# Partial differential equations from matrices with orthogonal columns

David Martínez Torres

###### Abstract.

We discuss a system of third order PDEs for strictly convex smooth functions
on domains of Euclidean space. We argue that it may be understood as a closure
of sorts of the first order prolongation of a family of second order PDEs. We
describe explicitly its real analytic solutions and all the solutions which
satisfy a genericity condition; we also describe a family of non-generic
solutions which has an application to Poisson geometry and Kahler structures
on toric varieties. Our methods are geometric: we use the theory of Hessian
metrics and symmetric spaces to link the analysis of the system of PDEs with
properties of the manifold of matrices with orthogonal columns.

##  1\. Introduction

Let $\phi$ be a strictly convex smooth function defined on a connected subset
$\Omega\subset\mathbb{R}^{n}$. Its Hessian $H\phi$ defines at each point an
inner product and therefore the inverse of the Hessian matrix is also a smooth
field of inner products. It is natural to ask whether this field is also the
Hessian of a function. If $g$ is a field of inner products which is the
Hessian of a function, then there must be an equality of partial derivatives:

$\frac{\partial g_{ij}}{\partial x_{k}}=\frac{\partial g_{ik}}{\partial
x_{j}},\quad 1\leq i,j,k\leq n.$

If $\Omega$ has trivial first homology group, then the agreement of the above
partial derivatives implies that $g$ is the Hessian of a function [3]. We
shall assume from now on that the domain $\Omega$ as trivial first homology
group.

######  Definition 1.0.1.

A strictly convex function $\phi\in C^{\infty}(\Omega)$ has property
$\mathcal{I}$ if it satisfies the system of third order PDEs:

$\frac{\partial}{\partial x_{k}}{H\phi^{-1}}_{ij}-\frac{\partial}{\partial
x_{j}}{H\phi^{-1}}_{ik}=0,\quad 1\leq i,j,k\leq n.$ (1)

The purpose of this paper is to analyze the system of third order PDEs (1) for
strictly convex functions.

To be more precise about our focus, we note that it is possible to construct
strictly convex solutions to (1) by elementary means: Every strictly convex
function of one variable has property $\mathcal{I}$. If $\phi_{1}(x_{1})$ and
$\phi_{2}(x_{2})$ are strictly convex functions of one variable, then
$\phi_{1}(x_{1})+\phi_{2}(x_{2})$ has property $\mathcal{I}$. Such a function
solves the second order hyperbolic PDE with constant coefficients

$\frac{\partial^{2}\phi}{\partial x_{1}\partial x_{2}}=0,$ (2)

and, conversely, all the solutions of (2) decompose (locally) as the sum of
two functions on each of the variables $x_{1}$ and $x_{2}$; the parallel
translates of the coordinate axis are the (constant) characteristics of the
solutions. If (2) is replaced by any second order hyperbolic PDE with constant
coefficients whose solutions have (constant) orthogonal characteristics, then
its strictly convex solutions will have property $\mathcal{I}$. There is a
natural generalization of this family of hyperbolic second order PDEs to
arbitrary dimensions. Its strictly convex solutions, which we refer to as
functions with (constant) orthogonal characteristics, will also have property
$\mathcal{I}$.

It is thus natural to study ‘how close’ a strictly convex function with
property $\mathcal{I}$ may be from having orthogonal characteristics.

Our main results describe sufficient conditions for a strictly convex function
with property $\mathcal{I}$ to have orthogonal characteristics. Among such
sufficient conditions there is a generic one:

######  Theorem 1.0.2.

If a strictly convex function has property $\mathcal{I}$ and at every point
the eigenvalues of its Hessian are simple, then it has orthogonal
characteristics.

Another sufficient condition refers to the regularity of the functions:

######  Theorem 1.0.3.

If a real analytic strictly convex function has property $\mathcal{I}$, then
it has orthogonal characteristics.

We shall prove Theorem 1.0.2 first for functions of two variables. For them
the system (1) has two equations and the theorem will follow from an algebraic
manipulation valid under the hypothesis on the Hessian. The algebraic
manipulation will have a geometric counterpart: The family of second order
hyperbolic PDEs with constant coefficients whose solutions have orthogonal
characteristics defines a pencil of hyperplanes on the space of jets of order
two; away from its base, and in the set where the Hessian is strictly
positive, it restricts to a foliation. Strictly convex functions with property
$\mathcal{I}$ define a subset111That strictly convex functions with orthogonal
characteristics have property $\mathcal{I}$ means that this subset contains
the prolongation of any of the previous hyperplanes. of the space of jets of
order three. The hypothesis on the eigenvalues of the Hessian singles out the
locus of smooth points for which the jet projection is a submersion; its image
is the aforementioned foliated open subset of the jets of order two. The
geometric manifestation of our algebraic manipulation will be that the Cartan
connection is tangent to the leaves of the pullback foliation. This is why one
may say that for strictly convex functions the system of third order PDEs (1)
is the closure of the prolongation of the aforementioned pencil of second
order hyperbolic PDEs.

To go to arbitrary dimensions we will not follow the jet space approach, as we
find the algebraic complexities difficult to manage. We shall switch our
viewpoint to that of Hessian metrics. In this language what we are asking is
when for a given Hessian metric on a domain of Euclidean space its inverse
metric is also Hessian. We refer to such metrics as Hessian metrics with
property $\mathcal{I}$. The first manifestation of the relevance of the metric
viewpoint will be the following:

######  Lemma 1.0.4.

A Hessian metric has property $\mathcal{I}$ if and only if its Christoffel
symbols (of the second kind) are symmetric in the three indices.

Lemma 1.0.4 suggests that property $\mathcal{I}$ could be described as a
feature of the tangent or the orthogonal frame bundle of the Hessian metric
with its Levi-Civita connection. A fundamental property of Hessian metrics on
domains of Euclidean space is that they posses a universal (positively
oriented) orthogonal frame bundle with connection [3],
$\pi:(\mathrm{Gl}(n)^{+},\nabla)\to\mathcal{P}$, where $\mathcal{P}$ denotes
positive matrices and $\pi$ and $\nabla$ are a natural map and connection,
respectively, which come from symmetric space theory. We shall argue that the
universal orthogonal frame bundle offers an appropriate replacement for the
jet space picture. Briefly, jet spaces of order two will be replaced by
positive matrices $\mathcal{P}$; the subset of the jet spaces of order three
defined by property $\mathcal{I}$ will be replaced by the submanifold of
matrices with orthogonal columns $\mathcal{C}\subset\mathrm{Gl}(n)^{+}$; the
restriction of the jet projection will correspond to
$\pi|_{\mathcal{C}}:\mathcal{C}\to\mathcal{P}$; the Cartan connection will
correspond to the universal Levi-Civita connection $\nabla$. Property
$\mathcal{I}$ for a Hessian metric will translate as follows:

######  Theorem 1.0.5.

A Hessian metric $H\phi$ in $\Omega$ has property $\mathcal{I}$ if and only if
for any point $x\in\Omega$ and any curve $\gamma$ at $x\in\Omega$ there exist
an orthonormal frame for $H\phi(x)$ in $\mathcal{C}$ such that the
corresponding horizontal lift of $\gamma$ at that frame is tangent to
$\mathcal{C}$.

There will be a property analogous to the tangency of the Cartan connection to
the pullback foliation coming from the pencil of degree two hyperbolic PDEs:

######  Proposition 1.0.6.

The restriction of the universal Levi-Civita connection to $\mathcal{C}$
defines a (regular) involutive distribution. Its leaves are the left
translates of the strictly positive matrices $\mathcal{D}$ which fit in the
Cartan factorization $\mathcal{C}=\mathrm{SO}(n)\mathcal{D}$.

The generic condition on eigenvalues in Theorem 1.0.2 is just the open stratum
of a natural stratification of $\mathcal{P}$. To describe more precise
sufficient conditions for a Hessian metric with property $\mathcal{I}$ to come
from a function with orthogonal characteristics, we will analyze the
interaction among this stratification, the foliation on $\mathcal{C}$ defined
by the universal Levi-Civita connection, and the map $\pi|_{\mathcal{C}}$.
Theorem 1.0.3 will hinge on real analytic features of these objects.

As we shall see, property $\mathcal{I}$ for strictly convex functions appears
in a problem of Poisson geometry in toric varieties. The so-called totally
real toric Poisson structures have properties analogous to that of Hamiltonian
Kahler forms. For instance, whereas the latter are encoded by appropriate
strictly convex functions [6], the former are encoded by the simplest strictly
convex functions: quadratic forms. The most natural Poisson-theoretic PDE for
a pair given by a totally real toric Poisson structure and a Hamiltonian
Kahler form will correspond to property $\mathcal{I}$:

######  Theorem 1.0.7.

Let $(X,\mathbb{T})$ be a (smooth) toric variety endowed with a totally real
toric Poisson structure $\Pi$ and a Kahler form $\sigma$ for with the action
of the maximal compact torus $T\subset\mathbb{T}$ is Hamiltonian. Let $P$
denote the inverse Poisson structure to $\sigma$. Then the following
statements are equivalent:

  1. (1)

The Poisson structures $\Pi$ and $P$ Poisson commute: $[\Pi,P]=0$.

  2. (2)

In a basis of the Lie algebra of $T$ for which $\Pi$ corresponds to the
standard quadratic from of $\mathbb{R}^{n}$, the strictly convex function
which corresponds to $\sigma$ has property $\mathcal{I}$.

In complex dimension one a totally real toric Poisson structure and (the
inverse of) a Hamiltonian Kahler form always Poisson commute because the
commutator is a field of trivectors on a surface; equivalently, if we use
Theorem 1.0.7 this corresponds to the fact that all strictly convex functions
of one variable have property $\mathcal{I}$. As it will turn out Theorem 1.0.3
will imply that in the real analytic category such a commuting pair is the
Cartesian product of one dimensional commuting pairs :

######  Theorem 1.0.8.

Let $(X,\mathbb{T})$ be a projective toric Poisson variety endowed with a
totally real toric Poisson structure $\Pi$ which Poisson commutes with a real
analytic Hamiltonian Kahler structure $\sigma$. Then $(X,\mathbb{T})$ is a
Cartesian product of projective lines and both $\Pi$ and $\sigma$ factorize.

We shall also describe a family of strictly convex functions which satisfy
property $\mathcal{I}$ but which do not have orthogonal characteristics. We
will use it to construct Hamiltonian Kahler forms in certain ($T$-invariant)
regions of toric varieties. These regions can be thought of as the result of
gluing to a ($T$-round) 0-handle several ($T$-round) 1-handles. An
illustration of the construction is the following:

######  Proposition 1.0.9.

Let $U\subset\mathbb{C}P^{2}$ be the complement of small $T^{2}$-invariant
neighborhoods of $[1:0:0]$ and $[1:0:1]$. Then there exist a totally real
toric Poisson structure on $\mathbb{C}P^{2}$ and a Hamiltonian Kahler form on
$U$ which Poisson commute.

The structure of this paper is as follows. Section 2 describes how matrices
with orthogonal columns are used to define the family of differential
relations of second order with constant coefficients whose solutions we call
functions with (constant) orthogonal characteristics; we also discuss why they
have property $\mathcal{I}$. In Section 3 we do the analysis of the system of
third order PDEs (1) for strictly convex functions of two variables using jet
spaces. The viewpoint of Hessian metrics is introduced in Section 4. Property
$\mathcal{I}$ is translated as symmetry of the Christoffel symbols, an
algebraically simpler condition which allows to analyze the interaction of
property $\mathcal{I}$ with the Legendre transform. Section 5 describes how
the universal orthogonal frame bundle offers the appropriate setting for the
geometric analysts of property $\mathcal{I}$. We analyze the map
$\pi:(\mathcal{C},\nabla)\to\mathcal{P}$ from the submanifold of orthogonal
matrices with the restriction of the universal Levi-Civita connection onto the
manifold of positive matrices; this is the our replacement of the subsets
defined by property $\mathcal{I}$ in the space of jets of order three and two
with the restriction of the Cartan connection. Section 6 contains our main
results: Firstly, the description of property $\mathcal{I}$ as a differential
relation related to the submanifold of orthogonal matrices
$\mathcal{C}\subset(\mathrm{Gl}(n)^{+},\nabla)\to\mathcal{P}$. Secondly,
sufficient conditions for a Hessian metric with property $\mathcal{I}$ to come
from a strictly convex function with orthogonal characteristics. In Section 7
we describe a family of strictly convex functions which have property
$\mathcal{I}$ but do not have in general orthogonal characteristics. The
domains of definition of its members are what we call polytopes with
1-handles. Despite polytopes with 1-handles not being convex in general, we
show that the family is invariant under Legendre transform. Section 8 contains
our applications to Poisson geometry. We explain how on a smooth toric variety
the Poisson commuting equation for a totally real toric Poisson structure and
for (the inverse of) a Hamiltonian Kahler form can be rewritten as property
$\mathcal{I}$ for either the Kahler or the symplectic potential [6] of the
latter form. That allows us to conclude that in the real analytic category any
such commuting pair must be the Cartesian product of commuting pairs on
projective lines. We also use the family introduced in Section 7 to construct
commuting pairs on certain topologically non-trivial regions of toric
varieties (which are not Cartesian products).

The author is grateful to Max Planck Institute for Mathematics in Bonn for its
hospitality and finantial support.

##  2\. Solutions with orthogonal characteristics

It is possible to construct strictly convex functions with property
$\mathcal{I}$ by elementary means.

  1. (a)

Every strictly convex function of one variable has property $\mathcal{I}$.

  2. (b)

If $\phi_{1}(x_{1}),\dots,\phi_{n}(x_{n})$ are strictly convex, then
$\phi_{1}(x_{1})+\cdots+\phi_{n}(x_{n})$ has property $\mathcal{I}$ in the
product of the corresponding intervals.

  3. (c)

If $\phi(x)$ has property $\mathcal{I}$ in $\Omega$ and $B\in\mathrm{O}(n)$ is
an orthogonal transformation, then $\phi(Bx)$ has property $\mathcal{I}$ in
$B^{-1}(\Omega)$. This is because

$H\phi(Bx)=B^{T}H\phi(x)B$

and, therefore, if $H\phi(x)^{-1}$ is the Hessian of $\psi(x)$, then
$(H\phi(Bx))^{-1}$ is the Hessian of $\psi(Bx)$.

A function is (locally) of the form
$\phi=\phi_{1}(x_{1})+\cdots+\phi_{n}(x_{n})$ if and only if it is a solution
of the system of second order PDEs

$\frac{\partial^{2}\phi}{\partial x_{i}\partial x_{j}}=0,\quad 1\leq i<j\leq
n.$ (3)

The solutions of the system have (constant) characteristics given by the
collection of axis. This information can be used to rewrite (3) in a more
geometric fashion. For any $n\times n$ matrix $A$ one can define a
differential operator of order two on functions with values on matrix-valued
functions by the following recipe:

$\mathcal{L}^{2}_{A}\phi:=A^{T}H\phi A.$ (4)

Equivalently, the $ij$-th component is the Lie derivative of $\phi$ with
respect to the (constant) vector field defined by the $i$-th column of $A$,
followed by the Lie derivative with respect to the vector field defined by the
$j$-th column.

Let $\mathcal{D}$ denote the set of diagonal matrices with strictly positive
entries and let $\mathcal{D}(\Omega)$ denote smooth functions on $\Omega$ with
values on $\mathcal{D}$. It follows that a function $\phi$ is strictly convex
and satisfies (3) if and only if
$\mathcal{L}^{2}_{\mathrm{I}}\phi\in\mathcal{D}(\Omega)$, where $\mathrm{I}$
is the identity matrix. Let $\mathcal{C}$ denote the set of matrices with
orthogonal columns.

######  Definition 2.0.1.

A function $\phi\in C^{\infty}(\Omega)$ has orthogonal characteristics if
there exists $C\in\mathcal{C}$ such that

$\mathcal{L}^{2}_{C}\phi\in\mathcal{D}(\Omega).$ (5)

Because $\mathcal{D}$ is invariant by conjugation by permutation matrices, in
Definition 2.0.1 we may assume that $C$ has positive determinant. We will
abuse notation and use $\mathcal{C}$ to refer to matrices with orthogonal
columns and positive determinant.

######  Lemma 2.0.2.

If $\phi\in C^{\infty}(\Omega)$ has orthogonal characteristics, then $\phi$ is
an strictly convex function with property $\mathcal{I}$. More precisely,
$\phi$ is the composition of an orthogonal transformation with a function with
trivial mixed partial derivatives.

###### Proof.

Because $C$ has orthogonal columns we can factor $C=B\Lambda$,
$\Lambda\in\mathcal{D}$, $B\in\mathrm{SO}(n)$. From
$\mathcal{L}^{2}_{C}\phi=C^{T}H\phi C=\Lambda B^{T}H\phi B\Lambda$ and (5) we
deduce that $B^{T}H\phi B\in\mathcal{D}(\Omega)$, or, equivalently, that
$H\phi(Bx)\in\mathcal{D}(\Omega)$. Thus, locally

$\phi(Bx)=\phi_{1}(x_{1})+\cdots+\phi_{n}(x_{n}),\quad
x=(x_{1},\dots,x_{n})\in B^{-1}(\Omega),$

and $\phi_{i}$ is strictly convex. Therefore $\phi(Bx)$ is strictly convex and
has property $\mathcal{I}$, and so the same occurs for
$\phi(x)=\phi(B^{-1}(Bx))$. ∎

##  3\. The two-dimensional case

We would like to know whether there exist strictly convex functions with
property $\mathcal{I}$ which do not have orthogonal characteristics. For that
we find convenient to discuss the algebraic structure of the system of PDEs
(1). This should be easier in the lowest non-trivial dimension.

We shall denote partial derivatives of a function $\phi(x)$,
$x=(x_{1},x_{2})\in\Omega\subset\mathbb{R}^{2}$, by means of subindices which
follow a comma. We introduce independent variables to parametrize
(homogeneous) jet spaces of order two and three:

$\chi=\phi_{,11},\,\tau=\phi_{,12},\,\zeta=\phi_{,22},\,\upsilon=\phi_{,111},\,\nu=\phi_{,112},\,\omega=\phi_{,122},\,\xi=\phi_{,222}.$

Strictly convex functions correspond to the open subset
$\chi\zeta-\tau^{2}>0$, $\chi+\zeta>0$. The system of PDEs (1) corresponds to
the common solutions of the degree three homogeneous polynomials:

$\begin{cases}\xi(\chi\zeta-\tau^{2})-\zeta(\nu\zeta+\chi\xi-2\tau\omega)+\nu(\chi\zeta-\tau^{2})-\tau(\upsilon\zeta+\chi\omega-2\tau\nu)&=0\\\
-\omega(\chi\zeta-\tau^{2})+\tau(\nu\zeta+\chi\xi-2\tau\omega)-\upsilon(\chi\zeta-\tau^{2})+\chi(\upsilon\zeta+\chi\omega-2\tau\nu)&=0\end{cases}$
(6)

######  Lemma 3.0.1.

Strictly convex functions with property $\mathcal{I}$ correspond in the space
of jets of order three of functions in the plane to an open subset of an
intersection of quadrics:

$\begin{cases}(\zeta-\chi)\nu+\tau(\upsilon-\omega)&=0\\\
(\zeta-\chi)\omega+\tau(\nu-\xi)&=0\end{cases},\quad\chi\zeta-\tau^{2}>0,\,\,\chi+\zeta>0$
(7)

###### Proof.

We interpret the equations of the cubics (6) as a (non-homogeneous) linear
system with indeterminates $\nu\zeta+\chi\xi-2\tau\omega$ and
$\upsilon\zeta+\chi\omega-2\tau\nu$:

$\begin{cases}\zeta(\nu\zeta+\chi\xi-2\tau\omega)+\tau(\upsilon\zeta+\chi\omega-2\tau\nu)&=(\xi+\nu)(\chi\zeta-\tau^{2})\\\
\tau(\nu\zeta+\chi\xi-2\tau\omega)+\chi(\upsilon\zeta+\chi\omega-2\tau\nu)&=(\upsilon+\omega)(\chi\zeta-\tau^{2})\end{cases}$

In the open subset defined by $\chi\zeta-\tau^{2}\neq 0$ we obtain the
equivalent relations

$\begin{cases}\nu\zeta+\chi\xi-2\tau\omega&=\begin{vmatrix}\xi+\nu&\tau\\\
\upsilon+\omega&\chi\end{vmatrix}\\\
\upsilon\zeta+\chi\omega-2\tau\nu&=\begin{vmatrix}\zeta&\xi+\nu\\\
\tau&\upsilon+\omega\end{vmatrix}\\\
\end{cases}\Longleftrightarrow\begin{cases}(\zeta-\chi)\nu+\tau(\upsilon-\omega)&=0\\\
(\zeta-\chi)\omega+\tau(\nu-\xi)&=0\end{cases}$

∎

To each $[a:b]\in\mathbb{R}P^{1}$ one can associate the following second order
PDE with constant coefficients for strictly convex functions:

$a\phi_{,11}-a\phi_{,22}=b\phi_{,12}.$ (8)

It corresponds to open subset of a hyperplane of the space of jets of order
two

$a(\chi-\zeta)-b\tau=0,\quad\chi\zeta-\tau^{2}>0,\,\,\chi+\zeta>0$ (9)

whose first prolongation is

$\begin{cases}a(\chi-\zeta)-b\tau&=0\\\ a(\upsilon-\omega)-b\nu&=0\\\
a(\nu-\xi)-b\omega&=0\end{cases},\quad\chi\zeta-\tau^{2}>0,\,\,\chi+\zeta>0.$
(10)

######  Proposition 3.0.2.

Let $\phi\in C^{\infty}(\Omega)$ be a strictly convex function.

  1. (1)

If $\phi$ satisfies (8) for some $[a:b]\in\mathbb{R}P^{1}$, then $\phi$ has
property $\mathcal{I}$.

  2. (2)

If $\phi$ satisfies (8) for more than one $[a:b]\in\mathbb{R}P^{1}$, then
$\phi$ is — up to a degree one polynomial — a multiple of the standard
quadratic form $x_{1}^{2}+x_{2}^{2}$.

  3. (3)

If $\phi$ has property $\mathcal{I}$ and its Hessian has simple eigenvalues,
then $\phi$ satisfies (8) for some $[a:b]\in\mathbb{R}P^{1}$.

###### Proof.

The set of equations (8) are exactly those second order PDEs whose solutions
have orthogonal characteristics. Therefore item (1) is the specialization of
Lemma 2.0.2 to the two-dimensional case. Alternatively, item (1) follows from
the inclusion of the solutions of (10) in the solutions of (7).

If $\phi$ satisfies (8) for more than one $[a:b]\in\mathbb{R}P^{1}$, then its
second jet belongs to the base of the pencil (9): $\chi-\zeta=\tau=0$. That is
to say $\phi_{,12}=0$ and $\phi_{,11}=\phi_{,22}$. Therefore
$0=\phi_{,221}=\phi_{,111}=\phi_{,112}=\phi_{,222}$. Hence $\phi$ is a degree
two polynomial whose homogeneous part of degree two equals
$k(x_{1}^{2}+x_{2}^{2})$, $k>0$.

The Hessian $H\phi$ has two eigenvalues if and only if it misses the base of
the pencil. Equivalently, the field of vectors in the plane
$(\phi_{,22}-\phi_{11},\phi_{,12})\in\mathbb{R}^{2}$ has no zeroes. Therefore
we can (locally) take the quotient of the components of the vector field to
get a well-defined slope function. Property $\mathcal{I}$ as in (7) can be
rewritten

$\begin{cases}\langle(\phi_{,22}-\phi_{11},\phi_{,12}),(\phi_{12},\phi_{,11}-\phi_{22})_{,1}\rangle&=0\\\
\langle(\phi_{,22}-\phi_{11},\phi_{,12}),(\phi_{12},\phi_{,11}-\phi_{22})_{,2}\rangle&=0,\end{cases}$

where $\langle\cdot,\cdot\rangle$ is the standard inner product. This implies
that the slope function is constant, which is exactly the second order PDE (8)
for some $[a:b]\in\mathbb{R}P^{1}$. ∎

Proposition 3.0.2 does not clarify whether strictly convex functions with
property $\mathcal{I}$ and which do not have orthogonal characteristics exist.
As we shall discuss in Section 7 such solutions exist: It is possible to start
from a multiple of the standard quadratic form in a subdomain of $\Omega$
which ‘bifurcates’ into solutions to different equations in (8) in other
subsets of the domain $\Omega$.

######  Remark 3.0.3.

(The Cartan connection on jet spaces) The algebraic manipulation in item (3)
in Proposition 3.0.2 has a geometric counter-part. The requirement on the
Hessian corresponds to the regularity condition needed to identify solutions
with holonomic sections with respect to the Cartan connection: On the one
hand, the subset of the jet spaces which corresponds to property $\mathcal{I}$
is not smooth; the 1-forms

$\begin{split}\Xi_{1}&=\nu(d\zeta-d\chi)+(\zeta-\chi)d\nu+\tau(d\upsilon-d\omega)+(\upsilon-\omega)d\tau,\\\
\Xi_{2}&=\omega(d\zeta-d\xi)+(\zeta-\xi)d\omega+\tau(d\nu-d\epsilon)+(\nu-\epsilon)d\tau\end{split}$

are colinear in the subset
$\omega(\upsilon-\omega)-\nu(\nu-\epsilon)=\chi-\zeta=\tau=0$. On the other
hand, the smooth locus of the intersection of quadrics fails to be transverse
to the fibers of the projection onto jets of order two in the points over the
base of the pencil.

The connection 1-forms one has to add when passing from jets of order two to
jets of order three are:

$\Theta_{1}=d\xi-\upsilon dx-\nu dy,\quad\Theta_{2}=d\tau-\nu dx-\omega
dy,\quad\Theta_{3}=d\chi-\omega dx-\zeta dy.$

The pullback foliation is defined by the 1-forms

$K_{1}=(\xi-\chi)d\theta-\tau(d\xi-d\chi),\quad
K_{2}=(\upsilon-\omega)d\nu-\nu(d\upsilon-d\omega),\quad
K_{3}=(\nu-\epsilon)d\omega-\omega(d\nu-d\epsilon).$

The equalities

$K_{1}=(\chi-\zeta)\Theta_{2}-\tau(\Theta_{1}-\Theta_{3}),\quad
K_{2}-\frac{\nu^{2}}{\tau^{2}}K_{1}=-\frac{\nu}{\tau}\Xi_{1},\quad
K_{3}-\frac{\omega^{2}}{\tau^{2}}K_{1}=-\frac{\omega}{\tau}\Xi_{2}$

hold in the intersection of (7) with the complement of the pullback of the
base of the pencil. Therefore holonomic sections in this subset are tangent to
the pullback foliation. Hence their order two jet must be inside a hyperplane
of the pencil.

######  Remark 3.0.4.

(Orthogonal characteristics and Legendre transform) Let $\phi$ be an strictly
convex function on a convex domain $\Omega\subset\mathbb{R}^{2}$. Its Legendre
transform is an strictly convex function $\phi^{*}$ on a convex domain
$\Omega^{*}$ which is related to $\Omega$ by a (Legendre) diffeomorphism. Let
us assume that $\phi$ satisfies the constant coefficients second order PDE:

$a\phi_{,11}+c\phi_{,22}-b\phi_{,12}=0,\quad[a:c:b]\in\mathbb{R}P^{2}.$

Because $H\phi^{*}$ at $x\in\Omega^{*}$ equals $H\phi^{-1}$ at its related
point in $\Omega$ we have the equality

$a\phi_{,22}^{*}+c\phi_{,11}^{*}+b\phi_{,12}^{*}=0.$

Thus the Legendre transform induces an involution in the parameter space of
constant coefficients degree two homogeneous PDEs: $[a:c:b]\mapsto[c:a:-b]$.
Its fixed point set is $[1:1:0]\cup[a:-a:b]\subset\mathbb{R}P^{2}$. To the
point $[1:1:0]$ corresponds the Laplace equation, which has no strictly convex
solutions. The projective line $[a:-a:b]$ parametrizes hyperbolic PDEs with
orthogonal characteristics (8). Therefore if $\phi$ is a function on the
convex domain $\Omega$ with orthogonal characteristics, so its Legendre
transform is. This invariance property holds regardless of the dimension:

$\mathcal{L}^{2}_{C}\phi\in\mathcal{D}(\Omega)\Longleftrightarrow
C^{T}(H\phi^{*})^{-1}C\in D(\Omega^{*})\Longleftrightarrow
C^{T}(H\phi^{*})C\in\mathcal{D}(\Omega^{*}),$

where the first equivalence uses the relation between Hessian matrices of the
original function and its Legendre transform, and in the second equivalence we
have inverted the matrices and we have used $C^{T}C\in\mathcal{D}$.

##  4\. Hessian metrics with symmetric Christoffel symbols

To generalize the results in Section 3 the complexities brought by the
increase of dimension shall be dealt with by shifting the perspective to that
of Hessian metrics.

A Hessian metric on $\Omega\subset\mathbb{R}^{n}$ is a Riemannian metric
obtained as the Hessian matrix of a (strictly convex) function on $\Omega$.
Property $\mathcal{I}$ for strictly convex functions can be translated to a
requirement for a Hessian metric: that its inverse metric be Hessian as well.
In such case we say that the given Hessian metric has property $\mathcal{I}$.

There is another natural differential condition on Hessian metrics which
allows to formulate in arbitrary dimensions the algebraic simplification of
property $\mathcal{I}$ described in Lemma 3.0.1. For a Hessian metric $H\phi$
the Christoffel symbols of the first kind equal the partial derivatives of
order three: $\Gamma_{ijk}=\phi_{,ijk}$. The Christoffel symbols (of the
second kind) are

$\Gamma_{kj}^{i}={H\phi^{-1}}_{il}\Gamma_{ljk}.$

Let $[H\phi]_{,k}$ denote the partial derivative with respect to $k$ of the
entries of the Hessian matrix. For each $1\leq k\leq n$ we define the
_Christoffel matrix_

$\Gamma_{k}:=\Gamma_{k\star}^{\bullet}={H\phi^{-1}}_{\bullet\circ}{[H\phi]_{,k}}_{\circ\star}=H\phi^{-1}[H\phi]_{,k}.$

######  Definition 4.0.1.

A Hessian metric $H\phi$ on $\Omega$ has symmetric Christoffel symbols if the
Christoffel symbols (of the second type) are symmetric on the three indices.
Equivalently, if its Christoffel matrices are symmetric.

Property $\mathcal{I}$ corresponds to an open subset of the solutions of a
system of polynomial equations of degree $2n-1$ in the space of jets of order
three. The symmetry of the Christoffel symbols is determined by a system of
polynomial equations of degree $n$; for $n=2$ it is exactly (7). The
generalization of Lemma 3.0.1 to arbitrary dimensions is that property
$\mathcal{I}$ translates into the symmetry of Christoffel symbols:

###### Proof of Lemma 1.0.4.

The Hessian metric is invertible if and only if the $i$-th and $j$-th lines of
$[H\phi^{-1}]_{,j}$ and $[H\phi^{-1}]_{,i}$ are equal. This is equivalent to
the same condition for the matrices $[H\phi^{-1}]_{,j}H\phi$ and
$[H\phi^{-1}]_{,i}H\phi$. If we prolong the identity $H\phi^{-1}H\phi=I$, then
the condition transforms on the same condition for the Christoffel matrices
$\Gamma_{j}$ and $\Gamma_{i}$. This amounts to symmetry of all Christoffel
matrices. ∎

The problem of the symmetry of Christoffel symbols of Hessian metrics is
amenable to Lie theoretic methods. A first instance of that is the following:

######  Proposition 4.0.2.

The following statements for a Hessian metric $H\phi$ on $\Omega$ are
equivalent:

  1. (1)

It has symmetric Christoffel symbols.

  2. (2)

The two matrices $H\phi$ and $[H\phi]_{,k}$ are in the same Cartan subalgebra
of the symmetric matrices, $1\leq k\leq n$. (The Cartan subalgebra may vary
with $k$).

  3. (3)

The two matrices $H\phi$ and $\Gamma_{k}$ are in the same Cartan subalgebra of
the symmetric matrices, $1\leq k\leq n$. (The Cartan subalgebra may vary with
$k$).

###### Proof.

Let $\mathfrak{s}$ be the vector subspace of symmetric matrices and let
$\mathfrak{d}\subset\mathfrak{s}$ denote the diagonal matrices; this is a
maximal Cartan subalgebra of the symmetric matrices.

The Christoffel matrix $\Gamma_{k}$ is the product of the symmetric matrices
$H\phi^{-1}$ and $[H\phi]_{,k}$. Therefore $H\phi$ has symmetric Christoffel
matrices if and only if the following commutators are trivial:

$[H\phi^{-1},[H\phi]_{,k}]=0,\quad 1\leq k\leq n.$

This is equivalent to require that $[H\phi]_{,k}$ be in the same maximal torus
of $\mathfrak{s}$ as $H\phi^{-1}$. If $B$ is a orthogonal matrix which
diagonalizes $H\phi$ then it also diagonallizes $H\phi^{-1}$:

$B^{T}H\phi B=\Lambda,\quad B^{T}H\phi^{-1}B=\Lambda^{-1}.$

Therefore if $H\phi,[H\phi]_{,k}$ are in the Cartan subalgebra
$\mathrm{Ad}_{B}(\mathfrak{d})\subset\mathfrak{s}$, then so is $H\phi^{-1}$.
This shows the equivalence between $(1)$ and $(2)$.

If $(2)$ holds then

$\Gamma_{k}=H\phi^{-1}[H\phi]_{,k}=B^{T}\Lambda_{1}BB^{T}\Lambda_{2}B=B^{T}\Lambda_{1}\Lambda_{2}B$

remains in the same Cartan subalgebra of the commuting factors, which proves
$(3)$. Condition $(3)$ by definition implies the symmetry of the Christoffel
matrices. ∎

By Lemma 1.0.4 Hessian metrics with symmetric Christoffel symbols are the same
as Hessian metrics with property $\mathcal{I}$. Thus by Lemma 2.0.2 strictly
convex functions with orthogonal characteristics define Hessian metrics with
symmetric Christoffel symbols. We can reprove this result with a Lie theoretic
approach:

######  Lemma 4.0.3.

Let $\phi\in C^{\infty}(\Omega)$. If
$\mathcal{L}^{2}_{C}\phi\in\mathcal{D}(\Omega)$ for some $C\in\mathcal{C}$,
then the Christoffel matrices of $H\phi$ for all points in $\Omega$ are in the
Cartan subalgebra $\mathrm{Ad}_{C}(\mathfrak{d})$. In particular $H\phi$ has
symmetric Christoffel symbols.

###### Proof.

By hypotheses for each $x\in\Omega$

$C^{T}H\phi C=\Lambda,\quad\Lambda=\Lambda(x)\in\mathcal{D}(\Omega),\quad
C\in\mathcal{C}.$

Hence $H\phi={(C^{T})}^{-1}\Lambda C^{{}^{-1}}$ and upon taking its first
order prolongation

$[H\phi]_{,k}={(C^{T})}^{-1}\Lambda_{,k}C^{-1}.$

Therefore both $H\phi$ and $[H\phi]_{,k}$ are in
$\mathrm{Ad}_{C}(\mathfrak{d})$. By item (2) in Proposition 4.0.2 the same
occurs for $\Gamma_{k}$ (the action by conjugation on $\mathfrak{d}$ of second
factor of $\mathcal{C}=\mathrm{SO}(n)\mathcal{D}$ is trivial). By Proposition
4.0.2 this implies symmetry of Christoffel matrices. ∎

######  Proposition 4.0.4.

The Legendre transform preserves the class of Hessian metrics with property
$\mathcal{I}$ on convex domains.

###### Proof.

Let $\mathcal{L}_{j}$ denote the Lie derivative with respect to
$\tfrac{\partial}{\partial x_{j}}$. We can rewrite property $\mathcal{I}$ for
$H\phi$ as

$\mathcal{L}_{\bullet}{(H\phi^{-1})}_{\star
k}-\mathcal{L}_{\star}{(H\phi^{-1})}_{\bullet k}=0.$ (11)

The differential of $\phi$ defines the Legendre diffeomorphism

$d\phi:\Omega\to\Omega^{*},\quad D(d\phi)=H\phi.$

If we push forward each equation in (11 by the Legendre diffeomorphism
$d\phi$, then the Lie derivative of the pushed forward functions — entries of
the inverse Hessian — by the pushed forward vector fields will subtract to
zero as well. The entries of the inverse Hessian matrix are pushed forward to
the entries of the Hessian of $\phi^{*}$; the coordinate vector fields are
pushed forward to the columns vector fields of the Jacobian matrix $H\phi$,
which at points in $\Omega^{*}$ is the matrix ${H\phi^{*}}^{-1}$. Therefore
property $\mathcal{I}$ for $H\phi$ is equivalent to

${{H\phi^{*}}^{-1}}_{\bullet\circ}{[H\phi^{*}]_{k}}_{\circ\star}-{{H\phi^{*}}^{-1}}_{\star\circ}{[H\phi^{*}]_{k}}_{\circ\bullet}=0,$

which is the symmetry of the Christoffel matrices of $H\phi^{*}$. Therefore by
Lemma 1.0.4 $H\phi^{*}$ has property $\mathcal{I}$. ∎

##  5\. The universal frame bundle for Hessian metrics and matrices with
orthogonal columns

To generalize Proposition 3.0.2 to arbitrary dimensions jet spaces will be
replaced by (a subset of) the principal orthogonal frame bundle of the Hessian
metric with its Levi-Civita connection. There are three reasons to do that:

  1. (a)

A function $\phi\in C^{\infty}(\Omega)$ has property $\mathcal{I}$ if and only
if $H\phi$ has symmetric Christoffel symbols. The Christoffel symbols are the
components of the Levi-Civita connection. Therefore one may expect a
reformulation of property $\mathcal{I}$ related to the tangent or orthogonal
frame bundle with the Levi-Civita connection.

  2. (b)

If a function $\phi\in C^{\infty}(\Omega)$ has orthogonal characteristics,
then the Hessian metric $H\phi$ splits (locally, but along the same
characteristics everywhere). In other words, the conclusion of the de Rham
Splitting Theorem holds. Therefore to study the relation between property
$\mathcal{I}$ and orthogonal characteristics it may be appropriate to look at
parallel transport on the principal frame bundle with its the Levi-Civita
connection.

  3. (c)

Hessian metric on domains of Euclidean space are characterized among
Riemannian metrics as those whose frame bundle is the pullback of a universal
principal bundle with connection coming from symmetric space theory [3].

For a function $\phi$ the information of the homogeneous part of its second
jet is the same as the one contained in its Hessian. Thus for our strictly
convex functions we shall be looking at the map $x\mapsto H\phi(x)$, which
takes values in the positive matrices $\mathcal{P}$. There, the pencil in (9)
defined by hyperbolic PDEs with orthogonal characteristics generalizes as
follows: The second order PDE equation (3) corresponds to Hessian metrics with
image in the (positive) diagonal matrices $\mathcal{D}$. Matrices with
orthogonal columns have a factorization into an orthogonal and a diagonal
matrix. Thus we may confine ourselves to the family of second order PDEs
$\mathcal{L}^{2}_{B}\phi\in\mathcal{D}$, $B\in\mathrm{SO}(n)$. To each of them
there corresponds the subset $\mathrm{Ad}_{B}(\mathcal{D})\subset\mathcal{P}$;
their union over $B\in\mathrm{SO}(n)$ fills $\mathcal{P}$ as any positive
matrix can be diagonalised by a special orthogonal transformation.

For a Riemannian metric defined on a subset of Euclidean space, its orthogonal
frame bundle — forgetting for the moment about the Levi-Civita connection — is
constructed via pullback: The map
$\pi:\mathrm{Gl}(n)^{+}\to\mathrm{Gl}(n)^{+}$, $A\mapsto{A^{-1}}^{T}A^{-1}$,
has image the closed embedded submanifold of positive matrices. The
restriction to its image

$\pi:\mathrm{Gl}(n)^{+}\to\mathcal{P}$ (12)

  * •

is a (right) principal bundle for $\mathrm{SO}(n)$;

  * •

intertwines the right action of $\mathrm{SO}(n)$ on $\mathrm{Gl}(n)^{+}$ and
the adjoint action of $\mathrm{SO}(n)$ on $\mathcal{P}$;

  * •

is the bundle of (positively oriented) orthogonal frames for metrics on
$\mathbb{R}^{n}$.

Let $\nabla$ be the $\mathrm{SO}(n)$-invariant principal connection on
$\pi:\mathrm{Gl}(n)^{+}\to\mathcal{P}$ which at the identity matrix has as
horizontal space the symmetric matrices222Its curvature there is
$[\mathfrak{s},\mathfrak{s}]$..

######  Proposition 5.0.1.

[3, Proposition 4.1] If $H\phi$ is a Hessian metric on $\Omega$, then the
pullback of $\nabla$ by $H\phi:\Omega\to\mathcal{P}$, $x\mapsto H\phi(x)$, is
the Levi-Civita connection on the orthogonal frame bundle of $H\phi$.
Furthermore, this property characterizes Hessian metrics among Riemmanian
metrics in domains of Euclidean space.

The appropriate replacement of the jets of order three will not be the full
bundle of orthogonal frames. It will be the subset of matrices with orthogonal
columns. The following result, of which Proposition 1.0.6 in the Introduction
follows, shows that it is well-behaved with respect to the universal Levi-
Civita connection:

######  Proposition 5.0.2.

The subset of matrices with orthogonal columns
$\mathcal{C}\subset\mathrm{Gl}(n)^{+}$ has the following properties:

  1. (1)

It is a closed embedded submanifold of $\mathrm{Gl}(n)^{+}$ on which the
Cartan decomposition $\mathrm{Gl}(n)^{+}=\mathrm{SO}(n)\mathcal{P}$ induces a
product structure $\mathcal{C}=\mathrm{SO}(n)\mathcal{D}$.

  2. (2)

The intersection of the horizontal distribution of $\nabla$ with the tangent
bundle $T\mathcal{C}$ is an involutive distribution on $\mathcal{C}$. Its
foliation $\mathcal{F}$ is the one associated to the Cartan decomposition,
with leaves the left $\mathrm{SO}(n)$-translates of $\mathcal{D}$.

###### Proof.

Let $\iota$ be the inversion map on $\mathrm{Gl}(n)^{+}$ and
$q=\pi\circ\iota:\mathrm{Gl}(n)^{+}\to\mathcal{P}$, $A\mapsto A^{T}A$. A
matrix $C$ has orthogonal columns if and only if $q(C)\in\mathcal{D}$.
Therefore $\mathcal{C}$ is the preimage under a submersion of the closed
embedded submanifold of positive diagonal matrices, thus a closed embedded
submanifold of $\mathrm{Gl}(n)^{+}$. We have already used the (unique)
factorisation of a matrix with orthogonal columns as a product of an
orthogonal and a diagonal matrix. It is straightforward that it gives rise to
a Cartesian product of manifolds $\mathcal{C}=\mathrm{SO}(n)\mathcal{D}$.

The product structure in (1) implies that its tangent space at
$C\in\mathcal{C}$ is

$\mathfrak{so}(n)\cdot
C\cdot\mathfrak{d}=C\cdot\mathrm{Ad}_{C^{-1}}(\mathfrak{so}(n))\cdot\mathfrak{d}.$

The horizontal space of $\nabla$ there is $C\cdot\mathfrak{s}$. Because the
conjugation of a skew orthogonal matrix by an orthogonal one can never be
symmetric, the intersection of the tangent spaces must be
$C\cdot\mathfrak{d}$. Therefore the intersection of the horizontal space of
$\nabla$ with $T\mathcal{C}$ is the distribution 333One could also deduce
involutivity by recalling that the curvature of the connection is
$C\cdot[\mathfrak{s},\mathfrak{s}]$, and, therefore, the abelian subalgebra
$\mathfrak{d}$ is flat. tangent to the left translates of $\mathcal{D}$ by
$\mathrm{SO}(n)$. ∎

Next, we argue how $\pi:(\mathcal{C},\mathcal{F})\to\mathcal{P}$ provides a
‘desingularization’ of the pencil $\operatorname{Ad}{{}_{B}}(\mathcal{D})$,
$B\in\mathrm{SO}(n)$.

######  Proposition 5.0.3.

The restriction $\pi|_{\mathcal{C}}:\mathcal{C}\to\mathcal{P}$ has the
following properties:

  1. (1)

It is a surjective map all whose values are clean.

  2. (2)

The restriction of the differential of $\pi|_{\mathcal{C}}$ to $T\mathcal{F}$
has trivial kernel and the restriction of $\pi|_{\mathcal{C}}$ to the leaf
$B\mathcal{D}$ is a diffeomorphism onto
$\operatorname{Ad}{{}_{B}}(\mathcal{D})$.

###### Proof.

Let $V\in\mathcal{P}$. Then it diagonalizes in an orthogonal basis:
$B^{T}VB=\Lambda$, $B\in\mathrm{SO}(n)$, $\Lambda\in\mathcal{D}$. Hence
$\pi(B\Lambda^{1/2})=V$, so $\pi|_{\mathcal{C}}$ is surjective. The fiber is

$\pi|_{\mathcal{C}}^{-1}(V)=B\Lambda^{1/2}\mathrm{SO}(n)_{\Lambda},$

where the latter subgroup is the stabilizer of $\Lambda$ for the adjoint
action. The kernel of the differential of $\pi$ at $B\Lambda^{1/2}$ is
$B\Lambda^{1/2}\cdot\mathfrak{so}(n)$. The tangent space of $\mathcal{C}$ at
$B\Lambda^{1/2}$ is
$B\Lambda^{1/2}\cdot\mathrm{ad}_{\Lambda}^{-1}(\mathfrak{d})$. Because the
adjoint orbit through $\Lambda$ intersects $\mathcal{D}$ cleanly at $\Lambda$,
their intersection — which is the kernel of of the differential of
$\pi|_{\mathcal{C}}$ at $B\Lambda^{1/2}$ — is
$B\Lambda^{1/2}\cdot\mathfrak{so}(n)_{\Lambda}$. Therefore all values of
$\pi|_{\mathcal{C}}$ are clean.

The tangent space to the leaf of $\mathcal{F}$ through $B\Lambda^{1/2}$ is
$B\Lambda^{1/2}\cdot\mathfrak{d}$. Its intersection with
$B\Lambda^{1/2}\cdot\mathfrak{so}(n)_{\Lambda}$ is trivial. Therefore the
restriction of $\pi$ to $B\mathcal{D}$ is a local diffeomorphism over its
image. That image is by construction $\mathrm{Ad}_{B}(\mathcal{D})$. To
conclude that it is a diffeomorphism one can either check that the map is
bijective or argue that the manifolds involved are contractible. ∎

The base of the pencil (9) corresponds to inner products in the plane which
have a unique eigenvalue. In arbitrary dimensions we have an analogous
subsets. For each symmetric matrix we can order its eigenvalues (with their
multiplicity) in an increasing sequence. To each partition $\kappa$ of
$\\{1,\dots,n\\}$ there correspond a subset $\Theta^{\kappa}_{\mathfrak{s}}$;
likewise, to each matrix with orthogonal columns we can order the norm of its
columns in an increasing sequence. In that way we obtain partitions of
$\mathcal{D}$, $\mathcal{P}$ and $\mathcal{C}$: $\Theta_{\mathcal{D}}$,
$\Theta_{\mathcal{P}}$, $\Theta_{\mathcal{C}}$.

######  Proposition 5.0.4.

The partitions $\Theta_{\mathcal{D}},\Theta_{\mathcal{P}}$ and
$\Theta_{\mathcal{C}}$ are stratifications of $\mathcal{D}$, $\mathcal{P}$ and
$\mathcal{C}$, respectively, and they interact with the map
$\pi|_{\mathcal{C}}:(\mathcal{C},\mathcal{F})\to\mathcal{P}$ as follows:

  1. (1)

The preimage of $\Theta_{\mathcal{P}}^{\kappa}$ is
$\Theta_{\mathcal{C}}^{\kappa}$ and the restriction is a principal bundle:

$\pi|_{\Theta^{\kappa}_{\mathcal{C}}}:\Theta^{\kappa}_{\mathcal{C}}\to\Theta^{\kappa}_{\mathcal{P}}.$
(13)

  2. (2)

The foliation $\mathcal{F}=\mathrm{SO}(n)\mathcal{D}$ of $\mathcal{C}$
intersects the stratum $\Theta^{\kappa}_{\mathcal{C}}$ cleanly and induces
there the foliation $\mathrm{SO}(n)\Theta^{\kappa}_{\mathcal{D}}$ of
$\Theta^{\kappa}_{\mathcal{C}}$.

  3. (3)

The foliation $\mathrm{SO}(n)\Theta^{\kappa}_{\mathcal{D}}$ of
$\Theta^{\kappa}_{\mathcal{C}}$ is projectable by the submersion
$\pi|_{\Theta^{\kappa}_{\mathcal{C}}}$. Its image is the foliation
$\mathrm{Ad}_{\mathrm{SO}(n)}(\Theta_{\mathcal{D}}^{\kappa})$ of
$\Theta^{\kappa}_{\mathcal{P}}$.

It is in this sense that
$\pi:(\mathcal{C},\mathcal{F},\Theta_{\mathcal{C}})\to(\mathcal{P},\mathrm{Ad}_{\mathrm{SO}(n)}(\mathcal{D}),\Theta_{\mathcal{P}})$
is a desingularization of the stratified pencil.

###### Proof.

The group $\mathrm{SO}(n)$ acts on $\mathfrak{s}$ by conjugation. As for any
proper action it produces a stratification of $\mathfrak{s}$ in orbit types
[4, Chapter 2]: two symmetric matrices are related if their isotropy subgroups
are conjugated. It is well known that upon passing to connected components the
outcome is a (Whitney B) stratification of $\mathfrak{s}$. The stratification
$\Theta_{\mathfrak{s}}$ is the result of possibly collecting some of the
strata of the orbit type stratification belonging to the same subset of the
orbit type partition; in any case, it is still a stratification for the
partial order associated to the partitions $\kappa$ of $\\{1,\dots,n\\}$. The
stratification $\Theta_{\mathfrak{s}}$ — made of adjoint orbits — intersects
the Cartan subalgebra $\mathfrak{d}$ cleanly, thus inducing a stratification
$\Theta_{\mathfrak{d}}$. Each strata there is a face of the positive Weyl
chamber of diagonal matrices with diagonal elements ordered increasingly; an
open convex polytope in a vector subspace $\mathfrak{d}^{\kappa}$. The
partition $\Theta_{\mathcal{P}}$ is obtained by intersecting
$\Theta_{\mathfrak{s}}$ with the open subset of positive matrices, thus it is
a stratification. The partition $\Theta_{\mathcal{D}}$ is also obtained upon
intersection; it is a stratification because for instance $\mathcal{D}$ is an
open subset of $\mathfrak{d}$. Finally, $\Theta_{\mathcal{C}}$ is the pull
back of $\Theta_{\mathcal{D}}$ by the submersion $q$, and therefore it is a
stratification as well.

Let $C\in\mathcal{C}$ with factorisation $C=B\Lambda$. Then
$q(C)=B\Lambda^{-2}B^{T}$ and therefore

$C\in\Theta^{\kappa}_{\mathcal{C}}\Longleftrightarrow\Lambda\in\Theta^{\kappa}_{\mathcal{D}}\Longleftrightarrow\Lambda^{-2}\in\Theta^{\kappa}_{\mathcal{D}}\Longleftrightarrow
q(C)\in\Theta^{\kappa}_{\mathcal{D}}.$

The stratum $\Theta_{k}^{\mathcal{D}}$ is an open subset of the vector
subspace $\mathfrak{d}^{\kappa}$ of all matrices whose stabilizer contains
$\mathrm{SO}(n)_{\kappa}$. Because the fiber of $\pi|_{\mathcal{C}}$ through
$C$ is $C\mathrm{SO}(n)_{\kappa}$ and $\pi|_{\Theta^{\kappa}_{\mathcal{C}}}$
is saturated by fibers of $\pi|_{\mathcal{C}}$, it is a principal
$\mathrm{SO}(n)_{\kappa}$-bundle. This proves (1).

The fibers of $q$ are the orbits of the left $\mathrm{SO}(n)$-action. The
restriction $q|_{\mathcal{D}}:\mathcal{D}\to\mathcal{D}$ is the square map,
which preserves the strata of $\Theta_{\mathcal{D}}$. Therefore the
factorization of $\mathcal{C}$ is compatible with the stratification:

$\Theta_{\mathcal{C}}=\mathrm{SO}(n)\Theta_{\mathcal{D}}.$

Thus the intersection of the leaf of $\mathcal{F}$ though
$C\in\Theta_{\mathcal{C}}^{\kappa}$ is $C\Theta_{\mathcal{D}}^{\kappa}$. At
$C$ the respective tangent spaces are $C\cdot\mathfrak{d}$ and
$C\cdot\mathfrak{d}^{\kappa}$. Therefore the intersection is clean and this
proves (2).

By item (1) above and by item (2) in Proposition 5.0.3 the restriction of
$\pi|_{\Theta^{\kappa}_{\mathcal{C}}}$ to the leaf
$C\cdot\Theta^{\kappa}_{\mathcal{D}}\subset\Theta^{\kappa}_{\mathcal{C}}$ is a
diffeomorphism over its image. Its image is
$\mathrm{Ad}_{B}(\Theta^{\kappa}_{\mathcal{D}})$
($\pi|_{\mathcal{D}}:\mathcal{D}\to\mathcal{D}$ is the inverse of the square
map); it is in fact the common image of all leaves through points of the fiber
$C\mathrm{SO}(n)_{\kappa}$. ∎

We can now sharpen Proposition 4.0.2.

######  Proposition 5.0.5.

Let $H\phi$ be a Hessian metric on $\Omega$ such that $H\phi(\Omega)$ is
contained in the stratum $\Theta^{\kappa}_{\mathcal{P}}$. Then its Christoffel
symbols are symmetric if and only if $H\phi$ and $[H\phi]_{k}$ can be
conjugated by an orthogonal matrix to a matrix in $\mathfrak{d}^{\kappa}$,
$1\leq k\leq n$.

###### Proof.

Because the image of $H\phi$ is contained in $\Theta^{\kappa}_{\mathcal{P}}$,
its partial derivatives must be in the tangent space to the stratum:
$[H\phi]_{k}\in T\Theta^{\kappa}_{\mathcal{P}}$. By item (2) in Proposition
4.0.2 there exists a special orthogonal matrix $B$ which conjugates $H\phi$
and $[H\phi]_{k}$ to a diagonal one: $B^{T}[H\phi]_{k}B\in\mathfrak{d}$.
Therefore

$B^{T}[H\phi]_{k}B\in\mathfrak{d}\cap
T\Theta^{\kappa}_{\mathcal{P}}=\mathfrak{d}^{\kappa}.$

∎

##  6\. Differential relations on the submanifold of matrices with orthogonal
columns

We want to transfer property $\mathcal{I}$ for Hessian metrics into a
differential condition for the orthogonal frame bundle at the submanifold of
matrices with orthogonal columns.

Let $H\phi$ be a Hessian metric on $\Omega\subset\mathbb{R}^{n}$. To every
curve $\gamma:(-\epsilon,\epsilon)$ based at $x\in\Omega$ we associate a curve
in $\mathcal{P}$ based at $H\phi(x)$:

$H\phi(\gamma):(-\epsilon,\epsilon)\to\mathcal{P},\quad t\mapsto
H\phi(\gamma(t)).$

Upon choosing an orthonormal frame for $H\phi(x)$, we can construct the
horizontal lift of $H\phi(\gamma)$ based at the orthonormal frame.

######  Definition 6.0.1.

A Hessian metric $H\phi$ in $\Omega$ has property $\mathcal{C}$ if for any
point $x\in\Omega$ and any curve $\gamma$ at $x\in\Omega$ there exist an
orthonormal frame $C\in\mathcal{C}$ for $H\phi(x)$ such that the corresponding
horizontal curve is tangent to $\mathcal{C}$ at $C$.

We now translate property $\mathcal{I}$ to the universal orthogonal frame
bundle setting:

###### Proof of Theorem 1.0.5.

We must show that a Hessian metric in $\Omega\subset\mathbb{R}^{n}$ has
property $\mathcal{C}$ if and only if it has symmetric Christoffel symbols.

Property $\mathcal{C}$ is linear in the velocity of the curve at $x$. Thus it
is enough to prove the equivalence for $\gamma(t)=x+te_{k}$, $1\leq k\leq n$.
Let us denote the horizontal lift at $A\in\mathcal{C}$ by $A(t)$. That $A$
belongs to $\mathcal{C}$ means that $A^{T}A=\Lambda\in\mathcal{D}$. By
Proposition 5.0.1 (taken from [3]) the pullback of
$\pi:(\mathrm{Gl}(n)^{+},\nabla)\to\mathcal{P}$ by $H\phi$ is the orthonormal
frame bundle of $H\phi$ with its Levi-Civita connection. Thus we have:

$0={A^{\prime}}_{\bullet\star}+\Gamma_{k\circ}^{\bullet}A_{\circ\star}(=A^{\prime}+\Gamma_{k}A).$

The image by the differential of $q$ of the vector of
$A^{\prime}=A^{\prime}(0)$ is ${A^{T}}^{\prime}A+A^{T}A^{\prime}$. Therefore
the Hessian metric satisfies property $\mathcal{C}$ at $A$ if and only if

$A^{T}\Gamma_{k}^{T}A+A^{T}\Gamma_{k}A\in\mathfrak{d}.$ (14)

We have $\Gamma_{k}={H\phi^{-1}}{[H\phi]_{,k}}$, $H\phi^{{-1}}=AA^{T}$, where
the latter identity uses that $A$ is an orthonormal frame for $H\phi$. Hence
we may rewrite $\Gamma_{k}=AA^{T}[H\phi]_{,k}$. Thus equation (14) is
equivalent to:

$A^{T}[H\phi]_{,k}AA^{T}A+A^{T}AA^{T}[H\phi]_{,k}A=A^{T}[H\phi]_{k}A\Lambda+\Lambda
A^{T}[H\phi]_{,k}A\in\mathfrak{d}.$

Because $\Lambda$ has non-zero positive entries if its anti-commutator with a
matrix is diagonal, then the matrix must be diagonal. The conclusion is that
property $\mathcal{C}$ is equivalent to:

$A^{T}[H\phi]_{,k}A\in\mathfrak{d},\quad A^{T}A\in\mathcal{D},\quad
AA^{T}=H\phi^{-1}.$

By item (2) in Proposition 4.0.2 this is exactly the symmetry of the
Christoffel matrices. ∎

We can verify that strictly convex functions with orthogonal characteristics
satisfy property $\mathcal{C}$.

######  Lemma 6.0.2.

If $\mathcal{L}^{2}_{C}\phi\in\mathcal{D}$, $C\in\mathcal{C}$, then $H\phi$
satisfies property $\mathcal{C}$.

###### Proof.

By definition $C^{T}H\phi C=\Lambda$, $\Lambda\in\mathcal{D}$. Therefore
$C^{T}[H\phi]_{,k}C\in\mathfrak{d}$. The matrix $C\Lambda^{-1/2}$ also belongs
to $\mathcal{C}$ and it is an orthonormal frame for $H\phi$. Therefore

${(C\Lambda^{-1/2}})^{T}[H\phi]_{,k}C\Lambda^{-1/2}\in\mathfrak{d},$

and thus property $\mathcal{C}$ holds. ∎

Next we analyze up to which extent Hessian metrics with property $\mathcal{C}$
are defined by functions with orthogonal characteristics.

######  Definition 6.0.3.

A Hessian metric $H\phi$ on $\Omega\subset\mathbb{R}^{n}$ has property
$\mathcal{CK}$ if there exists a point $x\in\Omega$ and an orthonormal frame
$C\in\mathcal{C}$ for $H\phi(x)$, such that for every curve in $\Omega$ based
at $x$ its horizontal lift at $C$ is contained in $\mathcal{C}$.

######  Theorem 6.0.4.

If a Hessian metric $H\phi$ on $\Omega$ has property $\mathcal{CK}$, then it
solves $\mathcal{L}^{2}_{C}\phi\in\mathfrak{d}$, $C\in\mathcal{C}$.

Furthermore, the following conditions are equivalent:

  1. (1)

The image $H\phi(\Omega)$ is contained in the stratum
$\Theta_{\mathcal{P}}^{\kappa}$.

  2. (2)

Property $\mathcal{CK}$ holds for all orthonormal frames in $\mathcal{C}$ over
all points of $H\phi(\Omega)$.

In either case $\phi$ restricts to the leaves of the (parallel) foliation
determined by the (rotation of) the subspace $\mathfrak{d}^{\kappa}$ to a
multiple of the standard quadratic form (up to an affine summand).

###### Proof.

Let $x$ and $C$ be a point and orthonormal frame for $H\phi(x)$ with respect
which condition $\mathcal{CK}$ is defined. Because $\Omega$ is (path)
connected for every point $y$ we have a curve $\gamma$ starting at $x$ and
ending at $y$. By hypotheses the horizontal lift of $H\phi(\gamma)$ is a curve
$A(t)$ contained in $\mathcal{C}$. Therefore it is in $T\mathcal{C}$ and
horizontal. By (2) in Proposition 5.0.2 $A(t)$ is contained in the leaf
$C\mathcal{D}$ of $\mathcal{F}$: $A(t)=CD(t)$. Because $A(t)$ is a curve of
orthogonal frames $A(t)^{T}H\phi(\gamma)A(t)=\mathrm{I}$. Therefore

$C^{T}H\phi C=D(t)^{-2}\in\mathcal{D}.$

Equivalently, if $C=B\Lambda$ then by (2) in Proposition 5.0.3
$\pi|_{\mathcal{C}}$ sends the leaf $C\mathcal{D}$ diffeomorphically onto
$\mathrm{Ad}_{B}(\mathcal{D})$, which is where the image of $H\phi$ must be
confined. The (local) splitting condition for $H\phi$ along the
characteristics of $C$ can be also argued as follows: that $A(t)\subset
C\mathcal{D}$ implies that line fields at $x$ spanned by each column of $C$
are invariant by parallel transport. Thus the de Rham Splitting Theorem
applies (and Hessian metrics restrict to Hessian metrics).

If $H\phi\subset\Theta^{\kappa}_{\mathcal{P}}$, then by item (1) in
Proposition 5.0.4 the frame $C\in\mathcal{C}$ with respect to which
$\mathcal{CK}$ is defined belongs to $\Theta^{\kappa}_{\mathcal{C}}$. By item
(2) in the same Proposition all horizontal curves based at $C$ must be
contained in the leaf $C\Theta^{\kappa}_{\mathcal{D}}$ of the foliation of
$\Theta^{\kappa}_{\mathcal{C}}$ induced by $\mathcal{F}$ upon clean
intersection. The principal $\mathrm{SO}(n)_{\kappa}$-action takes these
horizontal curves at $C$ to horizontal curves in
$\Theta_{\mathcal{C}}^{\kappa}$ based at any matrix in the fiber.

Conversely, let $C=B\Lambda$ and assume that the horizontal lifts at
$CB^{\prime}$, $B^{\prime}\in\mathrm{SO}(n)_{\Lambda}$, are contained in
$\mathcal{C}$. Then they are inside the corresponding leaf of $\mathcal{F}$
and therefore

$H\phi(\Omega)\subset\bigcap_{B^{\prime}\in\mathrm{SO}(n)_{\Lambda}}\mathrm{Ad}_{BB^{\prime}}(\mathcal{D})=\mathrm{Ad}_{B}\left(\bigcap_{B^{\prime}\in\mathrm{SO}(n)_{\Lambda}}\mathrm{Ad}_{B^{\prime}}(\mathcal{D})\right).$

Because the exponential intertwines the adjoint action the latter intersection
can be understood in $\mathfrak{s}$. If $\Lambda\in\mathfrak{d}^{\kappa}$ we
have

$\mathfrak{d}^{\kappa}=\bigcap_{B^{\prime}\in\mathrm{SO}(n)_{\Lambda}}\mathrm{Ad}_{B^{\prime}}(\mathfrak{d}),\quad\mathrm{SO}(n)_{\Lambda}=\mathrm{SO}(n)_{\kappa}.$

We used property $\mathcal{CK}$ with respect to an arbitrary point
$x\in\Omega$. If we select the point whose image lies in the stratum of
smallest dimension, then we conclude that $H\phi(\Omega)$ cannot leave that
stratum.

If $H\phi\subset\Theta^{\kappa}_{\mathcal{P}}$, then at any point $x\in\Omega$
all orthonormal frames in $\mathrm{SO}(n)_{\kappa}$ are parallel. This means
that the common eigendirections $\mathrm{Ad}_{B}(\mathfrak{d}^{\kappa})$ are
parallel. Therefore the restriction of $H\phi$ to the foliation given by the
parallel translates of $\mathrm{Ad}_{B}(\mathfrak{d}^{\kappa})$ in $\Omega$ is
flat. Hence on each such (affine) subspace it is a quadratic form with equal
eigenvalues. Therefore in the local splitting of $\phi$ along orthogonal
characteristics we shall have a multiple of the standard quadratic form along
$\mathrm{Ad}_{B}(\mathfrak{d}^{\kappa})$. ∎

The following result is more general than Theorem 1.0.2 in the Introduction.

######  Theorem 6.0.5.

If A Hessian metric $H\phi$ on $\Omega$ has property $\mathcal{C}$ and
$H\phi(\Omega)$ is contained in a stratum of $\Theta_{\mathcal{P}}$, then it
has property $\mathcal{CK}$. In particular $\phi$ has orthogonal
characteristics.

###### Proof.

Let $H\phi(\Omega)$ be contained in $\Theta^{\kappa}_{\mathcal{P}}$. This
stratum is foliated by $\mathrm{SO}(n)\Theta^{\kappa}_{\mathcal{D}}$ and we
want to show that for each curve $\gamma(t)$ in $\Omega$ the derivative of
$H\phi(\gamma(t))$ is tangent to this foliation. By property $\mathcal{C}$ for
each $t_{0}$ there exists an orthogonal frame $C\in\mathcal{C}$ such that the
horizontal curve $A(t)$ at $C$ has derivative at zero tangent to
$\mathcal{C}$:

$A^{\prime}(0)\in T\mathcal{C}\cap
T\pi^{-1}(\Theta^{\kappa}_{\mathcal{P}})=C\cdot\mathrm{ad}_{\Lambda}^{-1}(\mathfrak{d})\cap
C\cdot(\mathfrak{so}(n)+\mathfrak{d}^{\kappa})=C\cdot(\mathfrak{so}(n)_{\Lambda}+\mathfrak{d}^{\kappa})=T\pi|_{\mathcal{C}}^{-1}(\Theta_{\mathcal{P}}^{\kappa}).$

Therefore the intersection
$\mathcal{C}\cap\pi|_{\mathcal{C}}^{-1}(\Theta_{\mathcal{P}}^{\kappa})=\Theta^{\kappa}_{\mathcal{C}}$
is clean and $A^{\prime}(0)$ belongs to $\Theta^{\kappa}_{\mathcal{C}}$.
Because the vector is horizontal by item (2) in Proposition 5.0.4 it is
tangent to the foliation $\mathrm{SO}(n)\Theta^{\kappa}_{\mathcal{D}}$. Thus
by item (3) the tangent vector of $H\phi(\gamma)$ at $t_{0}$ is tangent to the
foliation $\mathrm{SO}(n)\Theta^{\kappa}_{\mathcal{D}}$. Because $\Omega$ is
connected this implies that $H\phi(\Omega)$ is contained in one of the leaves
of $\mathrm{SO}(n)\Theta^{\kappa}_{\mathcal{D}}$. ∎

The following result is a more precise statement that Theorem 1.0.3 in the
Introduction:

######  Theorem 6.0.6.

Let $H\phi$ be a real analytic Hessian metric on
$\Omega\subset\mathbb{R}^{n}$. Then $\phi$ has property $\mathcal{C}$ if and
only if it has property $\mathcal{CK}$. In such case $H\phi$ is the
restriction to $\Omega$ of a product Hessian metric on a (rotated) cube.

###### Proof.

The stratification $\Theta_{\mathcal{P}}$ has a finite number of strata.
Therefore there exists one stratum $\Theta^{\kappa}_{\mathcal{P}}$ whose
pullback by $H\phi$ has non-empty interior $\Omega^{\prime}\subset\Omega$. By
Theorem 6.0.5 and Theorem 6.0.4
$H\phi(\Omega^{\prime})\subset\mathrm{Ad}_{B}({\Theta^{\kappa}_{\mathcal{D}}})$,
$B\in\mathrm{SO}(n)$. In particular $H\phi(\Omega)$ must be contained in the
real analytic submanifold $\mathrm{Ad}_{B}(\mathcal{D})$444The real analytic
closure of $\mathrm{Ad}_{B}(\Theta^{\kappa}_{\mathcal{D}})$ is the exponential
of $\mathrm{Ad}_{B}(\mathfrak{d}^{\kappa})$. This means that $H\phi(\Omega)$
can only intersect strata of dimension equal or less to that of
$\Theta_{\mathcal{P}}^{\kappa}$; the equidimensional strata are those which
under permutations go to open subsets of $\mathfrak{d}^{\kappa}$).. By
Proposition 5.0.4 the restriction

$\pi|_{\mathcal{C}}:B\mathcal{D}\to\mathrm{Ad}_{B}({\mathcal{D}})$

is a diffeomorphism from a horizontal submanifold. Therefore all lifts of
curves in $\Omega$ at $C\in\Theta^{\kappa}_{\mathcal{C}}$ are contained in
$B\mathcal{D}\subset\mathcal{C}$, which is property $\mathcal{CK}$.

The image of $\Omega$ by the orthogonal projection onto a characteristic line
is connected, and hence an interval. The restriction of $\phi$ to the
foliation of $\Omega$ by affine lines parallel to the characteristic line is
locally projectable. Because the interval has trivial topology local
projections must agree on overlaps. ∎

##  7\. Bifurcation of orthogonal characteristics along quadratic forms

We shall construct a family of Hessian metrics with property $\mathcal{I}$
which do not have orthogonal characteristics, and, hence, by Theorem 6.0.4 do
not have property $\mathcal{CK}$. The family, despite being defined on domains
which are not necessarily convex, will be also invariant under Legendre
transform.

Firstly, we shall describe the domains $\Omega$ we are interested in. Let
$\Omega^{0}$ be an (open convex) _polytope_. By this we mean a domain defined
as the points where a finite number of affine maps are strictly positive; the
zero set of each such map is a _supporting hyperplane_. The closure of the
polytope need not be compact. Let
$\mathbb{R}^{1}_{l}\times\mathbb{R}^{n-1}_{l}$ be the result of applying an
orthogonal transformation $B_{l}$, $1\leq l\leq m$, to the splitting
$\mathbb{R}^{n}=\mathbb{R}\times\mathbb{R}^{n-1}$. We consider the polytope
$\Omega^{1}_{l}=I_{l}\times F_{l}$, where the factors are polytopes in
$\mathbb{R}_{l}^{1},\mathbb{R}^{n-1}_{l}$, respectively; we shall refer to
$\mathbb{R}^{1}_{l}$ as the _primary characteristic_ of $\Omega^{1}_{l}$. We
shall assume that $\mathbb{R}_{l}^{1}$ is oriented and we shall denote by
$p_{l}$ the infimum of the interval $I_{l}$ (the interval may not be bounded
from above). we shall refer to $H_{l}:=p_{l}\times\mathbb{R}^{n-1}_{l}$ as the
_primary supporting hyperplane_.

We shall assume that

  * •

the polytopes $\Omega^{0},\Omega^{1}_{1},\dots,\Omega^{1}_{m}$ are disjoint
and that $\Omega^{1}_{i}$ and $\Omega^{1}_{j}$, $i\neq j$, have disjoint
closure;

  * •

the primary supporting hyperplane $H_{l}$ for $\Omega^{1}_{l}$ is also a
supporting hyperplane for $\Omega^{0}$ and $\partial\Omega^{1}_{l}\cap
H_{l}\subset\partial\Omega^{0}\cap H_{l}$.

We define $\Omega$ to be

$\Omega=\Omega^{0}\bigcup(\Omega^{1}_{1}\cup p_{1}\times
F_{1})\bigcup\cdots\bigcup(\Omega^{1}_{m}\cup p_{m}\times F_{m}).$ (15)

We refer to $\Omega$ as in (15) as a polytope with 1-handles.

Secondly, we shall introduce appropriate strictly convex functions on the
polytope with 1-handles. Let $\phi_{0}$ be a multiple of the standard
quadratic form on $\mathbb{R}^{n}$:

$\phi_{0}(x)=k(x_{1}^{2}+\cdots+x_{n}^{2}).$

Let $y=(y_{1},\dots,y_{n})$ be the coordinates which correspond to the image
by $B_{l}$ of the canonical basis $e_{1},\dots,e_{n}$ and let
$q_{l}(y)=q_{l}(y_{2},\dots,y_{l})=k(y_{1}^{2}+\cdots+y_{n}^{2})$. Then we
have:

$\phi_{0}(y)=k(y_{1}^{2}+\cdots+y_{n}^{2})=ky_{1}^{2}+q_{l}(y_{2},\dots,y_{l}).$
(16)

######  Proposition 7.0.1.

Let $\phi$ be the function defined on the polytope with 1-handles $\Omega$ as
follows:

$\phi|_{\Omega^{0}\cup p_{1}\times F_{1}\cup\cdots\cup p_{m}\times
F_{m}}=\phi_{0},\quad\phi|_{\Omega^{1}_{l}}=\phi_{l}+q_{l},$

where $\phi_{l}(y)=\phi_{l}(y_{1})$ a strictly convex smooth function on
$I_{l}$ tangent at $p_{l}$ to $ky_{1}^{2}$ at infinite order. Then it has the
following properties:

  1. (1)

It is a smooth and strictly convex function on $\Omega$.

  2. (2)

It has property $\mathcal{I}$.

  3. (3)

The image $H\phi(\Omega)\subset\mathcal{P}$ is contained in the union of the
closed stratum and the two open strata of lowest dimension of
$\Theta_{\mathcal{P}}$.

  4. (4)

If there are two 1-handles on which $\phi_{l}$ is not a quadratic form and the
primary characteristic are neither equal not perpendicular, then $H\phi$ does
not have property $\mathcal{CK}$.

###### Proof.

That $\phi$ is smooth and strictly convex is a consequence of (16) and of the
definition of $\phi_{l}$.

The restriction of $\phi$ to $\Omega^{0}$ has property $\mathcal{I}$; the
restriction to each $\Omega^{1}_{l}$ has orthogonal characteristics given by
the columns of $B_{l}$. Therefore $\phi$ has property $\mathcal{I}$ on the
closure of the union, which is $\Omega$.

By construction $H\phi$ sends $\Omega^{0}$ to the closed stratum of
$\Theta_{\mathcal{P}}$, and thus so $\overline{\Omega^{0}}$; if $\phi_{l}$ is
not a quadratic form, then $H\phi$ sends an open subset of $\Omega^{1}_{l}$ to
the strata positive matrices with two eigenvalues so that one is simple:

$H\phi(\overline{\Omega^{0}})\subset\bigcap_{B\in\mathrm{SO}(n)}\mathrm{Ad}_{B}(\mathcal{D}),\quad
H\phi(\Omega^{1}_{l})\cap\mathrm{Ad}_{B_{l}}\left(\bigcap_{B^{\prime}\in
S(\mathrm{O}(1)\times\mathrm{O}(n-1))}\mathrm{Ad}_{B^{\prime}}(\mathcal{D})\right)\neq\emptyset.$

By Theorem 6.0.4 if $H\phi$ has property $\mathcal{CK}$ then $\phi$ has
orthogonal characteristics for some $C\in\mathcal{C}$ (or
$B\in\mathrm{SO}(n)$). On a 1-handle $\Omega_{l}^{1}$ with $\phi_{l}$
different from a quadratic form $\phi$ has orthogonal characteristics exactly
for all $B_{l}\mathrm{SO}(n)_{\kappa}\mathcal{D}$. The 1-handles
$\Omega_{i}^{1}$ and $\Omega_{j}^{1}$ have primary characteristics which are
neither equal nor orthogonal if and only if $B_{i}\mathrm{SO}(n)_{\kappa}\cap
B_{j}\mathrm{SO}(n)_{\kappa}=\emptyset$. ∎

######  Proposition 7.0.2.

Let $\Omega$ be a polytope with 1-handles and let $\phi\in C^{\infty}(\Omega)$
as in Proposition 7.0.1. Then the following holds:

  1. (1)

The Legendre map $d\phi$ on $\Omega$ is a diffeomorphism, its image
$\Omega^{*}$ is a polytope with 1-handles, and ${\Omega^{1}_{l}}^{*}$ and
$\Omega^{1}_{l}$ have the same primary characteristic.

  2. (2)

$\phi^{*}$ is a function as in Proposition 7.0.1.

###### Proof.

Because $\phi$ is strictly convex $d\phi:\Omega\to\mathbb{R}^{n}$ is a local
diffeomorphism. Because $\phi|_{\Omega^{0}}$ is a quadratic form
$d\phi(\Omega^{0})$ is another polytope. The restriction
$\phi|_{\Omega^{1}_{l}=I_{l}\times F_{l}}$ decomposes as a sum of strictly
convex functions $\phi_{l}+q_{l}$. The subset $d\phi(\Omega^{1}_{l})$ is
another 1-handle because $I_{l}$ is 1-dimensional, $F_{l}$ is a polytope and
$q_{l}$ is a quadratic form; furthermore, because the Legendre transform
commutes with orthogonal transformations
$d\phi(\Omega^{1}_{l})=I_{l}^{*}\times F_{l}^{*}$ where the product
decomposition is also with respect to
$\mathbb{R}_{l}^{1}\times\mathbb{R}^{n-1}_{l}$. The condition on the non-
overlap of the closures of the 1-handles can be restated as follows: if two
1-handles have common primary supporting hyperplane, then their polytopes
there have non-intersecting closure. This implies that if we prolongue each
$I_{l}\subset\mathbb{R}^{{}_{1}}_{l}$ across $p_{l}$ to a larger interval
$\tilde{I}_{l}$ so that $\tilde{I}_{l}\times F_{l}\subset\Omega^{0}\cup
p_{l}\times F_{l}\cup\Omega^{1}_{l}$, then

$d\phi(\Omega^{0})\cap d\phi(p_{l}\times
F_{l}\cup\Omega^{1}_{l})=\emptyset,\quad d\phi(d\phi(\tilde{I}_{i}\times
F_{i})\cap d\phi(\tilde{I}_{j}\times F_{j})\cap(\mathbb{R}^{n}\backslash
d\phi(\Omega^{0}))=\emptyset,\quad i\neq j.$

Therefore $d\phi:\Omega\to d\Omega$ is a bijection and thus a diffeomorphism
onto another polytope with 1-handles.

Because the Legendre transform of a multiple of the standard quadratic form is
a multiple of the standard quadratic form it follows that $\phi^{*}$ belongs
to the class of functions described in Proposition 7.0.1. ∎

##  8\. An application to Poisson geometry

We shall show that property $\mathcal{I}$ for strictly convex functions
condition is equivalent to the Poisson commuting equation for Poisson
structures related to Kahler forms on toric varieties. We shall use

  1. (a)

our classification of real analytic inversible Hessian metrics to deduce a
factorization result;

  2. (b)

the family of strictly convex functions with property $\mathcal{I}$ introduced
in Section 7 to produce pencils of Poisson structures on regions of projective
varieties which interpolate from a Kahler structure to a Poisson structure
with a finite number of Kahler leaves.

######  Definition 8.0.1.

[2, Section 4] A Poisson structure $\Pi$ on a toric variety $(X,\mathbb{T})$
is

  1. (1)

toric if the bivector field $\Pi$ is $\mathbb{T}$-invariant, of type $(1,1)$
and positive, and if the symplectic leaves of $\Pi$ equal the finitely many
orbits of the torus action;

  2. (2)

totally real if the orbits of the (maximal) compact torus $T$ are coisotropic
submanifolds.

######  Remark 8.0.2.

Totally real toric Poisson structures are good candidates to be limits of
Hamiltonian Kahler forms: for such a form its inverse Poisson bivector is
$T$-invariant, of type $(1,1)$ and positive; there is a unique symplectic leaf
of which the $T$-orbits are Lagrangian submanifolds. Thus it is natural to
look for converging sequences of such bivectors so that in the limit the
unique symplectic leaf breaks into the finitely many orbits and the
$T$-symmetry is enlarged to $\mathbb{T}$-symmetry. One possible source would
be a totally real toric Poisson structure which Poisson commutes with (the
inverse of) a Hamiltonian Kahler form. In such case the convex combination of
the bivectors would be a smooth family of (inverses of) Hamiltonian Kahler
forms converging to the totally real toric Poisson structure.

On a toric variety a $\mathbb{T}$-invariant Poisson structure has a simple
infinitesimal description: its restriction to the open dense orbit — which
upon fixing a point is identified with $\mathbb{T}$ — followed by the
logarithm map, defines a constant Poisson structure in the Lie algebra of
$\mathbb{T}$. The infinitesimal counter part of a toric Poisson structure is a
Hermitian inner product. If it is totally real it corresponds to an inner
product on $i\mathfrak{t}$, where $\mathfrak{t}$ denotes the Lie algebra of
$T$. In such case, we say that $e_{1},\dots,e_{n}\in i\mathfrak{t}$ is an
_adapted Darboux basis_ if the inner product becomes standard; equivalently,
$e_{1},\dots,e_{n},ie_{1},\dots,ie_{n}$ is a Darboux basis for the inverse
constant symplectic structure.

On a toric variety a Hamiltonian Kahler form can be described by (Legendre
dual) strictly convex functions: a _Kahler potential_ in logarithmic
coordinates and a _symplectic potential_ in momentum map coordinates [6].

The Poisson commuting equation for a totally real Poisson structure and a
Hamiltonian Kahler form corresponds — in appropriate coordinates — to property
$\mathcal{I}$:

######  Theorem 8.0.3.

Let $(X,\mathbb{T})$ be a toric variety endowed with a totally real toric
Poisson structure $\Pi$ and a Kahler structure $\sigma$ for which the action
of $T$ is Hamiltonian. Let $P$ be the inverse Poisson structure of $\sigma$.
Then the following statements are equivalent:

  1. (1)

$\Pi$ and $P$ Poisson commute: $[\Pi,P]=0$.

  2. (2)

In an adapted Darboux basis the Kahler potential $\phi$ has property
$\mathcal{I}$.

  3. (3)

In an adapted Darboux basis the symplectic potential $\phi^{*}$ has property
$\mathcal{I}$.

###### Proof.

We regard the equation $[\Pi,P]=0$ as the defining equation for degree
2-cocycles in the Poisson cohomology of $\Pi$: $d_{\Pi}P=0$. In logarithmic
coordinates $\exp^{*}\Pi$ has an inverse which is a (constant) symplectic
structure $\Xi$ on $\mathfrak{t}\oplus i\mathfrak{t}$. Therefore

$\Xi^{\\#}:(\mathfrak{X}^{\bullet},d_{\exp^{*}\Pi})\to(\Omega^{\bullet},d)$

is an isomorphism of chain complexes (see e.g. [5, Proposition 6.12]). Hence

$[\exp^{*}\Pi,\exp^{*}P]=0\Longleftrightarrow
d\omega=0,\quad\omega(X,Y):=\exp^{*}P(\Xi^{\\#}X,\Xi^{\\#}Y).$

Let $e_{1},\dots,e_{n},ie_{1},\dots,ie_{n}$ be an adapted Darboux basis for
the totally real toric Poisson structure. In the fixed coordinates and
associated frames of the complexified tangent and cotangent bundles the
matrices of $\Pi$ and $\Xi$ are:

$\Pi^{\\#}=\frac{2}{i}\begin{pmatrix}0&\mathrm{I}_{n}\\\
-\mathrm{I}_{n}&0\end{pmatrix},\quad\Xi^{\\#}=\frac{i}{2}\begin{pmatrix}0&-\mathrm{I}_{n}\\\
\mathrm{I}_{n}&0\end{pmatrix}$

If we let

$\frac{2}{i}\begin{pmatrix}0&g\\\ -g&0\end{pmatrix}$

denote the matrix of $\exp^{*}P^{\\#}$, then

$\omega^{\\#}=-\Xi^{\\#}\exp^{*}P^{\\#}\Xi^{\\#}=-\frac{i}{2}\begin{pmatrix}0&-\mathrm{I}_{n}\\\
\mathrm{I}_{n}&0\end{pmatrix}\frac{2}{i}\begin{pmatrix}0&g\\\
-g&0\end{pmatrix}\frac{i}{2}\begin{pmatrix}0&-\mathrm{I}_{n}\\\
\mathrm{I}_{n}&0\end{pmatrix}=\frac{i}{2}\begin{pmatrix}0&g\\\
-g&0\end{pmatrix}$

Hence

$\omega=\omega(x)=\frac{-i}{2}\sum_{j,k}g_{jk}(x)dz_{j}\wedge d\bar{z}_{k},$

Its exterior derivative is:

$\begin{split}d\omega&=-\frac{i}{2}\sum_{i,j=1}^{n}dg_{ij}\wedge dz_{i}\wedge
d\bar{z}_{j}=-\frac{i}{2}\sum_{i,j=1}^{n}\sum_{k=1}^{n}\left(\frac{\partial
g_{ij}}{\partial z_{k}}dz_{k}+\frac{\partial
g_{ij}}{\partial\bar{z}_{k}}d\bar{z}_{k}\right)\wedge dz_{i}\wedge
d\bar{z}_{j}=\\\ &=-\frac{i}{2}\left(\sum_{i,j,k=1}^{n}\left(\frac{\partial
g_{jk}}{\partial z_{i}}-\frac{\partial g_{ik}}{\partial z_{j}}\right)\wedge
dz_{i}\wedge dz_{j}\wedge d\bar{z}_{k}+\sum_{i,j,k=1}^{n}\left(\frac{\partial
g_{ij}}{\partial\bar{z}_{k}}-\frac{\partial
g_{ik}}{\partial\bar{z}_{j}}\right)\wedge dz_{i}\wedge d\bar{z}_{j}\wedge
d\bar{z}_{k}\right)\\\ \end{split}$

Because the entries of $g$ only depend on $x$ we have:

$\frac{\partial g_{jk}}{\partial z_{i}}-\frac{\partial g_{ik}}{\partial
z_{j}}=\frac{1}{2}\left(\frac{\partial g_{jk}}{\partial x_{i}}-\frac{\partial
g_{ik}}{\partial x_{j}}\right),\quad\frac{\partial
g_{ij}}{\partial\bar{z}_{k}}-\frac{\partial
g_{ik}}{\partial\bar{z}_{j}}=\frac{1}{2}\left(\frac{\partial g_{ij}}{\partial
x_{k}}-\frac{\partial g_{ik}}{\partial x_{j}}\right)$

The matrix of $\exp^{*}\sigma$ is

$\frac{i}{2}\begin{pmatrix}0&g^{-1}\\\ -g^{-1}&0\end{pmatrix},$

and $g^{-1}$ is the Hessian of the Kahler potential $\phi$. In particular $g$
and its inverse are symmetric matrices. Renaming the set of indices in the
first summand and using the symmetry of $g$ we obtain:

$d\omega=0\Longleftrightarrow\frac{\partial g_{ij}}{\partial
x_{k}}-\frac{\partial g_{ik}}{\partial x_{j}},\quad 1\leq i,j,k\leq n.$

This is exactly property $\mathcal{I}$ for the Hessian metric $g^{-1}=H\phi$.

The symplectic potential of $\sigma$ is the Legendre transform of $\phi$. By
Proposition 4.0.4 $\phi$ (in $\mathbb{R}^{n}$) has property $\mathcal{I}$ if
and only if $\phi^{*}$ (in $d\phi(\mathbb{R}^{n})$) has property
$\mathcal{I}$. ∎

######  Theorem 8.0.4.

Let $(X,\mathbb{T})$ be a projective toric Poisson structure endowed with a
toric Poisson structure $\Pi$ which Poisson commutes with a real analytic
Kahler structure $\sigma$ for which the action of $T$ is Hamiltonian. Then
$(X,\mathbb{T})$ is a Cartesian product of projective lines and both $\Pi$ and
$\sigma$ factorize.

###### Proof.

Because $\sigma$ is real analytic the Kahler potential $\phi$ is real
analytic; the Legendre transform preserves analytic (strictly convex)
functions. Therefore by Theorem 8.0.3 the symplectic potential $\phi^{*}$ has
property $\mathcal{I}$. By Theorem 6.0.6 $\phi^{*}$ is defined in a Cartesian
product of intervals $I_{1}\times\cdots\times I_{n}$ (we may dispense with the
rotation by changing accordingly the adapted Darboux basis). One must have the
equality $d\phi(\mathbb{R}^{n})=I_{1}\times\cdots\times I_{n}$ because
otherwise by repeating the Legendre transform we would get a domain for the
original Kahler potential strictly containing $\mathbb{R}^{n}$. Thus the
interior of the moment polytope $\Delta$ is a Cartesian product of intervals.
A product of intervals has a property invariant under affine transformations:
it is limited by pairs of parallel hyperplanes. Because $\Delta$ is a Delzant
polytope there is an affine transformation that takes the integral lattice of
$\mathfrak{t}^{*}$ to $\mathbb{Z}^{n}$, a vertex of $\Delta$ to the origin and
the facets containing this vertex to the coordinate hyperplanes. Because
$\Delta$ must be still described by parallel hyperplanes, is it actually a
Cartesian product of intervals in this integral affine coordinates of
$\mathfrak{t}^{*}$. The fan of the Delzant polytope determines the toric
variety $(X,\mathbb{T})$. The fan of a cube in
$(\mathbb{R}^{n},\mathbb{Z}^{n})$ corresponds to
$\mathbb{C}P^{1}\times\cdots\times\mathbb{C}P^{1}$.

To show that the Kahler form $\sigma$ also splits as a sum of Kahler forms on
each projective line we use toric charts for $(X,\mathbb{T})$. For that we
observe that the linear part of the above affine transformation must be a
permutation followed by a (signed) re-scaling of each Euclidean direction.
Therefore if we dispense the affine transformation we deduce that the in the
fixed compatible Darboux basis the subset $ie_{1},\dots,ie_{n}$ is — up to re-
scaling of its members — an integral basis of $\mathfrak{t}$. Let us re-scale
to a basis
$\epsilon_{1},\dots,\epsilon_{n},i\epsilon_{1},\cdots,i\epsilon_{n}$ so that
the second block is an integral basis of $i\mathfrak{t}$. In the corresponding
coordinates the Kahler potential of $\sigma$ is still a sum of strictly convex
functions on each coordinate and therefore the Legendre diffeomorphism still
sends $i\mathfrak{t}\cong\mathbb{R}^{n}$ to a cube. Let $v$ be the vertex of
its closure whose coordinates are smaller than those of the others. The basis
of inner pointing integral vectors normal to the facets containing $v$ is
exactly $i\epsilon_{1},\cdots,i\epsilon_{n}$. Therefore for the standard toric
chart associated to $v$ (see e.g. [1, Chapter2, Section 5])

$(\mathbb{C}^{*})^{n}\circlearrowright\mathbb{C}^{n}\subset\mathbb{C}P^{1}\times\cdots\times\mathbb{C}P^{1},\quad(0,\dots
0)=([0:1],\dots,[0:1]),$

the identification555Though we do not need it here, the toric chart could be
chosen compatible with the monoid structure, so that $(1,\dots,1)$ corresponds
to the fixed point in the open orbit. of $\mathbb{T}$ with
$(\mathbb{C}^{*})^{n}$ comes from the Lie algebra identification which sends
$i\epsilon_{1},\dots,i\epsilon_{n}$ to the canonical basis of
$\mathbb{R}^{n}\subset\mathbb{C}^{n}$. In other words, upon having identified
$\mathbb{T}$ with the open orbit of $X$, the product structure induced by
$X\cong\mathbb{C}P^{1}\times\cdots\times\mathbb{C}P^{1}$ on $\mathbb{T}$ is
exactly the factorisation of the torus coming from the (complex) basis
$e_{1},\cdots,e_{n}$ of its Lie algebra. The factorisation of $X$ decomposes
$\sigma=\sigma_{1}+\cdots+\sigma_{n}$, $\sigma_{k}\in\Omega^{2}(X)$. Each
$\sigma_{k}$ is projectable by

$X\to\mathbb{C}P^{1}\times\cdots\widehat{\mathbb{C}P^{1}}\times\cdots\times\mathbb{C}P^{1}$

because it has that property in the open dense subset $\mathbb{T}$. The
compatibility of $\Pi$ with the product structure is also immediate. Therefore

$(X,\mathbb{T},\Pi,\sigma)\cong(\mathbb{C}P^{1},\mathbb{C}^{*},\Pi_{1},\sigma_{1})\times\cdots\times(\mathbb{C}P^{1},\mathbb{C}^{*},\Pi_{n},\sigma_{n}),$

where symplectic forms are invariant under the action of $T$. ∎

Let $\Omega$ be a polytope with 1-handles. Its _outer boundary_ will be the
subset of the boundary lying in supporting hyperplanes for the 1-handles which
are parallel to the primary ones but not equal to them; its _inner boundary_
will be the subset of the boundary in supporting hyperplanes of the polytope
which are not primary supporting hyperplanes of some 1-handle.

######  Theorem 8.0.5.

Let $(X,\mathbb{T})$ be a toric variety with fan given by the polytope
$\Delta\subset(\mathbb{R}^{n},\mathbb{Z}^{n})$. Let $\Omega$ be a polytope
with 1-handles such that the intersection $\partial\Omega\cap\partial\Delta$
is contained in the union of the outer boundary of $\Omega$ and an orthogonal
subset of supporting hyperplanes of the inner boundary of $\Omega$.

Then there exist an open subset $X_{\Omega}\subset X$ invariant under the
action of $T$ and a Kahler form $\sigma\in\Omega^{2}(X_{\Omega})$ with the
following properties:

  1. (1)

The action of $T$ on $(X_{\Omega},\sigma)$ is Hamiltonian with momentum map
the union of $\Omega$ with the interior on each face of $\Delta$ of the outer
and inner boundaries of $\Omega$.

  2. (2)

The Poisson structure with corresponds to $\sigma$ Poisson commutes with any
toric Poisson structure for which the canonical basis of $\mathbb{R}^{n}$ is
an adapted Darboux basis up to scaling.

###### Proof.

We start with $\phi\in C^{\infty}(\Omega)$ a function as in Proposition 7.0.1
on which we shall impose natural boundary conditions 666The functions will
satisfy well-known boundary conditions to produce Kahler metrics (see e.g. [1,
Chapter 2]). These metrics/complex structures are constructed fixing the
symplectic structure. Because we are interested in keeping fixed the complex
structure, we are going to be very explicit with the computation of the Kahler
potentials and corresponding symplectic forms. Let $\Omega^{1}_{l}$ be a
1-handle whose supporting hyperplane in the outer boundary of $\Omega$
intersects $\partial\Delta$. Let $\alpha_{l}$ be the unique integral affine
map which vanishes in the supporting hyperplane and it is positive on
$\Delta$. We shall assume that $\phi_{l}$ equals
$\frac{1}{2}\alpha_{l}(\log(\alpha_{l})-1)$ near the end of $I_{l}$ opposite
to $p_{l}$. This is always possible because the existing constraint on
$\phi_{l}$ is near $p_{l}$.

The region $X_{\Omega}\subset X$ is the result of adding certain points to
$\exp(\Omega^{*}\oplus i\mathbb{R}^{n})\subset X$. By item (1) in Proposition
7.0.2 the primary characteristics of ${\Omega^{1}_{l}}^{*}$ and
$\Omega^{1}_{l}$ are the same and the orientation is also preserved. Because
$\Delta$ determines a fan the primary characteristic
$\mathbb{R}^{1}_{l}\subset\mathbb{R}^{n}\cong i\mathfrak{t}$ determines a
1-parameter subgroup $\mathbb{C}_{l}^{*}\subset\mathbb{T}$ together with an
isomorphism $\mathbb{C}^{*}_{l}\cong\mathbb{C}^{*}$ (the Lie algebra is
trivialized by a positive integral vector in $\mathbb{R}^{1}_{l}$). Because
the derivative of $\phi_{l}$ near the boundary point opposite to $p_{l}$ goes
to infinity $I^{*}_{l}\subset\mathbb{R}^{1}_{l}$ is a semi-infinite interval
in the positive half line; in particular it is a semigroup; let
$\mathbb{D}^{o}_{l}\subset\mathbb{C}_{l}^{*}$ be the semigroup
$\exp(I^{*}_{l}\oplus i\mathbb{R}^{1}_{l})$. This semigroup acts (freely) on
$\exp({\Omega^{1}_{l}}^{*}\oplus i\mathbb{R}^{n})$. If we let
$\mathbb{T}_{l}\subset\mathbb{T}$ be the subtorus which exponentiates the
complexification of $\mathbb{R}^{n-1}_{l}$, then we can factor

$\exp({\Omega^{1}_{l}}^{*}\oplus
i\mathbb{R}^{n})=\mathbb{D}^{o}_{l}\times\exp(F_{l}\oplus
i\mathbb{R}^{n-1}_{l})\subset\mathbb{T}=\mathbb{C}^{*}_{l}\times\mathbb{T}_{l}.$
(17)

We define $X_{\Omega}\subset X$ to be the union of $\exp(\Omega^{*}\oplus
i\mathbb{R}^{n})$ with the closure of every $\mathbb{D}^{o}_{l}$-orbit in
$\exp({\Omega^{1}_{l}}^{*}\oplus i\mathbb{R}^{n})$, for every 1-handle whose
supporting hyperplane in the outer boundary of $\Omega$ intersects
$\partial\Delta$ (for the moment we assume that no inner boundary components
are in $\partial\Delta$).

The function $\phi^{*}$ defines a Kahler form
$\sigma=i\partial\bar{\partial}\log(\phi^{*})$ on $\exp(\Omega^{*}\oplus
i\mathbb{R}^{n})$ for which the action of $T$ is Hamiltonian. We want to argue
that $\sigma$ extends to a Kahler form on $X_{\Omega}$.

Firstly, we show how upon adding the orbit closures to
$\exp({\Omega^{1}_{l}}^{*}\oplus i\mathbb{R}^{n})$ (and not in
$\exp(\Omega^{*}\oplus i\mathbb{R}^{n})$) we get an open subset $X_{l}\subset
X$ which is $T$-invariant and to which the product structure in (17) extends.
For that we use the toric atlas (as a monoid) determined by the polytope
$\Delta$ ([1, Chapter2, Section 5]): To each vertex $v\in\Delta$ there
correspond a toric chart which identifies the union of orbits of $X$ which
correspond to the star of $v$ with the standard affine toric variety:
$(\mathbb{C}^{n},{(\mathbb{C}^{*})}^{n})$. The standard integral basis of the
Lie algebra of ${(\mathbb{S}^{1})}^{n}$ comes from the integral linear forms
$\nu_{l_{1}},\dots,\nu_{l_{n}}$ associated to the affine forms
$\alpha_{l_{j}}$; in particular this describes how for each toric chart
$\mathfrak{t}\oplus i\mathfrak{t}$ — for which we already have picked a basis
— is identified with the Lie algebra of the standard complex torus
$\mathbb{R}^{n}\oplus i\mathbb{R}^{n}$. The point in the open orbit of $X$
which determines the monoid structure goes to the unit
$(1,\dots,1)\in\mathbb{C}^{n}$.

Let us fix a toric chart of a vertex $v$ which belongs to the supporting
hyperplane of $\Omega^{1}_{l}$ and $\Delta$. Under the identification of
$\mathbb{T}$ with the standard torus ${(\mathbb{C}^{*})}^{n}$ we can assume
that the (trivialized) subgroup $\mathbb{C}_{l}^{*}\cong\mathbb{C}^{*}$ maps
to the first factor of the standard torus so that on trivializations the
isomorphism is given by the inversion. Thus $\mathbb{D}^{o}_{l}$ maps to a
semigroup $\mathbb{D}^{i}_{l}\subset\mathbb{C}^{*}$ contained in the unit
disk. Let $W_{l}\subset\mathbb{C}^{n}$ be the image in the toric chart of the
second factor $\exp(F_{l}\oplus i\mathbb{R}^{n-1}_{l})$. Then the image of
$\exp({\Omega^{1}_{l}}^{*}\oplus i\mathbb{R}^{n})$ is

$(zw_{1},\dots,w_{n}),\quad z\in\mathbb{D}^{o}_{l},\quad
w=(w_{1},\dots,w_{n})\in W_{l}.$

Therefore the image of $X_{l}$ is also completely contained in the toric chart
and equals

$(zw_{1},\dots,w_{n}),\quad
z\in\mathbb{D}^{o}_{l}\cup\\{0\\},\quad(w_{1},\dots,w_{n})\in W_{l}.$

Because $W_{l}$ is a codimension 2 submanifold which intersects each complex
line parallel to the $z_{1}$-axis transversely in at most one point, we deduce
that $X_{l}$ is an open subset which extends the product structure. By
construction it is also $T$-invariant.

To show that $\sigma$ extends to a Kahler form on $X_{l}$ we shall work with
its inverse Poisson structure $P$. The decomposition
$\phi^{*}|_{{\Omega^{1}_{l}}^{*}}(y)=\phi^{*}_{l}(y_{1})+q^{*}_{l}(y_{2},\dots,y_{1})$
implies that on the exponentiation of the 1-handle $P$ decomposes as
$P_{l}+P^{\prime}_{l}$, where each summand is a field of bivectors tangent to
one of the foliations in the product decomposition (17). The second field of
bivectors is easier to describe: In the Lie algebra the foliation is given by
translates of $F_{l}\subset\mathbb{R}^{n-1}_{l}$. On each such leaf the Kahler
potential for the corresponding Kahler form is the quadratic form $q_{l}$.
Therefore $P_{l}^{\prime}$ corresponds to a constant bivector on
${\Omega^{1}}^{*}\times i\mathbb{R}^{n}$. The exponentiation of a constant
bivector to the (abelian) Lie group has an alternative description: it is the
field of bivectors obtained by replacing each vector in the decomposition in
$\wedge^{2}(\mathfrak{t}\oplus i\mathfrak{t})$ by its corresponding
infinitesimal vector field for the action by (left) multiplication. Because
the action of $\mathbb{T}$ on itself extends to an action on $X$ it follows
that $P_{l}^{\prime}$ is the restriction of a (Poisson) structure on $X$. To
describe $P_{l}$ on each semigroup orbit we may assume that $\phi_{l}(y_{1})$
equals $-\frac{1}{2}\alpha_{l}(\log(\alpha_{l}-1)$ everywhere in $I_{l}$. The
Legendre transform of $-\frac{1}{2}(-r)(\log(-r)-1)$ for $r<0$ is
$\frac{1}{2}\exp^{-2r}$. Using that the Legendre transform commutes with
orthogonal transformations and its behavior under translation and scaling we
obtain $\phi_{l}^{*}(y_{1})=\frac{1}{2}\exp^{-2y_{1}/|\nu_{l}|}-y_{1}d_{l}$,
where $d_{l}$ is the distance of the boundary point of $I_{l}$ different from
$p_{l}$. Since we are interested in Kahler forms/bivectors we may dispense
with the linear summand. Under the semigroup identification
$\log:\mathbb{D}_{l}^{o}\to I_{l}^{*}\oplus i\mathbb{R}$ the potential pull
backs to $\frac{1}{2z\bar{z}}$. Under the identification
$\mathbb{D}_{l}^{o}\to\mathbb{D}_{l}^{i}$ it maps to $\frac{1}{2}z\bar{z}$.
Under the action on the standard toric chart it maps to
$\frac{1}{2a_{1}^{2}}z_{1}\bar{z}_{1}$. Hence the bivector $P_{1}$ there
equals $\frac{2a_{1}^{2}}{i}\frac{\partial}{\partial
z_{1}}\wedge\frac{\partial}{\partial\bar{z}_{1}}$ near $z_{1}=0$, which
extends to $X_{l}$. Both $P_{l}$ and $P_{l}^{\prime}$ are non-degenerate in
the added points, and thus $\sigma$ extends to a Kahler form.

For boundary components in the inner boundary we change coordinates by a
rotation so that all supporting hyperplane involved are coordinate
hyperplanes. Then we impose the same boundary conditions as above on the
corresponding summands of the multiple of the standard quadratic form in these
coordinates. We may have supporting hyperplanes with non-empty intersection,
say $k$ of them. This means that we shall have work with the corresponding
coordinates and hence with a splitting into a vector subspace of dimension $k$
and its orthogonal complement. We shall work on a toric chart associated to a
vertex in the intersection of the supporting hyperplanes. There, the foliation
corresponding to the vector subspace will have leaves given by the action of
${(\mathbb{C}^{*})}^{k}$ on the first $k$ coordinates on an appropriate slice.
Hence by adding closure of (semigroup) orbits we shall obtain an open subset.
The computation of the Kahler potential for the inverse symplectic form of
$P_{l}$ on such leaves is analogous.

To computation of the image of the momentum map is straightforward.

∎

The proof of Proposition 1.0.9 in the Introduction is a minor variation of the
following:

######  Example 8.0.6.

(Attaching a toric 1-handle to the standard commuting pair) Let $\Delta$ be
the standard n-simplex in $\mathbb{R}^{n}$. Let $\Omega^{0}$ be the truncation
of the (open) cube of side $(0,\tfrac{1}{n})$ by the hyperplane
$x_{1}+\cdots+x_{n}=\frac{2}{3}$. Let $\Omega^{1}$ be the 1-handle with the
primary characteristic spanned by $(1,\cdots,1)$, and so that its primary
supporting hyperplane is $x_{1}+\cdots+x_{n}=\frac{2}{3}$, the parallel one is
$x_{1}+\cdots+x_{n}=1$ and the $n-1$-dimensional polytope is the (translation
of) the intersection of the cube and the primary supporting hyperplane.

We let $\Omega$ be the polytope with 1-handles determined by $\Omega^{0}$ and
$\Omega^{1}$ above. It is in the hypotheses of Theorem 8.0.5. By going through
the its proof we check that:

  * •

In the toric chart associated to the origin $\sigma$ (near the origin) will be
the standard (constant) Kahler form $\frac{i}{2}\sum_{j}dz_{j}\wedge
d\bar{z}_{j}$; the open cube is a (punctured) polydisk which is appropriately
truncated.

  * •

Near the truncation hypersurface $W$ the Kahler form is
$\frac{i}{2}\sum_{j}\frac{1}{z_{j}\bar{z}_{j}}dz_{j}\wedge d\bar{z}_{j}$.

  * •

Attaching the 1-handle amounts to the following: the truncating hypersurface
$W$ is stable under the diagonal action of $\mathbb{S}^{1}$. Then each such
orbit is being ‘capped’ by a (holomorphic) disk which is Kahler for $\sigma$;
the disk is nothing but (a part of) the projective line determined by the
orbit, its center being in the hyperplane at infinity
$\mathbb{C}P^{n}=\mathbb{C}^{n}\cup\mathbb{C}P^{n-1}$; this is done for the
whole $F_{1}\times{(\mathbb{S}^{1})}^{n-1}$-family.

  * •

The inverse of Kahler form $\sigma$ is a Poisson bivector field $P$ which
Poisson commutes with the totally real toric Poisson bivector field $\Pi$
which in the previous toric chart is
$\frac{2}{i}\sum_{j}z_{j}\bar{z}_{j}\frac{\partial}{\partial
z_{j}}\wedge\frac{\partial}{\partial\bar{z}_{j}}$.

## References

  * [1] Apostolov, V. The Kahler geometry of Kahler manifolds. http://www.cirget.uqam.ca/~apostolo/papers/toric-lecture-notes.pdf.
  * [2] Brambila, L; Frejlich, P.; Martínez Torres, D. Coregular submanifolds and Poisson submersions. Preprint arXiv:2010.09058.
  * [3] Duistermaat, J. J. On Hessian Riemannian structures. Asian J. Math. 5 (2001), no. 1, 79–91.
  * [4] Duistermaat, J. J.; Kolk, J. A. C. Lie groups. Universitext. Springer-Verlag, Berlin, 2000.
  * [5] Laurent-Gengoux, C.; Pichereau, A.; Vanhaecke, P. Poisson structures. Grundlehren der Mathematischen Wissenschaften, 347. Springer, Heidelberg, 2013.
  * [6] Guillemin, V. Kaehler structures on toric varieties. J. Differential Geom. 40 (1994), no. 2, 285–309.