorname{\mathsf{R}}^{\frac{n-k}{2}}\wedge\operatorname{\mathsf{g}}^{k},&n-k{\rm\
even}\\\ 0&{\rm otherwise.}\end{cases}$

######  Lemma 3.5.

The Lipschitz-Killing curvature measures are given by

$\Lambda_{k}^{M}=\left[\bm{\kappa}_{k},(k+1)\bm{\alpha}^{\vee}\wedge\operatorname{\mathsf{g}}^{k}\wedge\bm{\psi}_{n-k-1}\right]{=(k+1)\left[d_{n,k,\frac{n-k}{2}}\operatorname{\mathsf{R}}^{\frac{n-k}{2}}\wedge\operatorname{\mathsf{g}}^{k},\bm{\alpha}^{\vee}\wedge\operatorname{\mathsf{g}}^{k}\wedge\bm{\psi}_{n-k-1}\right],}$

$k=0,\dots,n$, under the identifications
$\Omega^{n,n}(M,M)\simeq\Omega^{n}(M),\
\Omega^{n-1,n}(SM,M)\simeq\Omega^{n-1}(SM)$ of (28).

###### Proof.

This is a direct translation of the Definitions 3.1 and 3.10 from
[fu_wannerer]. ∎

The following computation will be useful in Section 4.

######  Lemma 3.6.

$d_{\nabla}(\bm{\alpha}^{\vee}\wedge\bm{\psi}_{n-k-1})={d_{n,k,\frac{n-k-2}{2}}\bm{\alpha}^{\vee}\wedge\operatorname{\mathsf{R}}^{\frac{n-k-2}{2}}\wedge\operatorname{\mathsf{R}}_{0}}+\sum_{l}d_{n,k,l}\hat{\operatorname{\mathsf{R}}}^{l}\wedge\bm{\omega}^{n-2l-k}.$

As usual, constants with half-integer indices are taken to be zero.

###### Proof.

Applying Proposition 3.4, we compute

$\displaystyle
d_{\nabla}(\bm{\alpha}^{\vee}\wedge\operatorname{\mathsf{R}}^{l}\wedge\bm{\omega}^{n-2l-k-1})$
$\displaystyle=(d_{\nabla}\bm{\alpha}^{\vee})\wedge\operatorname{\mathsf{R}}^{l}\wedge\bm{\omega}^{n-2l-k-1}$
$\displaystyle\quad+(n-2l-k-1)\bm{\alpha}^{\vee}\wedge\operatorname{\mathsf{R}}^{l}\wedge
d_{\nabla}\bm{\omega}\wedge\bm{\omega}^{n-2l-k-2}$
$\displaystyle=\operatorname{\mathsf{R}}^{l}\wedge\bm{\omega}^{n-2l-k}+(n-2l-k-1)\bm{\alpha}^{\vee}\wedge\operatorname{\mathsf{R}}^{l}\wedge\operatorname{\mathsf{R}}_{0}\wedge\bm{\omega}^{n-2l-k-2}.$

By (25), the first term may be expressed as

$\displaystyle\operatorname{\mathsf{R}}^{l}\wedge\bm{\omega}^{n-2l-k}$
$\displaystyle=(\hat{\operatorname{\mathsf{R}}}-\bm{\alpha}^{\vee}\wedge\operatorname{\mathsf{R}}_{0})^{l}\wedge\bm{\omega}^{n-2l-k}$
$\displaystyle=\hat{\operatorname{\mathsf{R}}}^{l}\wedge\bm{\omega}^{n-2l-k}-l\bm{\alpha}^{\vee}\wedge\operatorname{\mathsf{R}}_{0}\wedge\operatorname{\mathsf{R}}^{l-1}\wedge\bm{\omega}^{n-2l-k}.$

Thus

$\displaystyle d_{\nabla}(\bm{\alpha}^{\vee}\wedge\bm{\psi}_{n-k-1})$
$\displaystyle=d_{\nabla}\sum_{l=0}^{\left\lfloor\frac{n-k-1}{2}\right\rfloor}d_{n,k,l}\bm{\alpha}^{\vee}\wedge\operatorname{\mathsf{R}}^{l}\wedge\bm{\omega}^{n-2l-k-1}$
$\displaystyle=\sum_{l=0}^{\left\lfloor\frac{n-k-1}{2}\right\rfloor}d_{n,k,l}\hat{\operatorname{\mathsf{R}}}^{l}\wedge\bm{\omega}^{n-2l-k}$
$\displaystyle\quad+\sum_{l=0}^{\left\lfloor\frac{n-k-1}{2}\right\rfloor-1}{\left[(n-2l-k-1)d_{n,k,l}-(l+1)d_{n,k,l+1}\right]}\bm{\alpha}^{\vee}\wedge\operatorname{\mathsf{R}}_{0}\wedge\operatorname{\mathsf{R}}^{l}\wedge\bm{\omega}^{n-2l-k-2}$
$\displaystyle\quad+{d_{n,k,\frac{n-k-2}{2}}\bm{\alpha}^{\vee}\wedge\operatorname{\mathsf{R}}^{\frac{n-k-2}{2}}\wedge\operatorname{\mathsf{R}}_{0}}.$

Now (38) yields the result. ∎

##  4\. Hermitian double forms

As we have seen above, the description of riemannian curvature measures in
terms of double forms is relatively simple. In the present section we give the
corresponding constructions in the Kähler case, which form the algebraic
nucleus of the present paper.

###  4.1. Algebraic curvature tensors

An algebraic curvature tensor on a real vector space $V$ is a symmetric double
form
$\operatorname{\mathsf{R}}=\operatorname{\mathsf{R}}^{\vee}\in{\bigwedge}^{2,2}(V,V)$
that satisfies the Bianchi identity

$\operatorname{\mathsf{R}}^{\prime}=0.$

Denote the space of all algebraic curvature tensors by
$\mathcal{R}=\mathcal{R}(V)$. Denote by
$\mathcal{K}=\mathcal{K}(\mathbb{C}^{n})$ the subspace of
$\mathcal{R}(\mathbb{C}^{n})\subset{\bigwedge}^{2,2}(\mathbb{C}^{n},\mathbb{C}^{n})$
consisting of all algebraic curvature tensors that satisfy the additional
condition

(41)
$\operatorname{\mathsf{R}}(JX,JY;Z,W)=\operatorname{\mathsf{R}}(X,Y;JZ,JW)=\operatorname{\mathsf{R}}(X,Y;Z,W),$

where $JX=\sqrt{-1}\cdot X.$ We refer to $\mathcal{K}$ as _the space of
algebraic Kähler curvature tensors_ , as the curvature tensor of a Kähler
manifold satisfies (41) (see e.g. [kobayashi_nomizu_vol2, p. 165]).

###  4.2. Double forms on Kähler manifolds

Let $M$ be an $n$-dimensional Kähler manifold, with riemannian metric
$\operatorname{\mathsf{g}}\in\Omega^{1,1}(M)$, complex structure $J$, and
Kähler form $F\in\Omega^{2}(M)$ related by

$F(X,Y)=\operatorname{\mathsf{g}}(JX;Y).$

Let $\bm{\beta}\in\Omega^{1,0}(SM,M)$ be given by

$\bm{\beta}_{\xi}(X)=\operatorname{\mathsf{g}}(J\xi;d\pi X).$

Note that $(\bm{\beta}^{\vee})^{\prime}=\bm{\beta}.$

Define the Gray double form $\operatorname{\mathsf{G}}\in\Omega^{2,2}(M)$ by

$\displaystyle\operatorname{\mathsf{G}}(W,X;Y,Z)$
$\displaystyle:=\operatorname{\mathsf{g}}(W;Y)\operatorname{\mathsf{g}}(X;Z)-\operatorname{\mathsf{g}}(W;Z)g(X;Y)+\operatorname{\mathsf{g}}(JW;Y)\operatorname{\mathsf{g}}(JX;Z)$
(42)
$\displaystyle\quad-\operatorname{\mathsf{g}}(JW;Z)\operatorname{\mathsf{g}}(JX;Y)+2\operatorname{\mathsf{g}}(JW;X)\operatorname{\mathsf{g}}(JY;Z).$

Note that if $M$ is a space of constant holomorphic sectional curvature
$4\lambda$, then $\operatorname{\mathsf{R}}=\lambda\operatorname{\mathsf{G}}$
(see e.g. [gray_book]*Lemma 6.9).

######  Definition 4.1.

Define for $k<2n$ and $0\leq 2p\leq k$ the elements

(43) $\displaystyle\bm{\gamma}_{k,p}$
$\displaystyle=\bm{\alpha}^{\vee}\wedge\operatorname{\mathsf{G}}^{p}\wedge\operatorname{\mathsf{g}}^{k-2p}\wedge\bm{\psi}_{2n-k-1}$
(44) $\displaystyle\tilde{\bm{\gamma}}_{k,p}$
$\displaystyle=\bm{\alpha}^{\vee}\wedge\bm{\beta}\wedge\bm{\beta}^{\vee}\wedge\operatorname{\mathsf{G}}^{p}\wedge\operatorname{\mathsf{g}}^{k-2p-1}\wedge\tilde{\bm{\psi}}_{2n-k-1}$

of $\Omega^{2n-1,2n}(SM,M)\simeq\Omega^{2n-1}(SM)$, and put
$\bm{\gamma}_{2n,p}=\widetilde{\bm{\gamma}}_{2n,p}=0$. Let also

$\bm{\varphi}_{k,p}=d_{2n,k,n-\frac{k}{2}}\operatorname{\mathsf{G}}^{p}\wedge\operatorname{\mathsf{g}}^{k-2p}\wedge\operatorname{\mathsf{R}}^{n-\frac{k}{2}}$

in $\Omega^{2n,2n}(M)\simeq\Omega^{2n}(M)$.

Define the curvature measures

(45) $\displaystyle\Gamma_{k,p}$
$\displaystyle=\left[\bm{\varphi}_{k,p},\bm{\gamma}_{k,p}\right]$
$\displaystyle\tilde{\Gamma}_{k,p}$
$\displaystyle:=[0,\tilde{\bm{\gamma}}_{k,p}],$

where we identify these double forms with ordinary forms via (28).

######  Remark 4.2.

Note that the $\Gamma_{k,p}$ are not the curvature measures denoted by this
symbol in [bernig_fu_hig, bernig_fu_solanes].

As we will see below, the $\Gamma_{k,p},\tilde{\Gamma}_{k,p}$ represent Kähler
curvature measures that behave well with respect to Kähler inclusions, and (in
the case of $\Gamma_{k,p}$) with respect to Alesker multiplication. This is
essentially guaranteed by the particular linear combinations of canonical
double forms that constitute the $\bm{\psi}_{j},\tilde{\bm{\psi}}_{j}$. Our
ultimate objects of interest, however, will be the curvature measures
$\Delta_{k,p}^{M},B_{k,p}^{M}$—linear combinations of the
$\Gamma_{k,p},\tilde{\Gamma}_{k,p}$, respectively, that carry special
geometric significance—and the valuations $\mu_{k,p}$ that are their common
globalizations.

###  4.3. More relations among hermitian double forms

As in (25), let $\operatorname{\mathsf{G}}_{0}\in\Omega^{2,1}(SM,M)$ be given
by

$(\operatorname{\mathsf{G}}_{0})_{\xi}(X_{1},X_{2};Y)=\operatorname{\mathsf{G}}_{x}(d\pi
X_{1},d\pi X_{2};Y,\xi).$

Since $\operatorname{\mathsf{G}}_{0}^{\prime}=0$ and
$(\operatorname{\mathsf{G}}_{0}^{\vee})^{\prime}=\operatorname{\mathsf{G}}_{0}$
on complex space forms by (26) and (27), the same holds on any Kähler manifold
$M$.

We will need another operator on double forms on a Kähler manifold.

######  Proposition 4.3.

Let $\bm{\varphi}$ be a spherical double form of type $(p,q)$ on a Kähler
manifold $M$. Define a new spherical double form of the same type by

$J\bm{\varphi}(X_{1},\ldots,X_{p};Y_{1},\ldots,Y_{q}):=\left.\frac{d}{dt}\right|_{t=0}\bm{\varphi}(X_{1},\ldots,X_{p};e^{it}Y_{1},\ldots,e^{it}Y_{q}),$

for $X_{1},\ldots,X_{p}\in T_{\xi}SM$ and $Y_{1},\ldots,Y_{q}\in T_{\pi\xi}M$,
where on the right hand side we use the given complex structure on $TM$. Then
$J$ is a derivation, i.e. for double forms $\bm{\varphi},\bm{\theta}$ we have

$J(\bm{\varphi}\wedge\bm{\theta})=J\bm{\varphi}\wedge\bm{\theta}+\bm{\varphi}\wedge
J\bm{\theta}$

and $J$ commutes with the exterior derivative,

$d_{\nabla}\circ J=J\circ d_{\nabla}.$

Moreover, we have for $\operatorname{\mathsf{R}}\in\mathcal{K}$

(46) $J\bm{\alpha}^{\vee}=-\bm{\beta}^{\vee},\quad
J\bm{\beta}^{\vee}=\bm{\alpha}^{\vee},\quad J\operatorname{\mathsf{R}}=0,$

and

(47)
$\operatorname{\mathsf{G}}=\frac{1}{2}(\operatorname{\mathsf{g}}\wedge\operatorname{\mathsf{g}}+J\operatorname{\mathsf{g}}\wedge
J\operatorname{\mathsf{g}}+4F\wedge F^{\vee}).$

###### Proof.

Straightforward. ∎

######  Proposition 4.4.

On $\Omega^{\bullet,\bullet}(SM,M)$ we have the following differential
identities

(48) $d_{\nabla}F^{\vee}=0,\quad d_{\nabla}\operatorname{\mathsf{G}}=0,\quad
d_{\nabla}\bm{\beta}^{\vee}=-J\bm{\omega},$

and

(49)
$d_{\nabla}(\operatorname{\mathsf{G}}_{0})^{\vee}=-\bm{\omega}\wedge\operatorname{\mathsf{g}}-J\bm{\omega}\wedge
J\operatorname{\mathsf{g}}-2d_{\nabla}\bm{\beta}\wedge F^{\vee}.$

###### Proof.

If the orthonormal frame $e_{0},\ldots,e_{2n+1}$ is hermitian (i.e.
$e_{2i+1}=Je_{2i}$), we have

$J(1\otimes\theta_{2i})=-1\otimes\theta_{2i+1},\quad
J(1\otimes\theta_{2i+1})=1\otimes\theta_{2i}.$

Moreover, since $M$ is Kähler we have $\nabla J=0$, which translates into

$\omega_{2i,2j}=\omega_{2i+1,2j+1},\qquad\omega_{2i,2j+1}=-\omega_{2i+1,2j}.$

Since

$F=\sum_{i=0}^{n-1}\theta_{2i}\wedge\theta_{2i+1}\quad\text{and}\quad\beta=\theta_{1},$

using these relations and (35), the first and last identity in (48) are
straightforward. As for $d_{\nabla}G=0$, it follows immediately from (47)
since $dF=0$ on Kähler manifolds.

The identity (49) then follows from

$\operatorname{\mathsf{G}}_{0}^{\vee}=-\bm{\alpha}^{\vee}\wedge\operatorname{\mathsf{g}}+\bm{\beta}^{\vee}\wedge
J\operatorname{\mathsf{g}}-2\bm{\beta}\wedge F^{\vee}.$

∎

Given a complex linear isometry $\sigma\colon\mathbb{C}^{n}\to T_{x}M$, the
isomorphism (10)

$\bar{\sigma}^{*}\colon\left.\Omega^{\bullet}(SM)\right|_{S_{x}M}\to{\bigwedge}^{\bullet}(\mathbb{C}^{n})^{*}\otimes\Omega^{\bullet}(S^{2n-1})\simeq\left.\Omega^{\bullet}(S\mathbb{C}^{n})\right|_{\\{0\\}\times
S^{2n-1}}$

extends to an algebra isomorphism

(50) $\displaystyle\bar{\sigma}^{*}\colon$
$\displaystyle\left.\Omega^{\bullet,\bullet}(SM,M)\right|_{S_{x}M}\to\left.\Omega^{\bullet,\bullet}(S\mathbb{C}^{n},\mathbb{C}^{n})\right|_{\\{0\\}\times
S^{2n-1}}$

given by

$\bar{\sigma}^{*}(\omega\otimes
s)=\bar{\sigma}^{*}(\omega)\otimes\sigma^{*}(s),$

where $\omega\in\left.\Omega^{\bullet}(SM)\right|_{S_{x}M}$, $s$ is a map
${S_{x}M}\to{\bigwedge}^{\bullet}T_{x}^{*}M$, and $\sigma^{*}(s)$ is the
corresponding map ${S^{2n-1}}\to{\bigwedge}^{\bullet}(\mathbb{C}^{n})^{*}$.

######  Lemma 4.5.

The images of
$\bm{\alpha},d\bm{\alpha},\bm{\beta},d\bm{\beta},\operatorname{\mathsf{g}},J\operatorname{\mathsf{g}},\bm{\omega},J\bm{\omega},\operatorname{\mathsf{G}},\operatorname{\mathsf{G}}_{0}$
under $\bar{\sigma}^{*}$ do not depend on $M,x$ or $\sigma$.

###### Proof.

Let us check the statement for $\bm{\omega}$, the other cases being similar.
Given $X=(X_{1},X_{2})\in\mathbb{C}^{n}\oplus v^{\bot}\simeq
T_{(0,v)}S\mathbb{C}^{n}$ and $Y\in\mathbb{C}^{n}$ one has

(51) $\bar{\sigma}^{*}\bm{\omega}(X;Y)=\operatorname{\mathsf{g}}(\sigma
X_{2};\sigma Y)=\langle X_{2},Y\rangle,$

where $\langle\ ,\ \rangle$ is the $\mathbb{R}$-valued euclidean inner
product. ∎

###  4.4. The Rumin differential

The product of valuations is given by the formula of [alesker_bernig], which
involves the Rumin differential $D$ of the boundary term. In this section
(Proposition 4.7) we show that the product of any two valuations
$\operatorname{glob}(\Gamma^{M}_{k,p})$ on a Kähler manifold $M$ may be
expressed in terms of differential forms canonically constructed from the
forms defining the two factors. Thus the key point is that the Rumin
differential of the boundary term $\bm{\gamma}_{k,p}$ is canonical in the
model algebra.

For convenience we extend the definition of the Rumin differential to double
forms
$\bm{\omega}=\omega\otimes\operatorname{vol}_{M}\in\Omega^{2n-1,2n}(SM,M)$ on
an $n$-dimensional Kähler manifold $M$ by

(52) $D\bm{\omega}:=(D\omega)\otimes\operatorname{vol}_{M}.$

Recalling Lemma 3.1, this operation may be defined directly by

(53) $D\bm{\omega}=d_{\nabla}(\bm{\omega}+\bm{\alpha}\wedge\bm{\xi}),$

where $\bm{\xi}\in\Omega^{2n-2,2n}(SM,M)$ is uniquely determined, modulo
$\bm{\alpha}$, by the condition that the expression (53) is a multiple of
$\bm{\alpha}$. Recall also that $\mathcal{K}(\mathbb{C}^{n})$ denotes the
space of algebraic Kähler curvature tensors and that (6) defines a map
$\tau\colon\mathcal{V}(M)\to\Omega^{2n}(SM)$.

######  Proposition 4.6.

There exist polynomial maps

$P_{k,j},\widetilde{P}_{k,j}:\mathcal{K}(\mathbb{C}^{n})\to{\bigwedge}^{\bullet}(\mathbb{C}^{n})^{*}\otimes\Omega^{\bullet}(S^{2n-1}),$

such that, for any Kähler manifold $M$ of dimension $n$, any point $x\in M$,
and any complex linear isometry $\sigma\colon\mathbb{C}^{n}\to T_{x}M$, we
have

(54) $\displaystyle\left.D\bm{\gamma}_{k,j}^{M}\right|_{S_{x}M}$
$\displaystyle=(\bar{\sigma}^{*})^{-1}\widetilde{P}_{k,j}(\sigma^{*}\operatorname{\mathsf{R}}^{M}_{x})$
(55)
$\displaystyle\left.\tau(\operatorname{glob}\Gamma_{k,j}^{M})\right|_{S_{x}M}$
$\displaystyle=(\bar{\sigma}^{*})^{-1}P_{k,j}(\sigma^{*}\operatorname{\mathsf{R}}^{M}_{x}).$

###### Proof.

Taking $\bm{\omega}=\bm{\gamma}_{k,j}$ in (53) above, we will show that we may
take

$\bm{\xi}:={-}j\bm{\alpha}^{\vee}\wedge\operatorname{\mathsf{G}}^{j-1}\wedge\operatorname{\mathsf{G}}_{0}^{\vee}\wedge\operatorname{\mathsf{g}}^{k-2j}\wedge\widetilde{\bm{\psi}}_{2n-k-1}.$

Since the $\widetilde{\bm{\psi}}_{i}$ are polynomial in
$\operatorname{\mathsf{R}}$, in view of Lemma 3.1, Propositions 3.4 and 4.4,
and Lemma 4.5, we have that ${\bar{\sigma}^{*}}(d_{\nabla}\bm{\gamma}_{k,j})$
and ${\bar{\sigma}^{*}}(d_{\nabla}\bm{\xi})$ depend polynomially on
${\sigma^{*}}(\operatorname{\mathsf{R}}_{x}^{M})$, which will yield (54).
Recalling (5), since ${\bar{\sigma}^{*}(\pi^{*}\bm{\varphi}_{k,p})}$ is
polynomial in $\sigma^{*}(\operatorname{\mathsf{R}}_{x}^{M})$, this will give
(55).

Let us write, similarly as in (25)

$\displaystyle\operatorname{\mathsf{g}}$
$\displaystyle=\bm{\alpha}\wedge\bm{\alpha}^{\vee}+\hat{\operatorname{\mathsf{g}}},$

and put
$\widehat{\operatorname{\mathsf{G}}}=\operatorname{\mathsf{G}}+\bm{\alpha}^{\vee}\wedge\operatorname{\mathsf{G}}_{0}$.
Therefore

$\displaystyle\operatorname{\mathsf{G}}^{j}\wedge\operatorname{\mathsf{g}}^{k-2j}$
$\displaystyle=-j\bm{\alpha}^{\vee}\wedge\operatorname{\mathsf{G}}^{j-1}\wedge\operatorname{\mathsf{G}}_{0}\wedge\operatorname{\mathsf{g}}^{k-2j}$
$\displaystyle\quad+(k-2j)\bm{\alpha}\wedge\bm{\alpha}^{\vee}\wedge\operatorname{\mathsf{G}}^{j}\wedge\operatorname{\mathsf{g}}^{k-2j-1}+\widehat{\operatorname{\mathsf{G}}}^{j}\wedge\hat{\operatorname{\mathsf{g}}}^{k-2j}.$

By Lemma 3.6 and (25),

$\displaystyle d_{\nabla}\bm{\gamma}_{k,j}$
$\displaystyle=d_{\nabla}(\bm{\alpha}^{\vee}\wedge\operatorname{\mathsf{G}}^{j}\wedge\operatorname{\mathsf{g}}^{k-2j}\wedge\bm{\psi}_{2n-k-1})$
$\displaystyle=d_{\nabla}(\bm{\alpha}^{\vee}\wedge\bm{\psi}_{2n-k-1})\wedge\operatorname{\mathsf{G}}^{j}\wedge\operatorname{\mathsf{g}}^{k-2j}$
$\displaystyle=-\bm{\alpha}^{\vee}\wedge\sum_{l\geq
0}jd_{2n,k,l}\operatorname{\mathsf{G}}^{j-1}\wedge\operatorname{\mathsf{G}}_{0}\wedge\operatorname{\mathsf{g}}^{k-2j}\wedge\operatorname{\mathsf{R}}^{l}\wedge\bm{\omega}^{2n-2l-k}$
$\displaystyle\quad+\bm{\alpha}\wedge\bm{\alpha}^{\vee}\wedge\sum_{l\geq
0}(k-2j)d_{2n,k,l}\operatorname{\mathsf{R}}^{l}\wedge\operatorname{\mathsf{G}}^{j}\wedge\operatorname{\mathsf{g}}^{k-2j-1}\wedge\bm{\omega}^{2n-2l-k}$
$\displaystyle\quad+\sum_{l\geq
0}d_{2n,k,l}\widehat{\operatorname{\mathsf{R}}}^{l}\wedge\bm{\omega}^{2n-2l-k}\wedge\widehat{\operatorname{\mathsf{G}}}^{j}\wedge\widehat{\operatorname{\mathsf{g}}}^{k-2j}$
$\displaystyle\quad+d_{2n,k,n-\frac{k}{2}-1}\bm{\alpha}^{\vee}\wedge\operatorname{\mathsf{R}}^{n-\frac{k}{2}-1}\wedge\operatorname{\mathsf{R}}_{0}\wedge\operatorname{\mathsf{G}}^{j}\wedge\operatorname{\mathsf{g}}^{k-2j}$
$\displaystyle\equiv-\bm{\alpha}^{\vee}\wedge\sum_{l\geq
0}jd_{2n,k,l}\operatorname{\mathsf{G}}^{j-1}\wedge\operatorname{\mathsf{G}}_{0}\wedge\operatorname{\mathsf{g}}^{k-2j}\wedge\operatorname{\mathsf{R}}^{l}\wedge\bm{\omega}^{2n-2l-k}\mod\alpha\otimes
1.$

Note that the last term in the next to last line is a multiple of
$\operatorname{vol}\otimes\operatorname{vol}$, hence $\equiv
0\mod\alpha\otimes 1$. As for the sum in the previous line, it vanishes as

$\widehat{\operatorname{\mathsf{R}}}_{\xi}(\cdot,\cdot\,;\cdot,\xi)=\widehat{\operatorname{\mathsf{G}}}_{\xi}(\cdot,\cdot\,;\cdot,\xi)=0,\qquad\widehat{\operatorname{\mathsf{g}}}_{\xi}(\cdot\,;\,\xi)=\bm{\omega}_{\xi}(\cdot\,;\,\xi)=0.$

A double form of type $(p,2n+1)$ necessarily vanishes. Using Proposition 3.4
and (27) we thus find

$\displaystyle 0$
$\displaystyle=\left(\bm{\alpha}^{\vee}\wedge\operatorname{\mathsf{G}}^{j-1}\wedge\operatorname{\mathsf{G}}_{0}^{\vee}\wedge\operatorname{\mathsf{g}}^{k-2j}\wedge\sum_{l\geq
0}d_{2n,k,l}\operatorname{\mathsf{R}}^{l}\wedge\bm{\omega}^{2n-2l-k}\right)^{\prime}$
$\displaystyle=\bm{\alpha}\wedge\operatorname{\mathsf{G}}^{j-1}\wedge\operatorname{\mathsf{G}}_{0}^{\vee}\wedge\operatorname{\mathsf{g}}^{k-2j}\wedge\sum_{l\geq
0}d_{2n,k,l}\operatorname{\mathsf{R}}^{l}\wedge\bm{\omega}^{2n-2l-k}$
$\displaystyle\quad-\bm{\alpha}^{\vee}\wedge\operatorname{\mathsf{G}}^{j-1}\wedge\operatorname{\mathsf{G}}_{0}\wedge\operatorname{\mathsf{g}}^{k-2j}\wedge\sum_{l\geq
0}d_{2n,k,l}\operatorname{\mathsf{R}}^{l}\wedge\bm{\omega}^{2n-2l-k}$
$\displaystyle\quad+\bm{\alpha}^{\vee}\wedge\operatorname{\mathsf{G}}^{j-1}\wedge\operatorname{\mathsf{G}}_{0}^{\vee}\wedge\operatorname{\mathsf{g}}^{k-2j}\wedge\sum_{l\geq
0}(2n-2l-k)d_{2n,k,l}\operatorname{\mathsf{R}}^{l}\wedge\bm{\omega}^{2n-2l-k-1}\wedge
d_{\nabla}\bm{\alpha}.$

It follows that

$d_{\nabla}\bm{\gamma}_{k,j}\equiv-
d_{\nabla}\bm{\alpha}\wedge\bm{\xi}\mod\bm{\alpha},$

as claimed. ∎

This brings us to the main result of this section, a direct consequence of
Proposition 2.1 and Proposition 4.6. Together with the observation that the
contributions of the interior terms of the valuations in question in the
multiplication formulas of Proposition 2.1 introduce no complications, this
will imply that the multiplication tables for these families of valuations and
curvature measures are independent of the particular Kähler manifold $M$ where
they live.

In the statement of the proposition below, $\tau$ refers to the expression
(13) in Proposition 2.1. Put $W_{\mathbb{C}}=\mathbb{C}^{n}\oplus
e_{0}^{\bot}$ where $\perp$ refers to the $\mathbb{R}$-valued euclidean inner
product.

######  Proposition 4.7.

There exist polynomial maps

$P_{k,p,l,q}\colon\mathcal{K}(\mathbb{C}^{n})\to\bigwedge\nolimits^{2n}{W^{*}_{\mathbb{C}}},$

such that, for any Kähler manifold $M$ of dimension $n$, any $\xi\in
SM,x=\pi(\xi)$, and any complex linear isometry $\sigma\colon\mathbb{C}^{n}\to
T_{x}M$ with $\sigma e_{0}=\xi$,

(56)
$\tau(\operatorname{glob}(\Gamma_{k,p}^{M})\cdot\operatorname{glob}(\Gamma_{l,q}^{M}))_{\xi}=(\bar{\sigma}_{\xi}^{*})^{-1}P_{k,p,l,q}(\sigma^{*}\operatorname{\mathsf{R}}^{M}_{x}).$

Moreover, there exist polynomial maps
$R_{k,p,l,q},\widetilde{R}_{k,p,l,q}\colon\mathcal{K}(\mathbb{C}^{n})\to\bigwedge\nolimits^{2n-1}{W^{*}_{\mathbb{C}}}$
such that, for any Kähler manifold $M$ of dimension $n$, any $\xi\in
SM,x=\pi(\xi)$, and any complex linear isometry isometry
$\sigma\colon\mathbb{C}^{n}\to T_{x}M$ with $\sigma e_{0}=\xi$,

$(\bar{\sigma}_{\xi}^{*})^{-1}R_{k,p,l,q}(\sigma^{*}\operatorname{\mathsf{R}}^{M}_{x})\quad\text{and}\quad(\bar{\sigma}_{\xi}^{*})^{-1}\widetilde{R}_{k,p,l,q}(\sigma^{*}\operatorname{\mathsf{R}}^{M}_{x})$

give the boundary terms of
$\operatorname{glob}(\Gamma_{k,p}^{M})\cdot\Gamma_{l,q}^{M}$ and
$\operatorname{glob}(\Gamma_{k,p}^{M})\cdot\widetilde{\Gamma}_{l,q}^{M}$ at
$\xi$, respectively.

The following identity is a special case of Proposition 6.12 below.

######  Proposition 4.8.

We have for every Kähler manifold $M$

$\operatorname{glob}_{M}(\Gamma_{1,0})=\operatorname{glob}_{M}(\tilde{\Gamma}_{1,0}).$

###### Proof.

Put

$\displaystyle\overline{\bm{\psi}}_{2n-3}$
$\displaystyle=\sum_{l=0}^{n-2}(2n-2l-2)d_{2n,1,l}\operatorname{\mathsf{R}}^{l}\wedge\bm{\omega}^{2n-2l-3}$
$\displaystyle\bm{\eta}$
$\displaystyle=\bm{\alpha}^{\vee}\wedge\bm{\beta}^{\vee}\wedge\overline{\bm{\psi}}_{2n-3}\wedge
J\operatorname{\mathsf{g}}.$

Recalling the Definition 4.1, we claim that

(57)
$d\bm{\eta}\equiv\widetilde{\bm{\gamma}}_{1,0}-\bm{\gamma}_{1,0}\mod\bm{\alpha},d\bm{\alpha}.$

Since exact forms and multiples of $\bm{\alpha},d\bm{\alpha}$ induce curvature
measures that lie in the kernel of the globalization map, this will imply the
statement.

By Lemma 3.1, (36), and (48),

(58)
$\displaystyle\begin{split}d_{\nabla}\bm{\eta}=&\bm{\omega}\wedge\bm{\beta}^{\vee}\wedge\overline{\bm{\psi}}_{2n-3}\wedge
J\operatorname{\mathsf{g}}\\\ \quad&-\bm{\alpha}^{\vee}\wedge
J\bm{\omega}\wedge\overline{\bm{\psi}}_{2n-3}\wedge
J\operatorname{\mathsf{g}}+\bm{\alpha}^{\vee}\wedge\bm{\beta}^{\vee}\wedge
d_{\nabla}\overline{\bm{\psi}}_{2n-3}\wedge
J\operatorname{\mathsf{g}}.\end{split}$

Note that $J\bm{\psi}_{2n-2}=J\bm{\omega}\wedge\overline{\bm{\psi}}_{2n-3}$
and note also that $J(\rho\otimes\operatorname{vol})=0$ for any form
$\rho\in\Omega^{\bullet}(SM)$. Therefore

$\displaystyle 0=J(\bm{\alpha}^{\vee}\wedge\bm{\psi}_{2n-2}\wedge
J\operatorname{\mathsf{g}})=-\bm{\beta}^{\vee}\wedge\bm{\psi}_{2n-2}\wedge
J\operatorname{\mathsf{g}}+\bm{\alpha}^{\vee}\wedge
J\bm{\omega}\wedge\overline{\bm{\psi}}_{2n-3}\wedge
J\operatorname{\mathsf{g}}-\bm{\alpha}^{\vee}\wedge\bm{\psi}_{2n-2}\wedge\operatorname{\mathsf{g}},$

whence

(59) $\displaystyle d_{\nabla}\bm{\eta}$
$\displaystyle=-\bm{\beta}^{\vee}\wedge(\bm{\omega}\wedge\overline{\bm{\psi}}_{2n-3}+\bm{\psi}_{2n-2}+\bm{\alpha}^{\vee}\wedge
d_{\nabla}\overline{\bm{\psi}}_{2n-3})\wedge
J\operatorname{\mathsf{g}}-\bm{\alpha}^{\vee}\wedge\bm{\psi}_{2n-2}\wedge\operatorname{\mathsf{g}}.$

By (36) and (38) we get

$\displaystyle d_{\nabla}\overline{\bm{\psi}}_{2n-3}$
$\displaystyle=\sum_{l}(l+1)(2n-2l-3)d_{2n,1,l+1}\operatorname{\mathsf{R}}^{l}\wedge\operatorname{\mathsf{R}}_{0}\wedge\bm{\omega}^{2n-2l-4}$
$\displaystyle=\sum_{l}l(2n-2l-1)d_{2n,1,l}\operatorname{\mathsf{R}}^{l-1}\wedge\operatorname{\mathsf{R}}_{0}\wedge\bm{\omega}^{2n-2l-2}.$

Using the decomposition
$\operatorname{\mathsf{R}}=\operatorname{\mathsf{R}}_{0}\wedge\bm{\alpha}^{\vee}+\widehat{\operatorname{\mathsf{R}}}$,
we rewrite

$\displaystyle\bm{\omega}\wedge\overline{\bm{\psi}}_{2n-3}=$
$\displaystyle\sum_{l}(2n-2l-2){d}_{2n,1,l}\operatorname{\mathsf{R}}^{l}\wedge\bm{\omega}^{2n-2l-2}$
$\displaystyle=$
$\displaystyle\sum_{l}l(2n-2l-2){d}_{2n,1,l}\operatorname{\mathsf{R}}_{0}\wedge\bm{\alpha}^{\vee}\wedge\operatorname{\mathsf{R}}^{l-1}\wedge\bm{\omega}^{2n-2l-2}$
$\displaystyle+\sum_{l}(2n-2l-2)d_{2n,1,l}\widehat{\operatorname{\mathsf{R}}}^{l}\wedge\bm{\omega}^{2n-2l-2},$

and similarly

$\displaystyle\bm{\psi}_{2n-2}$
$\displaystyle=\sum_{l}d_{2n,1,l}\operatorname{\mathsf{R}}^{l}\wedge\bm{\omega}^{2n-2l-2}$
$\displaystyle=\sum_{l}ld_{2n,1,l}\operatorname{\mathsf{R}}_{0}\wedge\bm{\alpha}^{\vee}\wedge\operatorname{\mathsf{R}}^{l-1}\wedge\bm{\omega}^{2n-2l-2}+\sum_{l}d_{2n,1,l}\widehat{\operatorname{\mathsf{R}}}^{l}\wedge\bm{\omega}^{2n-2l-2}.$

It follows that

(60)
$\bm{\omega}\wedge\overline{\bm{\psi}}_{2n-3}+\bm{\psi}_{2n-2}+\bm{\alpha}^{\vee}\wedge
d_{\nabla}\overline{\bm{\psi}}_{2n-3}=\sum_{l}(2n-2l-1)d_{2n,1,l}\widehat{\operatorname{\mathsf{R}}}^{l}\wedge\bm{\omega}^{2n-2l-2}.$

Combining (59) and (60), and using the identity

$J\operatorname{\mathsf{g}}=\bm{\alpha}^{\vee}\wedge\bm{\beta}+\widehat{J\operatorname{\mathsf{g}}},$

where $\widehat{(J\operatorname{\mathsf{g}})}_{\xi}(\,\cdot\,;\,\xi)=0$, we
conclude

$\displaystyle d_{\nabla}\bm{\eta}$
$\displaystyle=-\bm{\beta}^{\vee}\wedge\sum_{l}(2n-2l-1)d_{2n,1,l}\widehat{\operatorname{\mathsf{R}}}^{l}\wedge\bm{\omega}^{2n-2l-2}\wedge
J\operatorname{\mathsf{g}}-\bm{\alpha}^{\vee}\wedge\bm{\psi}_{2n-2}\wedge\operatorname{\mathsf{g}}$
$\displaystyle=-\bm{\beta}^{\vee}\wedge\sum_{l}(2n-2l-1)d_{2n,1,l}\widehat{\operatorname{\mathsf{R}}}^{l}\wedge\bm{\omega}^{2n-2l-2}\wedge(\bm{\alpha}^{\vee}\wedge\bm{\beta}+\widehat{J\operatorname{\mathsf{g}}})-\bm{\alpha}^{\vee}\wedge\bm{\psi}_{2n-2}\wedge\operatorname{\mathsf{g}}$
$\displaystyle=\bm{\alpha}^{\vee}\wedge\bm{\beta}\wedge\bm{\beta}^{\vee}\wedge\widetilde{\bm{\psi}}_{2n-2}-\bm{\alpha}^{\vee}\wedge\bm{\psi}_{2n-2}\wedge\operatorname{\mathsf{g}},$

which is our claim (57). ∎

###  4.5. Restrictions

The main result of this subsection is

######  Theorem 4.9.

If $M\subset\tilde{M}$ is an inclusion of Kähler manifolds then

(61) $\displaystyle\left.\Gamma_{k,p}^{\tilde{M}}\right|_{M}$
$\displaystyle=\Gamma^{M}_{k,p},$ (62)
$\displaystyle\left.\tilde{\Gamma}_{k,p}^{\tilde{M}}\right|_{M}$
$\displaystyle=\tilde{\Gamma}^{M}_{k,p}$

We will prove this theorem by relating the double forms
$\bm{\gamma}^{M}_{k,p},\tilde{\bm{\gamma}}^{M}_{k,p},\bm{\varphi}_{k,p}^{M}$
defining the curvature measures on $M$ to the corresponding forms
$\bm{\gamma}^{\tilde{M}}_{k,p},\tilde{\bm{\gamma}}^{\tilde{M}}_{k,p}$ for
$\tilde{M}$. Denote the complex dimensions of $M,\tilde{M}$ by $m,n$
respectively. The case $m=n$ is trivial so we assume $d:=n-m>0$. Since a
subset $A\subset M$ has empty interior as a subset of $\tilde{M}$, the
interior term $\bm{\varphi}^{\tilde{M}}_{k,p}$ is irrelevant.

This task reduces to expressing the normal cycle
$\operatorname{nc}(A,\tilde{M})$ of $A$, considered as a subset of
$\tilde{M}$, in terms of its normal cycle $\operatorname{nc}(A,M)$ as a subset
of $M$. Since the statement is local, we may fix a hermitian orthonormal local
frame

$b_{2m},\ldots,b_{2n-1},\quad{Jb_{2r}=b_{2r+1}}$

of the normal bundle $NM$ of $M$ in $\tilde{M}$. Denote the normal space to
$M$ at a point $x$ by $N_{x}M\subset T_{x}\tilde{M}$.

For $(x,y)\in M\times S^{2d-1}$, $y=(y_{2m},\ldots,y_{2n-1})$, put

$yb:=\sum_{r=2m}^{2n-1}y_{r}b_{r}(x)$

lying in the unit sphere in the normal fiber $N_{x}M$ at $x$, and define
similarly $Yb$ for $Y\in T_{y}S^{2d-1}\subset\mathbb{C}^{d}$.

Consider the maps

$\displaystyle F\colon$ $\displaystyle
SM\times\left(0,\frac{\pi}{2}\right)\times S^{2d-1}\to S\tilde{M},$
$\displaystyle(x,v,\theta,y)$
$\displaystyle\mapsto\left(x,\cos\theta\,v+\sin\theta\,yb\right)$
$\displaystyle F_{0}\colon$ $\displaystyle M\times S^{2d-1}\to NM\subset
S\tilde{M}$ $\displaystyle(x,y)$ $\displaystyle\mapsto(x,yb),$

together with the projections

$\displaystyle P:SM\times\left(0,\frac{\pi}{2}\right)\times S^{2d-1}$
$\displaystyle\to SM,$ $\displaystyle P_{0}:M\times S^{2d-1}$
$\displaystyle\to M.$

It then holds for every $A\in\mathcal{P}(M)$ that

(63)
$\operatorname{nc}(A\subset\tilde{M})=(F_{0})_{*}\left([[A]]\times[[S^{2d-1}]]\right)+F_{*}\left(\operatorname{nc}(A)\times\left[\left[0,\frac{\pi}{2}\right]\right]\times[[S^{2d-1}]]\right),$

where the double brackets mean that we consider the corresponding oriented
manifold as a current. In other words, if
$[\phi,\omega]\in\mathcal{C}(\tilde{M})$ then

(64)
$\mathcal{C}(M)\owns\left.[\phi,\omega]\right|_{M}=\left[P_{0*}F_{0}^{*}\omega,P_{*}F^{*}\omega\right],$

where $P_{0*},P_{*}$ denote fiber integration with respect to $P_{0},P$.
(Observe that since the codimension $d>0$, any restriction
$\left.[\phi,0]\right|_{M}=0$.)

We will apply (64) to the forms $\omega$ representing the boundary terms of
$\Gamma^{\tilde{M}}_{k,p},\tilde{\Gamma}^{\tilde{M}}_{k,p}$. Formally, these
are double forms
$\bm{\gamma}^{\tilde{M}}_{k,p},\tilde{\bm{\gamma}}^{\tilde{M}}_{k,p}\in\Omega^{2n-1,2n}(S\tilde{M},\tilde{M})$,
which we identify with elements of $\Omega^{2n-1}(S\tilde{M})$ under the
convention (28).

Following the usual formalism of double forms, the pullback and pushforward
operations above are applied to the first factor. This will result in elements
of $\Omega^{2m-1,2n}(i\circ\pi)$ and $\Omega^{2m,2n}(i)$, where
$i:M\to\tilde{M}$ denotes the inclusion map and $\pi:SM\to M$ is the
projection. Denote by $\operatorname{vol}_{NM}\in\Omega^{0,d}(i)$ the section
consisting of the volume forms of the normal fibers $N_{x}M$. Pulling it back
in the obvious way, we obtain injections

$\Omega^{\bullet,\bullet}(SM,M)\hookrightarrow\Omega^{\bullet,\bullet+d}(i\circ\pi),\quad\Omega^{\bullet,\bullet}(M)\hookrightarrow\Omega^{\bullet,\bullet+d}(i)$

by $\bm{\gamma}\mapsto\bm{\gamma}\wedge\operatorname{vol}_{NM}$. In these
terms, Theorem 4.9 will follow from the identities

(65) $\displaystyle P_{*}F^{*}\bm{\gamma}_{k,p}^{\tilde{M}}$
$\displaystyle=\bm{\gamma}_{k,p}^{M}\wedge\operatorname{vol}_{NM},$ (66)
$\displaystyle P_{0*}F_{0}^{*}\bm{\gamma}_{k,p}^{\tilde{M}}$
$\displaystyle=\bm{\varphi}_{k,p}^{M}\wedge\operatorname{vol}_{NM},$ (67)
$\displaystyle P_{*}F^{*}{\tilde{\bm{\gamma}}_{k,p}}^{\tilde{M}}$
$\displaystyle=\tilde{\bm{\gamma}}_{k,p}^{M}\wedge\operatorname{vol}_{NM},$
(68) $\displaystyle P_{0*}F_{0}^{*}{\tilde{\bm{\gamma}}_{k,p}}^{\tilde{M}}$
$\displaystyle=0.$

###  4.6. Some identities for pullbacks of double forms

Let us first compute the pullbacks of
$\bm{\alpha},\bm{\alpha}^{\vee},\bm{\beta}^{\vee},\bm{\omega}$ under
$F,F_{0}$, which live in the spaces $\Omega^{\bullet,\bullet}(\pi\circ
F),\Omega^{\bullet,\bullet}(\pi\circ F_{0})$, respectively. We may consider
$\Omega^{\bullet,\bullet}(\pi\circ F_{0})$ to be a subspace of
$\Omega^{\bullet,\bullet}(\pi\circ F)$ via pullback under the projection map
$SM\times\left[0,\frac{\pi}{2}\right]\times S^{2d-1}\to M\times S^{2d-1}$.
Denote by $K$ the subspace of $\Omega^{1,1}(\pi\circ F_{0})$ (or of
$\Omega^{1,1}(\pi\circ F)$) consisting of sections of the bundle with fibers
$T_{x}^{*}M\otimes N_{x}^{*}M$.

Define

$\bm{\tau}:=F_{0}^{*}\bm{\alpha}^{\vee}\in\Omega^{0,1}(\pi\circ F_{0}),$

and observe that

(69) $\displaystyle F^{*}\bm{\alpha}$ $\displaystyle=\cos\theta\,\bm{\alpha},$
$\displaystyle
F^{*}\bm{\alpha}^{\vee}=\cos\theta\,\bm{\alpha}^{\vee}+\sin\theta\,\bm{\tau},$
(70) $\displaystyle F^{*}\bm{\beta}$ $\displaystyle=\cos\theta\,\bm{\beta},$
$\displaystyle
F^{*}\bm{\beta}^{\vee}=\cos\theta\bm{\beta}^{\vee}-\sin\theta\,J\bm{\tau}.$

Define also

$\bm{H},\bm{\sigma}\in\Omega^{1,1}(\pi\circ F_{0})$

by taking for $(x,y)\in M\times S^{2d-1}$ and for $X\in T_{x}M,\ Y\in
T_{y}S^{2d-1},\ Z\in T_{x}\tilde{M}$

$\bm{H}(x,y)((X,Y);Z):=\operatorname{\mathsf{g}}_{x}(\nabla_{X}(yb);Z^{\top}),$

where $Z=Z^{\top}+Z^{\perp}\in T_{x}M\oplus N_{x}M$, a pullback of the second
fundamental form of $M$ in $\tilde{M}$ at the normal vector $yb\in N_{x}M$,
and

$\bm{\sigma}((X,Y);Z):=\operatorname{\mathsf{g}}(Yb;Z).$

We claim that

(71) $\displaystyle F_{0}^{*}\bm{\omega}^{\tilde{M}}$
$\displaystyle\equiv\bm{H}+\bm{\sigma}\quad\mod K.$

Indeed, let $X=x^{\prime}(0)\in T_{x}M$ and $Y=y^{\prime}(0)\in T_{y}S^{2d-1}$
for some smooth curve in $M\times S^{2d-1}$. The tautological section $\xi$
along the curve $F_{0}(x(t),y(t))$ is given by $\sum y_{r}(t)b_{r}(x(t))$.
Putting $V:=dF_{0}(X,Y)$, and using (22) one gets

$(\pi^{*}\nabla)_{V}\xi=\sum_{r}y_{r}^{\prime}(0)b_{r}+y_{r}(0)(\pi^{*}\nabla)_{V}\pi^{*}b_{r}=Yb+\nabla_{X}yb.$

Equation (71) then follows easily.

Define $\operatorname{vol}_{S}\in\Omega^{2d-1,0}(\pi\circ F_{0})$ to be the
volume form of the $S^{2d-1}$ factor, and observe that

(72)
$\bm{\tau}\wedge\bm{\sigma}^{2d-1}=(2d-1)!\operatorname{vol}_{S}\wedge\operatorname{vol}_{NM}.$

Finally, we observe that similarly to (71) we have

(73)
$\displaystyle\begin{split}F^{*}\bm{\omega}^{\tilde{M}}&\equiv\cos\theta\,\bm{\omega}^{M}+\sin\theta\,(\bm{H}+\bm{\sigma})+(d\theta\otimes
1)\wedge(\cos\theta\,\bm{\tau}-\sin\theta\,\bm{\alpha}^{\vee})\quad\mod
K\end{split}.$

####  4.6.1. The Weyl lemma

Consider the second fundamental forms associated to the $b_{r}$ as the double
forms

$\bm{l}_{r}\in\Omega^{1,1}(M),\quad r=2m,\ldots,2n-1,$

$\bm{l}_{r}(X;Y):=\operatorname{\mathsf{g}}\left(\nabla_{X}b_{r},Y\right),\qquad
X,Y\in T_{x}M.$

Thus

$\bm{H}(x,y)=y\cdot(\bm{l}_{1},\ldots,\bm{l}_{r})_{x}^{t}=:y\,\bm{l}_{x}.$

Define also the complex-valued version

$\displaystyle\bm{\lambda}_{s}(X;Y)$
$\displaystyle:=\bm{l}_{2s}(X;Y)+\sqrt{-1}\bm{l}_{2s+1}(X;Y),\quad
s=m,\dots,n-1,$

and observe that at each point $x\in M$ the value $(\bm{\lambda}_{s})_{x}$ is
a symmetric bilinear form over the complex vector space $T_{x}M$.

Given a double form
$\operatorname{\mathsf{C}}\in\Omega^{\bullet,\bullet}(\tilde{M})$, the
pullback $i^{*}\operatorname{\mathsf{C}}$ by the inclusion map $i$ consists of
sections of the bundle over $M$ with fibers
${\bigwedge}^{\bullet,\bullet}(T_{x}M,T_{x}\tilde{M}),\ x\in M$. Denote by

$\left.\operatorname{\mathsf{C}}\right|_{M}\in\Omega^{\bullet,\bullet}(M)$

the image of $i^{*}\operatorname{\mathsf{C}}$ under the pullback
${\bigwedge}^{\bullet,\bullet}(T_{x}M,T_{x}\tilde{M})\to{\bigwedge}^{\bullet,\bullet}(T_{x}M,T_{x}M)$
induced by $i$ on the second factor.

######  Lemma 4.10 (Weyl lemma).

For even exponents $2e$ we have for each $x\in M$

(74)
$\int_{S^{2d-1}}\bm{H}(x,y)^{2e}dy=\frac{2^{e+1}s_{2d+2e-1}}{s_{2e}}(\operatorname{\mathsf{R}}^{M}_{x}-\operatorname{\mathsf{R}}_{x}^{\tilde{M}}|_{M})^{e},$

while the integral vanishes for odd exponents.

###### Proof.

If the exponent is odd then the integrand is odd in $y$. If the exponent is
even, apply [bernig_curv_tensors, Proposition A1] and use the Gauss equation

(75)
$\sum_{s=m}^{n-1}\bm{\lambda}_{s}\wedge\overline{\bm{\lambda}_{s}}=\sum_{r=2m}^{2n-1}\bm{l}_{r}^{2}=2(\operatorname{\mathsf{R}}^{M}-\operatorname{\mathsf{R}}^{\tilde{M}}|_{M}).$

∎

####  4.6.2. Proof of (61)

This amounts to proving (65) and (66). Since
$\operatorname{\mathsf{g}},\operatorname{\mathsf{G}}$, are horizontal, with
$\left.\operatorname{\mathsf{g}}^{\tilde{M}}\right|_{M}=\operatorname{\mathsf{g}}^{M},\left.\operatorname{\mathsf{G}}^{\tilde{M}}\right|_{M}=\operatorname{\mathsf{G}}^{M}$,
referring to (45) it is enough to show that

(76)
$\displaystyle\begin{split}P_{*}F^{*}(\bm{\alpha}^{\vee}\wedge\bm{\psi}^{\tilde{M}}_{2n-k-1})&=\bm{\alpha}^{\vee}\wedge\bm{\psi}^{M}_{2m-k-1}\wedge\operatorname{vol}_{NM},\\\
P_{0*}F_{0}^{*}(\bm{\alpha}^{\vee}\wedge\bm{\psi}^{\tilde{M}}_{2n-k-1})&=d_{2m,k,m-\frac{k}{2}}(\operatorname{\mathsf{R}}^{M})^{m-\frac{k}{2}}\wedge\operatorname{vol}_{NM}.\end{split}$

Since the curvature double form $\operatorname{\mathsf{R}}^{\tilde{M}}$ is
horizontal, we observe that

$P_{*}F^{*}(\bm{\alpha}^{\vee}\wedge(\bm{\omega}^{\tilde{M}})^{a}\wedge(\operatorname{\mathsf{R}}^{\tilde{M}})^{b})=(i^{*}\operatorname{\mathsf{R}}^{\tilde{M}})^{b}\wedge
P_{*}F^{*}(\bm{\alpha}^{\vee}\wedge(\bm{\omega}^{\tilde{M}})^{a}).$

Since the double forms appearing in (76) are sums of terms of this form, we
first evaluate the expressions
$P_{*}F^{*}(\bm{\alpha}^{\vee}\wedge(\bm{\omega}^{\tilde{M}})^{a})$. Expanding
this using (73), it is clear from (72) that any terms that are not  multiples
of

$F^{*}\bm{\alpha}^{\vee}\wedge(d\theta\otimes
1)\wedge(\cos\theta\,\bm{\tau}-\sin\theta\,\bm{\alpha}^{\vee})·\wedge\bm{\sigma}^{2d-1}=(d\theta\otimes
1)\wedge\bm{\alpha}^{\vee}\wedge\bm{\tau}\wedge\bm{\sigma}^{2d-1}$

map to $0$ under $P_{*}$. Modulo these irrelevant terms we compute

$\displaystyle F^{*}(\bm{\alpha}^{\vee}\wedge(\bm{\omega}^{\tilde{M}})^{a})$
$\displaystyle\equiv\bm{\alpha}^{\vee}\wedge\sum_{k}\frac{a!}{(a-k-2d)!k!(2d-1)!}$
$\displaystyle\quad(\cos\theta)^{a-k-2d}\bm{\omega}^{a-k-2d}(\sin\theta)^{k+2d-1}\bm{H}^{k}$
$\displaystyle\quad\wedge(d\theta\otimes
1)\wedge\bm{\tau}\wedge\bm{\sigma}^{2d-1}$
$\displaystyle=\bm{\alpha}^{\vee}\wedge\sum_{k}\frac{a!}{(a-k-2d)!k!}(\cos\theta)^{a-k-2d}(\sin\theta)^{k+2d-1}$
$\displaystyle\quad\bm{\omega}^{a-k-2d}\wedge\bm{H}^{k}\wedge(d\theta\otimes
1)\wedge\operatorname{vol}_{S}\wedge\operatorname{vol}_{NM}.$

Recalling the identity

(77) $\int_{0}^{\frac{\pi}{2}}\cos^{a}\theta\sin^{b}\theta
d\theta=\frac{s_{a+b+1}}{s_{a}s_{b}}=\frac{(a+b+2)\omega_{a+b+2}}{(a+1)(b+1)\omega_{a+1}\omega_{b+1}}$

and invoking (20) and Lemma 4.10, integration yields

$\displaystyle P_{*}F^{*}(\bm{\alpha}^{\vee}\wedge\bm{\omega}^{a})$
$\displaystyle=\bm{\alpha}^{\vee}\wedge\sum_{e}\frac{1}{2^{e}\pi^{e}e!}\frac{\omega_{a+1}(a+1)!}{\omega_{a-2e-2d+1}(a-2e-2d+1)!}\cdot$
$\displaystyle\quad\cdot\bm{\omega}^{a-2e-2d}\wedge(\operatorname{\mathsf{R}}^{M}-\operatorname{\mathsf{R}}^{\tilde{M}}|_{M})^{e}\wedge\operatorname{vol}_{NM}.$

Setting $a=2n-2l-k-1$, recalling that $n-d=m$, and applying (39),

$\displaystyle
P_{*}F^{*}(d_{2n,k,l}\bm{\alpha}^{\vee}\wedge\bm{\omega}^{2n-2l-k-1})$
$\displaystyle=\bm{\alpha}^{\vee}\wedge\sum_{e}\frac{d_{2n,k,l}}{2^{e}\pi^{e}e!}\frac{\omega_{2n-2l-k}(2n-2l-k)!}{\omega_{2m-2l-2e-k}(2m-2l-2e-k)!}\cdot$
$\displaystyle\quad\cdot\bm{\omega}^{2m-2l-2e-k-1}\wedge(\operatorname{\mathsf{R}}^{M}-\operatorname{\mathsf{R}}^{\tilde{M}}|_{M})^{e}\wedge\operatorname{vol}_{NM}$
$\displaystyle=\bm{\alpha}^{\vee}\wedge\sum_{e}{d_{2m,k,l+e}}\binom{l+e}{e}\cdot\bm{\omega}^{2m-2l-2e-k-1}\wedge(\operatorname{\mathsf{R}}^{M}-\operatorname{\mathsf{R}}^{\tilde{M}}|_{M})^{e}$
$\displaystyle\quad\wedge\operatorname{vol}_{NM}.$

Thus

$\displaystyle
P_{*}F^{*}(\bm{\alpha}^{\vee}\wedge\bm{\psi}^{\tilde{M}}_{2n-k-1})$
$\displaystyle=\sum_{l}d_{2n,k,l}(\operatorname{\mathsf{R}}^{\tilde{M}}|_{M})^{l}\wedge
P_{*}F^{*}(\bm{\alpha}^{\vee}\wedge\bm{\omega}^{2n-2l-k-1})$ (78)
$\displaystyle=\bm{\alpha}^{\vee}\wedge\operatorname{vol}_{NM}$
$\displaystyle\quad\wedge\sum_{f}d_{2m,k,f}\bm{\omega}^{2m-2f-k-1}\wedge\sum_{e}\binom{f}{e}\wedge(\operatorname{\mathsf{R}}^{\tilde{M}}|_{M})^{f-e}\wedge(\operatorname{\mathsf{R}}^{M}-\operatorname{\mathsf{R}}^{\tilde{M}}|_{M})^{e}$
$\displaystyle=\bm{\alpha}^{\vee}\wedge\sum_{f}d_{2m,k,f}\bm{\omega}^{2m-2f-k-1}\wedge(\operatorname{\mathsf{R}}^{M})^{f}\wedge\operatorname{vol}_{NM}$
$\displaystyle=\bm{\alpha}^{\vee}\wedge\bm{\psi}^{M}_{2m-k-1}\wedge\operatorname{vol}_{NM},$

which is the first relation of (76).

We next consider the interior terms for $\Gamma_{k,p}$.

Since $F^{*}_{0}\bm{\alpha}^{\vee}=\bm{\tau}$, we conclude that, modulo the
kernel of the fiber integral map,

$\displaystyle F_{0}^{*}(\bm{\alpha}^{\vee}\wedge\bm{\omega}^{a})$
$\displaystyle\equiv\binom{a}{2d-1}\bm{\tau}\wedge\bm{H}^{a-2d+1}\wedge\bm{\sigma}^{2d-1}$
$\displaystyle=(-1)^{a-2d+1}\binom{a}{2d-1}\bm{H}^{a-2d+1}(2d-1)!\operatorname{vol}_{S}|_{y}\wedge\operatorname{vol}_{NM}.$

By Lemma 4.10, the fiber integral vanishes if $a$ is even, hence
$P_{0*}F_{0}^{*}\bm{\psi}_{k,p}=0$ if $k$ is odd. If $a$ is odd, we use Lemma
4.10 to compute

$\displaystyle P_{0*}F_{0}^{*}(\bm{\alpha}^{\vee}\wedge\bm{\omega}^{a})$
$\displaystyle=\frac{a!}{(a-2d+1)!}\frac{2^{\frac{a+1}{2}-d+1}s_{a}}{s_{a-2d+1}}(\operatorname{\mathsf{R}}^{M}-\operatorname{\mathsf{R}}^{\tilde{M}}|_{M})^{\frac{a-2d+1}{2}}\wedge\operatorname{vol}_{NM}.$

We thus have for even $k$, using (40),

$\displaystyle
P_{0*}F_{0}^{*}(\bm{\alpha}^{\vee}\wedge\bm{\psi}^{\tilde{M}}_{2n-k-1})$
$\displaystyle=\sum_{l}d_{2n,k,l}\frac{(2n-2l-k-1)!}{(2m-2l-k)!}\frac{2^{m-l-\frac{k}{2}+1}s_{2n-2l-k-1}}{s_{2m-2l-k}}(\operatorname{\mathsf{R}}^{\tilde{M}}|_{M})^{l}\wedge(\operatorname{\mathsf{R}}^{M}-\operatorname{\mathsf{R}}^{\tilde{M}}|_{M})^{m-\frac{k}{2}-l}\wedge\operatorname{vol}_{NM}$
$\displaystyle=\frac{\pi^{k}}{(k+1)!\omega_{k}}\frac{1}{\left(\frac{k}{2}\right)!\left(m-\frac{k}{2}\right)!}\frac{1}{2^{m-\frac{k}{2}}\pi^{m}}\operatorname{\mathsf{R}}^{m-\frac{k}{2}}\wedge\operatorname{vol}_{NM},$

which is the second relation of (76).

####  4.6.3. Proof of (62)

Recalling (70), and the identity

$\frac{s_{a+2}}{s_{a-2e-2d+2}}=\frac{a-2e-2d+1}{a+1}\frac{s_{a}}{s_{a-2e-2d}},$

an argument as above shows that

$\displaystyle
P_{*}F^{*}(\bm{\beta}\wedge\bm{\alpha}^{\vee}\wedge\bm{\beta}^{\vee}\wedge\bm{\omega}^{a})$
$\displaystyle\equiv\sum_{e}\frac{1}{2^{e}\pi^{e}e!}\frac{a-2e-2d+1}{a+1}\frac{s_{a}a!}{s_{a-2e-2d}(a-2e-2d)!}$
$\displaystyle\quad\bm{\alpha}^{\vee}\wedge\bm{\omega}^{a-2e-2d}\wedge\bm{\beta}\wedge\bm{\beta}^{\vee}\wedge(\operatorname{\mathsf{R}}^{M}-\operatorname{\mathsf{R}}^{\tilde{M}}|_{M})^{e}\wedge\operatorname{vol}_{NM}$

and therefore, using (39),

$\displaystyle P_{*}F^{*}((2n-2l-k)$ $\displaystyle
d_{2n,k,l}\bm{\beta}\wedge\bm{\alpha}^{\vee}\wedge\bm{\beta}^{\vee}\wedge\bm{\omega}^{2n-2l-k-1})=$
$\displaystyle=\sum_{e}\binom{l+e}{e}(2m-2l-2e-k)d_{2m,k,l+e}$
$\displaystyle\quad\bm{\beta}\wedge\bm{\alpha}^{\vee}\wedge\bm{\beta}^{\vee}\wedge\bm{\omega}^{2m-2l-2e-k-1}\wedge(\operatorname{\mathsf{R}}^{M}-\operatorname{\mathsf{R}}^{\tilde{M}}|_{M})^{e}\wedge\operatorname{vol}_{NM}.$

As above, we conclude that

$P_{*}F^{*}(\bm{\beta}\wedge\bm{\alpha}^{\vee}\wedge\bm{\beta}^{\vee}\wedge\tilde{\psi}^{\tilde{M}}_{2n-k-1})=\bm{\beta}\wedge\bm{\alpha}^{\vee}\wedge\bm{\beta}^{\vee}\wedge\tilde{\psi}^{M}_{2m-k-1}\wedge\operatorname{vol}_{NM},$

which yields (67).

To prove (68), observe that $\tilde{\bm{\gamma}}_{k,p}$ is divisible by
$\bm{\beta}$, and note that $F_{0}^{*}\bm{\beta}=0$.

###  4.7. Embedded Kähler curvature tensors

It is convenient to record here the following fundamental fact.

######  Definition 4.11.

An algebraic Kähler curvature tensor
$\operatorname{\mathsf{R}}\in\mathcal{K}(\mathbb{C}^{m})$ is called _embedded_
if there exists a complex submanifold $M^{m}\subset\mathbb{C}^{n+m}$, a point
$p\in M$, and a $J$-linear isometry $f\colon\mathbb{C}^{m}\to T_{p}M$ such
that $\operatorname{\mathsf{R}}=f^{*}\operatorname{\mathsf{R}}_{p}^{M}$.

######  Theorem 4.12 (Pointwise embedding lemma).

The set of embedded algebraic Kähler curvature tensors is a full-dimensional
convex cone in $\mathcal{K}(\mathbb{C}^{m})$.

###### Proof.

Setting $\operatorname{\mathsf{R}}^{\tilde{M}}=0$ since
$\tilde{M}=\mathbb{C}^{n}$, the relation (75) characterizes this set
explicitly. Let $B$ be the complex vector space of symmetric complex-bilinear
forms on $\mathbb{C}^{m}$. Note that any finite collection
$\bm{\lambda}_{m},\dots,\bm{\lambda}_{n-1}\in B$ occur as second fundamental
forms as above, e.g. for
$M=\\{\left(\vec{z},\frac{1}{2}\bm{\lambda}_{m}(\vec{z},\vec{z}),\dots,\frac{1}{2}\bm{\lambda}_{n-1}(\vec{z},\vec{z})\right):\vec{z}=(z_{0},\dots,z_{m-1})\in\mathbb{C}^{m}\\}\subset\mathbb{C}^{n}$.

Polarizing, in order to prove the Theorem it is enough to show that the map

$\bm{\lambda}\otimes_{\mathbb{R}}\bm{\mu}\mapsto\bm{\lambda}\wedge\overline{\bm{\mu}}+\bm{\mu}\wedge\overline{\bm{\lambda}}$

constitutes a surjective map
$B\otimes_{\mathbb{R}}B\to\mathcal{K}(\mathbb{C}^{m})$.

By [sitaramayya, Theorem 3.6],

$\dim_{\mathbb{R}}\mathcal{K}(\mathbb{C}^{m})=\binom{m+1}{2}^{2},$

so it is enough to show that the dimension of the image is at least this
large. Selecting a basis $e_{1},\dots,e_{m}$ of $\mathbb{C}^{m}$, let
$\theta_{1},\dots,\theta_{m}$ be the corresponding dual basis, and observe
that the images of

$(\theta_{i}\odot\theta_{j})\otimes(\theta_{i}\odot\theta_{j}),\quad(\theta_{i}\odot\theta_{j})\otimes(\theta_{k}\odot\theta_{l}),\quad(\sqrt{-1}\theta_{i}\odot\theta_{j})\otimes(\theta_{k}\odot\theta_{l}),$

with $i\leq j$, $k\leq l$, and $(i,j)<(k,l)$ with respect to the lexicographic
order, are linearly independent, as the reader can readily confirm by
evaluating them on 4-tuples of vectors selected from $e_{i},e_{j},e_{k},e_{l}$
and their images under $J$. Therefore the dimension of the image of
$B\otimes_{\mathbb{R}}B$ under the map above is
$\geq\binom{m+1}{2}+2\binom{\binom{m+1}{2}}{2}=\binom{m+1}{2}^{2}$, as
claimed. ∎

###  4.8. Angularity of the $\Gamma_{k,p}^{M}$

Recall that among all curvature measures the ones called angular are
distinguished by enjoying the simplest possible behavior on polytopes. More
precisely, in $\mathbb{R}^{n}$ a translation invariant curvature measure
$\Psi$ is called angular (see [bernig_fu_solanes, wannerer_angular]), if there
exist functions $f_{k}$ on the Grassmannian
$\operatorname{Gr}_{k}(\mathbb{R}^{n})$ of $k$-dimensional linear subspaces of
$\mathbb{R}^{n}$ such that

$\Psi(P,U)=\sum_{k=0}^{n}\sum_{\dim
F=k}f_{k}(\overline{F})\gamma(F,P)\operatorname{vol}_{k}(F\cap U)$

for every polytope $P\subset\mathbb{R}^{n}$ and Borel set
$U\subset\mathbb{R}^{n}$. Here the inner sum is over all $k$-dimensional faces
of $P$, $\overline{F}$ is the linear subspace parallel to the affine hull of
$F$, and $\gamma(F,P)$ denotes the external angle of $P$ at $F$. On a
riemannian manifold $M$ a smooth curvature measure $\Psi$ is called angular,
if $\tau_{x}\Psi$ is angular for all $x\in M$ where $\tau_{x}$ is the transfer
map (16).

######  Proposition 4.13.

The curvature measures $\Gamma_{k,p}^{M}$ are angular.

###### Proof.

It suffices to show that the curvature measures defined by the $(2n-1)$-forms

(79)
$\bm{\gamma}_{k,p,l}:=\bm{\alpha}^{\vee}\wedge\operatorname{\mathsf{G}}^{p}\wedge\operatorname{\mathsf{g}}^{k-2l-2p}\wedge\operatorname{\mathsf{R}}^{l}\wedge\bm{\omega}^{2n-k-1},\quad
l,p\geq 0,\quad 2p+2l\leq k$

are angular.

Fix a point $x\in M$ and a $k$-dimensional linear subspace $E\subset T_{x}M$.
Let $\xi\in S_{x}M$ be perpendicular to $E$. Let

$\bar{\tau}_{x}\colon\left.\Omega^{\bullet,\bullet}(SM,M)\right|_{S_{x}M}\to\Omega^{\bullet,\bullet}(ST_{x}M,T_{x}M)^{T_{x}M}$

be the natural extension of (15) to double forms. At a point $(v,\xi)\in
ST_{x}M$, the restriction of $\bar{\tau}_{x}\bm{\gamma}_{k,p,l}$ to
$E\oplus(\xi^{\perp}\cap E^{\perp})\subset T_{(v,\xi)}ST_{x}M$ is given by

${c_{n,k}}\bar{\tau}_{x}(\operatorname{\mathsf{G}}^{p}\wedge\operatorname{\mathsf{g}}^{k-2l-2p}\wedge\operatorname{\mathsf{R}}^{l})|_{E}\wedge(1\otimes\operatorname{vol}_{E^{\perp}})\wedge(\operatorname{vol}_{S(E^{\perp})}|_{\xi}\otimes
1)=c(E)\wedge(\operatorname{vol}_{S(E^{\perp})}|_{\xi}\otimes 1),$

where $c_{n,k}$ is a constant and
$c(E)\in\mbox{\Large$\wedge$}^{k}E^{*}\otimes\mbox{\Large$\wedge$}^{2n}T_{x}^{*}M$
does not depend on $\xi\in E^{\perp}$. This shows that
$\bar{\tau}_{x}\bm{\gamma}_{k,p,l}$ defines an angular translation invariant
curvature measure on $T_{x}M$ and concludes the proof. ∎

##  5\. The Gray algebra

The $\Gamma_{k,p},\tilde{\Gamma}_{k,p}$ are not linearly independent when
$k>n$. In this section we identify certain canonical linear combinations that
are linearly independent, and which moreover carry a geometric meaning
(Proposition 6.3). They arise through an analysis of the following
algebraic/combinatorial structure.

######  Definition 5.1.

The _Gray algebra_ of degree $n$ is the complex subalgebra
$\mathcal{G}(n)\subset{\bigwedge}^{\bullet,\bullet}(\mathbb{C}^{n},\mathbb{C}^{n})\otimes_{\mathbb{R}}\mathbb{C}$
generated by $\operatorname{\mathsf{g}},\operatorname{\mathsf{G}}$. Clearly
$\mathcal{G}(n)$ is a graded commutative algebra over $\mathbb{C}$:

$\mathcal{G}(n)=\bigoplus_{k=0}^{2n}\mathcal{G}_{k}(n),$

where $\mathcal{G}_{k}(n)$ denotes the subspace of double forms of type
$(k,k)$.

###  5.1. Canonical elements of $\mathcal{G}(n)$

The elements

(80) $\bm{\phi}_{k,p}:=\sum_{j\geq
0}c_{k,p,j}\operatorname{\mathsf{G}}^{j}\wedge\operatorname{\mathsf{g}}^{k-2j}=g_{k,p}(\operatorname{\mathsf{G}},\operatorname{\mathsf{g}})\in\mathcal{G}_{k}(n),\qquad
2p\leq k$

will play an important role, where

$\displaystyle g_{k,p}(s,t)$ $\displaystyle:=\sum_{j\geq
0}c_{k,p,j}s^{j}t^{k-2j}\in\mathbb{C}[s,t]$ $\displaystyle c_{k,p,j}$
$\displaystyle:=(-1)^{p+j}2^{j}\sum_{l\geq
0}\binom{k+1}{2p+2l+1}\binom{p+l}{l}\binom{p+l}{j}.$

An alternative description of the constants in terms of generating function is
given by the following lemma.

######  Lemma 5.2.

$G(x,y,z):=\sum_{k,p,j\geq
0}c_{k,p,j}x^{j}y^{p}z^{k}=\bigg{[}(1-z)^{2}-(1-2x)(1-y)z^{2}\bigg{]}^{-1}.$

###### Proof.

Using the binomial series

$\sum_{n\geq k}\binom{n}{k}x^{n}=\frac{x^{k}}{(1-x)^{k+1}}$

we compute

$\displaystyle\sum_{k,p,j\geq 0}c_{k,p,j}x^{j}y^{p}z^{k}$
$\displaystyle=\sum_{k,p,j\geq 0}(-1)^{p+j}2^{j}\sum_{l\geq
0}\binom{k+1}{2p+2l+1}\binom{p+l}{l}\binom{p+l}{j}x^{j}y^{p}z^{k}$
$\displaystyle=\sum_{l,p,j\geq
0}(-1)^{p+j}2^{j}\binom{p+l}{l}\binom{p+l}{j}x^{j}y^{p}\frac{z^{2p+2l}}{(1-z)^{2p+2l+2}}$
$\displaystyle=\sum_{l,p\geq
0}(-1)^{p}\binom{p+l}{l}(1-2x)^{p+l}y^{p}\frac{z^{2p+2l}}{(1-z)^{2p+2l+2}}$
$\displaystyle=(1-z)^{-2}\sum_{r,l\geq
0}(-1)^{r-l}\binom{r}{l}(1-2x)^{r}y^{r-l}\frac{z^{2r}}{(1-z)^{2r}}$
$\displaystyle=(1-z)^{-2}\sum_{r\geq
0}\bigg{[}\frac{(1-2x)(1-y)z^{2}}{(1-z)^{2}}\bigg{]}^{r}$
$\displaystyle=\bigg{[}(1-z)^{2}-(1-2x)(1-y)z^{2}\bigg{]}^{-1}.$

∎

The generating function $G$ yields the following relations among the
$g_{k,p}$:

######  Lemma 5.3.

(81) $(k+2)tg_{k,p}(s,t)=2(p+1)g_{k+1,p+1}(s,t)+(k-2p+1)g_{k+1,p}(s,t),$ (82)
$(j+1)c_{k,p,j+1}+(k-2j)c_{k,p,j}=(k+1)c_{k-1,p,j}.$

###### Proof.

Note that

$\sum_{k,p\geq
0}g_{k,p}(s,t)y^{p}z^{k}=G\left(\frac{s}{t^{2}},y,tz\right),\quad g_{0,p}=0,\
p>0.$

Thus the relations (81) are equivalent to the functional identity

$\left(2t+tz\frac{\partial}{\partial
z}\right)G(s/t^{2},y,tz)=\left(\frac{2(1-y)}{z}\frac{\partial}{\partial
y}+\frac{\partial}{\partial z}\right)G(s/t^{2},y,tz),$

which is easily checked.

Relations (82) follow by equating coefficients in the identity

$(1-2x)\frac{\partial G}{\partial x}=z(z-1)\frac{\partial G}{\partial z}+2zG.$

∎

The Gray algebra interacts nicely with Kähler curvature tensors:

######  Theorem 5.4.

Suppose $\operatorname{\mathsf{R}}\in\mathcal{K}(\mathbb{C}^{n})$. Then, for
$p+q\leq n$,

$\operatorname{\mathsf{G}}^{p}\wedge\operatorname{\mathsf{R}}^{q}\wedge\operatorname{\mathsf{g}}^{2(n-p-q)}=2^{p}\frac{C_{n-p-q}}{C_{n-q}}\operatorname{\mathsf{R}}^{q}\wedge\operatorname{\mathsf{g}}^{2(n-q)},$

where

$C_{n}=\frac{(2n)!}{n!(n+1)!}$

denotes the $n$-th Catalan number.

###### Proof.

Note that by Lemma 3.2

$\displaystyle\operatorname{\mathsf{G}}^{p}\wedge\operatorname{\mathsf{R}}^{q}\wedge\operatorname{\mathsf{g}}^{2(n-p-q)}$
$\displaystyle=(-1)^{p}\frac{q!}{(p+q)!}\left.\frac{d^{p}}{d\lambda^{p}}\right|_{\lambda=0}(\operatorname{\mathsf{R}}-\lambda\operatorname{\mathsf{G}})^{p+q}\wedge\operatorname{\mathsf{g}}^{2(n-p-q)}$
$\displaystyle=(-1)^{p}\frac{q!(2n-2p-2q)!}{(p+q)!(2p+2q)!}\left.\frac{d^{p}}{d\lambda^{p}}\right|_{\lambda=0}C^{2(p+q)}((\operatorname{\mathsf{R}}-\lambda\operatorname{\mathsf{G}})^{p+q}).$

Given $\operatorname{\mathsf{R}}\in\mathcal{K}(\mathbb{C}^{n})$, the total
Chern form $\gamma\in{\bigwedge}^{\bullet}(\mathbb{C}^{n})^{*}$ of
$\operatorname{\mathsf{R}}$ is defined by

$\gamma:=\det\left(I-\frac{1}{2\pi\sqrt{-1}}\Xi\right),$

where

${\bigwedge}^{2}(\mathbb{C}^{n})^{*}\owns\Xi_{ij}:=\operatorname{\mathsf{R}}(\cdot,\cdot\,;e_{2i},e_{2j})-\sqrt{-1}\operatorname{\mathsf{R}}(\cdot,\cdot\,;e_{2i},e_{2j+1}),\quad
i,j=0,\ldots,n-1,$

see [gray_book]*p. 88. Denote the degree $2k$ component of $\gamma$ by
$\gamma_{k}$. According to [gray_book, Lemma 7.11] we have

$\frac{(n-p-q)!}{(p+q)!(2p+2q)!}C^{2(p+q)}((\operatorname{\mathsf{R}}-\lambda\operatorname{\mathsf{G}})^{p+q})=(2\pi)^{p+q}(\widetilde{\gamma}_{p+q}\wedge
F^{n-p-q})(e_{1},\ldots,e_{2n}),$

where

$\widetilde{\gamma}_{p+q}=\sum_{k=0}^{n}\binom{n-k+1}{p+q-k}\left(-\frac{\lambda}{\pi}F\right)^{p+q-k}\wedge\gamma_{k}$

by [gray_book, Lemma 7.15].

Therefore

$\displaystyle\operatorname{\mathsf{G}}^{p}$
$\displaystyle\wedge\operatorname{\mathsf{R}}^{q}\wedge\operatorname{\mathsf{g}}^{2(n-p-q)}$
$\displaystyle=(-1)^{p}(2\pi)^{p+q}\frac{q!(2n-2p-2q)!}{(n-p-q)!}\left.\frac{d^{p}}{d\lambda^{p}}\right|_{\lambda=0}(\widetilde{\gamma}_{p+q}\wedge
F^{n-p-q})(e_{1},\ldots,e_{2n})$
$\displaystyle=2^{p+q}\pi^{q}\frac{q!(2n-2p-2q)!(n-q+1)!}{(n-p-q)!(n-p-q+1)!}(\gamma_{q}\wedge
F^{n-q})(e_{1},\ldots,e_{2n}).$

Since

$\displaystyle(2\pi)^{q}(\gamma_{q}\wedge F^{n-q})(e_{1},\ldots,e_{2n})$
$\displaystyle=\frac{(n-q)!}{q!(2q)!}C^{2q}(\operatorname{\mathsf{R}}^{q})$
$\displaystyle=\frac{(n-q)!}{q!(2n-2q)!}\operatorname{\mathsf{R}}^{q}\wedge\operatorname{\mathsf{g}}^{2(n-q)}$

by [gray_book, Lemma 7.6] and Lemma 3.2, the claim follows. ∎

###  5.2. Structure of the algebra $\mathcal{G}(n)$

We will need to determine the structure of the Gray algebra. For every finite-
dimensional real vector space $U$, denote by $K^{p}(U)$ the kernel of the
linear map

$\
\operatorname{Sym}^{2}(\mbox{\Large$\wedge$}^{p}U^{*})\to\mbox{\Large$\wedge$}^{p+1}U^{*}\otimes\mbox{\Large$\wedge$}^{p-1}U^{*},\quad
A\mapsto A^{\prime}.$

Thus $K^{2}(U)=\mathcal{R}(U)$. Put
$K_{\mathbb{C}}^{p}(U):=K^{p}(U)\otimes_{\mathbb{R}}\mathbb{C}$

Put

(83) $\displaystyle g_{k}(s,t):=g_{k,0}(s,t)$ $\displaystyle=\sum_{i\geq
0}(-1)^{i}2^{k-i}\binom{k-i}{i}s^{i}t^{k-2i},$ (84) $\displaystyle\sum_{k\geq
0}g_{k}(s,t)x^{k}$ $\displaystyle=\left(1-2tx+2sx^{2}\right)^{-1}.$

These identities follow by setting $x=s/t^{2}$, $y=0$, and $z=tx$ in Lemma
5.2. Note that the polynomials $g_{k}$ satisfy

$g_{0}=1,\quad g_{1}=2t,$

and the relation

(85) $2sg_{k}-2tg_{k+1}+g_{k+2}=0$

for $k\geq 0$.

######  Theorem 5.5.

Let $p=0,1,\ldots,2n$.

  1. a)

The component $\mathcal{G}_{p}(n)$ of the Gray algebra coincides with the
subspace of $\operatorname{U}(n)$-invariant elements of
$K_{\mathbb{C}}^{p}(\mathbb{C}^{n})$, with

(86)
$\dim\mathcal{G}_{p}(n)=1+\left\lfloor\frac{\min(p,2n-p)}{2}\right\rfloor.$

  2. b)

The multiplication pairing
$P\colon\mathcal{G}_{p}(n)\times\mathcal{G}_{2n-p}(n)\to\mathcal{G}_{2n}(n)\simeq\mathbb{C}$
is perfect.

  3. c)

Consider the graded polynomial algebra $\mathbb{C}[s,t]$, where $\deg t=1$ and
$\deg s=2$. Then the map $\varphi\colon\mathbb{C}[s,t]\to\mathcal{G}(n)$ of
graded algebras given by

$\varphi(h(s,t))=h(\operatorname{\mathsf{G}},\operatorname{\mathsf{g}})$

covers an isomorphism

$\mathbb{C}[s,t]/(g_{n+1},g_{n+2})\cong\mathcal{G}(n).$

The rest of this section is devoted to the proof.

######  Remark 5.6.

It follows from c) above that the Gray algebra $\mathcal{G}(n)$ is isomorphic
to the cohomology algebra of the Grassmann manifold
$\mathrm{Gr}_{2}(\mathbb{C}^{n+2})$: by [fulton97, p. 152], [fulton98] the
cohomology algebra of $\mathrm{Gr}_{2}(\mathbb{C}^{n+2})$ is isomorphic to
$\mathbb{C}[\sigma_{2},\sigma_{1}]/(\tilde{g}_{n+1},\tilde{g}_{n+2})$, where

$\sum_{k\geq
0}\tilde{g}_{k}(\sigma_{2},\sigma_{1})x^{k}=\frac{1}{1-x\sigma_{1}+x^{2}\sigma_{2}}.$

Hence $t\mapsto\frac{1}{2}\sigma_{1},s\mapsto\frac{1}{2}\sigma_{2}$ covers an
isomorphism $\mathcal{G}(n)\to H^{*}(\mathrm{Gr}_{2}(\mathbb{C}^{n+2}))$.

Let $\operatorname{SO}(n,F)$ and $\operatorname{SL}(n,F)$ denote the
orthogonal and special linear groups over a field $F$. Recall that the finite
dimensional representations of $\operatorname{SO}(n,\mathbb{R})$ and
$\operatorname{SL}(n,\mathbb{R})$ are in bijection with the finite dimensional
analytic representations of $\operatorname{SO}(n,\mathbb{C})$ and
$\operatorname{SL}(n,\mathbb{C})$, since the latter groups are the
complexifications of the former (see, e.g., [bump13, Chapter 24]). In the
following we denote by $\varpi_{p}$, $p=0,1,\ldots,n-1$ the fundamental
weights of $\operatorname{SL}(n,\mathbb{C})$, i.e., the highest weights of the
irreducible representations $\mbox{\Large$\wedge$}^{p}\mathbb{C}^{n}$.

######  Lemma 5.7.

For $p=1,\ldots,n-1$, $K^{p}_{\mathbb{C}}(\mathbb{R}^{n})$ is irreducible as a
representation of $\operatorname{SL}(n,\mathbb{C})$ and has highest weight
$2\varpi_{n-p}$.

###### Proof.

Fix a basis $e_{1},\ldots,e_{n}$ of
$W=\mathbb{R}^{n}\otimes_{\mathbb{R}}\mathbb{C}$ and let
$\alpha_{1},\ldots\alpha_{n}$ denote the dual basis. Consider the diagram

${\mbox{\Large$\wedge$}^{p}W^{*}\otimes\mbox{\Large$\wedge$}^{p}W^{*}}$${\mbox{\Large$\wedge$}^{p+1}W^{*}\otimes\mbox{\Large$\wedge$}^{p-1}W^{*}}$${\mbox{\Large$\wedge$}^{p}W^{*}\otimes\mbox{\Large$\wedge$}^{n-p}W}$${\mbox{\Large$\wedge$}^{p+1}W^{*}\otimes\mbox{\Large$\wedge$}^{n-p+1}W}$$f_{0}$′$f_{1}$$(-1)^{p+1}L$

where the vertical arrows are induced by the
$\operatorname{SL}(n,\mathbb{C})$-equivariant isomorphisms
$\mbox{\Large$\wedge$}^{k}W^{*}\cong\mbox{\Large$\wedge$}^{n-k}W$ and

$L(\xi\otimes x)=\sum_{i=1}^{n}(\alpha_{i}\wedge\xi)\otimes(e_{i}\wedge x),$

analogous to multiplication by the symplectic form. We claim that the above
diagram is commutative.

Let
$\phi\otimes\psi\in\mbox{\Large$\wedge$}^{p}W^{*}\otimes\mbox{\Large$\wedge$}^{p}W^{*}$.
A direct computation shows that

(87)
$(\phi\otimes\psi)^{\prime}=\sum_{i=1}^{n}(\alpha_{i}\wedge\phi)\otimes(\iota_{e_{i}}\psi).$

Let $I=(i_{1},\ldots,i_{p})$ and $J=(j_{1},\ldots,j_{p})$ be $p$-tuples with
$1\leq i_{1}<\cdots<i_{p}\leq n$ and $1\leq j_{1}<\cdots<j_{p}\leq n$ and let
us write $\alpha_{I}$ for $\alpha_{i_{1}}\wedge\cdots\wedge\alpha_{i_{p}}$. We
denote by $J^{c}$ the complement of $J$, ordered in such a way that
$(J,J^{c})$ is an even permutation. Then

$\displaystyle f_{1}((\alpha_{I}\otimes\alpha_{J})^{\prime})$
$\displaystyle=\sum_{i=1}^{n}f_{1}(\alpha_{i}\wedge\alpha_{I}\otimes\iota_{e_{i}}\alpha_{J})$
$\displaystyle=(-1)^{p+1}\sum_{i=1}^{n}\alpha_{i}\wedge\alpha_{I}\otimes{e_{i}}\wedge
e_{J^{c}}$ $\displaystyle=(-1)^{p+1}L(\alpha_{I}\otimes e_{J^{c}})$
$\displaystyle=(-1)^{p+1}L(f_{0}(\alpha_{I}\otimes\alpha_{J})).$

This proves that the diagram is commutative.

By symplectic linear algebra, the map $L$ is surjective. As
$\operatorname{SL}(n,\mathbb{C})$-representations, the spaces
$\mbox{\Large$\wedge$}^{p}W^{*}\otimes\mbox{\Large$\wedge$}^{n-p}W$ and
$\mbox{\Large$\wedge$}^{p+1}W^{*}\otimes\mbox{\Large$\wedge$}^{n-p+1}W$ differ
by precisely one irreducible representation, namely the one with highest
weight $2\varpi_{n-p}$, see, e.g., [wannerer_angular]*Lemma 2.9 for a proof.
It follows that the kernel of $L$ coincides with this irreducible
subrepresentation. The commutativity of the diagram now implies the lemma. ∎

######  Remark 5.8.

The kernel of the map
${}^{\prime}:\mbox{\Large$\wedge$}^{p}W^{*}\otimes\mbox{\Large$\wedge$}^{p}W^{*}\to\mbox{\Large$\wedge$}^{p+1}W^{*}\otimes\mbox{\Large$\wedge$}^{p-1}W^{*}$
is irreducible and contains the kernel of the restriction
${}^{\prime}:\operatorname{Sym}^{2}\mbox{\Large$\wedge$}^{p}W^{*}\to\mbox{\Large$\wedge$}^{p+1}W^{*}\otimes\mbox{\Large$\wedge$}^{p-1}W^{*}$,
hence both kernels must coincide. This shows that if a double form
$\omega\in\mbox{\Large$\wedge$}{}^{p,p}(\mathbb{R}^{n},\mathbb{R}^{n})$
satisfies $\omega^{\prime}=0$, then it is also symmetric, i.e.
$\omega^{\vee}=\omega$. For a different proof of this fact see
[gray69]*Proposition (2.3).

######  Lemma 5.9.

For $p=1,\ldots,n-1$, $K^{p}_{\mathbb{C}}(\mathbb{R}^{n})$ and
$K^{n-p}_{\mathbb{C}}(\mathbb{R}^{n})$ are isomorphic as representations of
$\operatorname{SO}(n,\mathbb{C})$.

###### Proof.

Fix a basis $e_{1},\ldots,e_{n}$ of
$W=\mathbb{R}^{n}\otimes_{\mathbb{R}}\mathbb{C}$ and let
$\alpha_{1},\ldots\alpha_{n}$ denote the dual basis. Let
$*\colon\mbox{\Large$\wedge$}^{k}W^{*}\to\mbox{\Large$\wedge$}^{n-k}W^{*}$
denote the Hodge star operator, normalized so that
$\phi\wedge{*}\psi=\langle\phi,\psi\rangle\operatorname{vol}$. Moreover, let
$I_{a,b}\colon\mbox{\Large$\wedge$}^{a}W^{*}\otimes\mbox{\Large$\wedge$}^{b}W^{*}\to\mbox{\Large$\wedge$}^{b}W^{*}\otimes\mbox{\Large$\wedge$}^{a}W^{*}$
be the map that interchanges the first and second factor. Since

$\displaystyle*(\phi\wedge\alpha_{i})$ $\displaystyle=\iota_{e_{i}}{*}\phi,$
$\displaystyle*(\iota_{e_{i}}\psi)$
$\displaystyle=(-1)^{n-1}{*}\psi\wedge\alpha_{i},$

a comparison with (87) shows that the diagram

${\mbox{\Large$\wedge$}^{p}W^{*}\otimes\mbox{\Large$\wedge$}^{p}W^{*}}$${\mbox{\Large$\wedge$}^{p+1}W^{*}\otimes\mbox{\Large$\wedge$}^{p-1}W^{*}}$${\mbox{\Large$\wedge$}^{n-p}W^{*}\otimes\mbox{\Large$\wedge$}^{n-p}W^{*}}$${\mbox{\Large$\wedge$}^{n-p+1}W^{*}\otimes\mbox{\Large$\wedge$}^{n-p-1}W^{*}}$$(*\otimes*)\circ
I_{p,p}$′$(-I_{n-p-1,n-p+1})\circ(*\otimes*)$′

commutes. Hence $(*\otimes*)\circ I_{p,p}\colon
K^{p}_{\mathbb{C}}(\mathbb{R}^{n})\to K^{n-p}_{\mathbb{C}}(\mathbb{R}^{n})$ is
an isomorphism intertwining the action of $\operatorname{SO}(n,\mathbb{C})$. ∎

###  5.3. Proof of Theorem 5.5

###### Proof of Theorem 5.5, items (a) and (b).

Let
$\operatorname{Res}^{\operatorname{SL}(n,\mathbb{C})}_{\operatorname{SO}(n,\mathbb{C})}$
denote the restriction of representations functor. We denote by $\Gamma_{\nu}$
the irreducible representation with highest weight $\nu$. It is well-known,
see, e.g., [wannerer_angular]*Lemma 2.10, that, whenever $p\leq n/2$,

$\operatorname{Res}^{\operatorname{SL}(n,\mathbb{C})}_{\operatorname{SO}(n,\mathbb{C})}\Gamma_{2\varpi_{p}}=\bigoplus_{\lambda}\Gamma_{\mu},$

where the sum is over all highest weights of $\operatorname{SO}(n,\mathbb{C})$
of the form

(88) $\mu=(\underbrace{2,\ldots,2}_{k\text{ times}},0,\ldots,0)$

with $0\leq k\leq p$. Using Lemma 5.9, we conclude that
$K^{p}_{\mathbb{C}}(\mathbb{R}^{n})$ decomposes as a representation of
$\operatorname{SO}(n,\mathbb{C})$ into irreducible representations of weight
(88) with $0\leq k\leq\min(p,n-p)$.

Next we recall the fact that the
$\operatorname{SO}(2n,\mathbb{R})$-representation $\Gamma_{\mu}$ contains a
non-zero vector that is fixed under the action of the unitary group
$\operatorname{U}(n)\subset\operatorname{SO}(2n,\mathbb{R})$ if and only if
$k$ is even. This can be deduced from the Cartan-Helgason theorem for the
symmetric pair $(\operatorname{SO}(2n,\mathbb{R}),\operatorname{U}(n))$ or
from the branching rule for this pair. We refer the reader to
[alesker_mcullenconj01, boroczky_domokos_solanes] for further discussion and
earlier applications of this fact to valuation theory.

Combining the above two facts we can thus conclude that

$\dim
K^{p}_{\mathbb{C}}({\mathbb{C}^{n}})^{\operatorname{U}(n)}=1+\left\lfloor\frac{\min(p,2n-p)}{2}\right\rfloor.$

Since $\mathcal{G}_{p}(n)\subset
K^{p}_{\mathbb{C}}(\mathbb{C}^{n})^{\operatorname{U}(n)}$ as
$\operatorname{\mathsf{G}}^{\prime}=0,g^{\prime}=0$, we obtain that the right-
hand side of (86) is an upper bound on the dimension of $\mathcal{G}_{p}(n)$.

Assume first that $p=2m\leq n$, and consider the pairing matrix for the
elements

$2^{-m}\operatorname{\mathsf{G}}^{m},\
2^{1-m}\operatorname{\mathsf{G}}^{m-1}\operatorname{\mathsf{g}}^{2},\ldots,\
\operatorname{\mathsf{g}}^{2m}\in\mathcal{G}_{2m}(n),$
$2^{m-n}\operatorname{\mathsf{G}}^{n-m},\
2^{m+1-n}\operatorname{\mathsf{G}}^{n-m-1}\operatorname{\mathsf{g}}^{2},\ldots,\
\operatorname{\mathsf{G}}^{n-2m}\operatorname{\mathsf{g}}^{2m}\in\mathcal{G}_{2n-2m}(n),$

viz.

$C_{n}^{-1}\begin{pmatrix}C_{0}&C_{1}&\cdots&C_{m}\\\
C_{1}&C_{2}&\cdots&C_{m+1}\\\ \vdots&&\cdots&\vdots\\\
C_{m}&C_{m+1}&\cdots&C_{2m}\end{pmatrix}$

by the case $q=0$ of Theorem 5.4. This matrix is nonsingular; in fact,
dropping the factor $C_{n}^{-1}$, it is well known to have determinant 1
[aigner99]. The perfectness assertion (b) and the linear independence of the
considered elements of $\mathcal{G}_{p}(n)$ follows.

The case of odd $p$ is treated in the same way using the identity

$\det\begin{pmatrix}C_{1}&C_{2}&\cdots&C_{m}\\\ C_{2}&C_{3}&\cdots&C_{m+1}\\\
\vdots&&\cdots&\vdots\\\ C_{m}&C_{m+1}&\cdots&C_{2m-1}\end{pmatrix}=1.$

∎

To complete the proof of Theorem 5.5, item (c), we will need the following
combinatorial identity.

######  Lemma 5.10.

For all non-negative integers $k,n$,

$\sum_{i\geq 0}(-1)^{i}\binom{n+1-i}{i}C_{n-k-i}=(-1)^{n-k}\binom{k}{n-k}.$

In particular, the left-hand side is zero if $n>2k$.

###### Proof.

In terms of generating functions, we wish to show that

$\sum_{n,k,i\geq
0}(-1)^{i}\binom{n+1-i}{i}C_{n-k-i}x^{n}y^{k}=\frac{1}{1-xy(1-x)}.$

Indeed, using that

$\sum_{n\geq 0}C_{n}x^{n}=\frac{1-\sqrt{1-4x}}{2x},$

the left-hand side may be expressed as

$\displaystyle\sum_{n,k,i\geq 0}(-1)^{i}\binom{n-i+1}{i}C_{n-k-i}x^{n}y^{k}$
$\displaystyle=\sum_{m,k,i\geq 0}(-1)^{i}\binom{m+k+1}{i}C_{m}x^{m+k+i}y^{k}$
$\displaystyle=(1-x)\sum_{m,k\geq 0}C_{m}(1-x)^{m+k}x^{m+k}y^{k}$
$\displaystyle=\frac{1-\sqrt{1-4x(1-x)}}{2x(1-xy(1-x))}$
$\displaystyle=\frac{1}{1-xy(1-x)}.$

∎

###### Proof of Theorem 5.5, item (c).

Let us first show that
$g_{n+1}(\operatorname{\mathsf{G}},\operatorname{\mathsf{g}})=g_{n+2}(\operatorname{\mathsf{G}},\operatorname{\mathsf{g}})=0$
in $\mathcal{G}(n)$. By (83), Theorem 5.4, and Lemma 5.10 we have

$\displaystyle
P(\operatorname{\mathsf{G}}^{k}\operatorname{\mathsf{g}}^{n-2k-1},g_{n+1}(\operatorname{\mathsf{G}},\operatorname{\mathsf{g}}))$
$\displaystyle=2^{n+1}\sum_{i\geq
0}(-2)^{-i}\binom{n+1-i}{i}P(\operatorname{\mathsf{G}}^{k}\operatorname{\mathsf{g}}^{n-2k-1},\operatorname{\mathsf{G}}^{i}\operatorname{\mathsf{g}}^{n+1-2i})$
$\displaystyle=\frac{2^{n+k+1}}{C_{n}}\left(\sum_{i\geq
0}(-1)^{i}\binom{n+1-i}{i}C_{n-k-i}\right)\operatorname{\mathsf{g}}^{2n}$
$\displaystyle=0$

for $n>2k$. Since the pairing is perfect and the double forms
$\operatorname{\mathsf{G}}^{k}\operatorname{\mathsf{g}}^{n-2k-1}$, $0\leq
2k<n$, span $\mathcal{G}_{n-1}(n)$, the relation
$g_{n+1}(\operatorname{\mathsf{G}},\operatorname{\mathsf{g}})=0$ follows. The
same argument shows
$g_{n+2}(\operatorname{\mathsf{G}},\operatorname{\mathsf{g}})=0$.

Since $t\cdot g_{n+1}\neq g_{n+2}$, repeating the argument of [fu06]*Lemma 3.7
shows that the ideal of relations in $\mathcal{G}(n)$ is generated by
$g_{n+1}(\operatorname{\mathsf{G}},\operatorname{\mathsf{g}})$ and
$g_{n+2}(\operatorname{\mathsf{G}},\operatorname{\mathsf{g}})$. ∎

Theorem 5.5 has the following crucial consequences.

######  Proposition 5.11.

If $k-p>n$ then, as an element of $\mathcal{G}(n)$,

$\bm{\phi}_{k,p}=0.$

###### Proof.

Relation (85) and part c) of Theorem 5.5 imply the statement for $p=0$. The
case $p>0$ is proved by induction. Assume $p>0$ and $k-p>n$ and that the
proposition has been proved for $p-1$. Relation (81) and the inductive
assumption imply

$\bm{\phi}_{k,p}=g_{k,p}(\operatorname{\mathsf{G}},\operatorname{\mathsf{g}})=\frac{k+1}{2p}\operatorname{\mathsf{g}}g_{k-1,p-1}(\operatorname{\mathsf{G}},\operatorname{\mathsf{g}})-\frac{k-2p+2}{2p}g_{k,p-1}(\operatorname{\mathsf{G}},\operatorname{\mathsf{g}})=0.$

∎

######  Proposition 5.12.

Let $\operatorname{\mathsf{R}}\in\mathcal{K}(\mathbb{C}^{n})$. Then, for even
$k$,

$\bm{\phi}_{k,p}\wedge\operatorname{\mathsf{R}}^{n-\frac{k}{2}}=\begin{cases}(2p+1)\operatorname{\mathsf{R}}^{n-p}\wedge\operatorname{\mathsf{g}}^{2p},&k=2p,\\\
0,&k>2p.\end{cases}$

###### Proof.

By Theorem 5.4,

(89)
$\displaystyle\bm{\phi}_{k,p}\wedge\operatorname{\mathsf{R}}^{n-\frac{k}{2}}$
$\displaystyle=\sum_{j=0}^{\frac{k}{2}}c_{k,p,j}\operatorname{\mathsf{G}}^{j}\wedge\operatorname{\mathsf{g}}^{k-2j}\wedge\operatorname{\mathsf{R}}^{n-\frac{k}{2}}$
(90)
$\displaystyle=\left(\sum_{j=0}^{\frac{k}{2}}c_{k,p,j}2^{j}\frac{C_{\frac{k}{2}-j}}{C_{\frac{k}{2}}}\right)\operatorname{\mathsf{g}}^{k}\wedge\operatorname{\mathsf{R}}^{n-\frac{k}{2}}.$

The numerical factor in the last line is independent of $n$. Denote it by
$A_{k,p}$. We claim that $A_{k,p}=0$ if $k>2p$ and $=2p+1$ if $k=2p$. To prove
this it is enough to consider the case $n=\frac{k}{2}$, so that the final
expression above is $A_{k,p}\operatorname{\mathsf{g}}^{2n}$, where
$\operatorname{\mathsf{g}}^{2n}\neq 0$.

If $k>2p$ then Proposition 5.11 implies that then $\bm{\phi}_{k,p}=0$ as an
element of $\mathcal{G}(n)$, so $A_{k,p}=0$.

For the case $k=2p=2n$ we recall that $\bm{\phi}_{2p,q}=0$ for $q<p$ by
Proposition 5.11. Since

(91) $\sum_{p}c_{k,p,j}=\begin{cases}0,&j>0,\\\ k+1,&j=0\end{cases}$

by Lemma 5.2, we conclude

$\displaystyle\bm{\phi}_{2p,p}$ $\displaystyle=\sum_{q=0}^{p}\bm{\phi}_{2p,q}$
$\displaystyle=\sum_{q,j=0}^{p}c_{2p,q,j}\operatorname{\mathsf{G}}^{j}\wedge\operatorname{\mathsf{g}}^{2(p-j)}$
$\displaystyle=(2p+1)\operatorname{\mathsf{g}}^{2p}.$

∎

##  6\. The Kähler-Lipschitz-Killing algebra

###  6.1.  The Kähler-Lipschitz-Killing curvature measures and the hermitian
intrinsic volumes

######  Definition 6.1.

Given a Kähler manifold $M$ we define the  Kähler-Lipschitz-Killing curvature
measures on $M$ by

$\displaystyle\Delta^{M}_{k,p}$
$\displaystyle:=\sum_{j}c_{k,p,j}\Gamma_{k,j}=[\bm{\kappa}_{k,p},{\bm{\omega}}_{k,p}],\quad
0\leq p\leq\frac{k}{2}\leq n$ $\displaystyle B^{M}_{k,p}$
$\displaystyle:=\sum_{j}c_{k,p,j}\frac{k+1}{k-2p}\tilde{\Gamma}_{k,j}=[0,\tilde{\bm{\omega}}_{k,p}],\quad
0\leq p<\frac{k}{2}<n$

where

(92) $\displaystyle\bm{\omega}_{k,p}$
$\displaystyle:=\bm{\alpha}^{\vee}\wedge\bm{\phi}_{k,p}\wedge\bm{\psi}_{2n-k-1}$
(93) $\displaystyle\tilde{\bm{\omega}}_{k,p}$
$\displaystyle:=\frac{k+1}{k-2p}\bm{\alpha}^{\vee}\wedge\bm{\beta}\wedge\bm{\beta}^{\vee}\wedge\bm{\phi}_{k-1,p}\wedge\tilde{\bm{\psi}}_{2n-k-1}$
(94) $\displaystyle\bm{\kappa}_{k,p}$
$\displaystyle:=d_{2n,k,n-\frac{k}{2}}\bm{\phi}_{k,p}\wedge\operatorname{\mathsf{R}}^{n-\frac{k}{2}}=\delta_{k,2p}\cdot\bm{\kappa}_{k}$

by Proposition 5.12.

Define also the _hermitian intrinsic volumes_ on $M$ to be the valuations

$\mu_{k,p}^{M}:=[\Delta_{k,p}^{M}]\in\mathcal{V}(M).$

We denote by $\mathcal{KLK}(M)\subset\mathcal{V}(M)$ the vector space spanned