# On Lyubeznik type invariants

Thomas Reichelt  Thomas Reichelt
Mathematisches Institut
Universität Heidelberg
Im Neuenheimer Feld 205
69120 Heidelberg
Germany<EMAIL_ADDRESS>,  Uli Walther  Uli Walther
Purdue University
Dept. of Mathematics
150 N. University St.
West Lafayette, IN 47907
USA<EMAIL_ADDRESS>and  Wenliang Zhang  Wenliang Zhang
Department of Mathematics, Statistics, and Computer Science, University of
Illinois at Chicago, Chicago, IL 60607<EMAIL_ADDRESS>
###### Abstract.

We discuss for an affine variety $Y$ embedded in affine space $X$ two sets of
integers attached to $Y\subseteq X$ via local and de Rham cohomology spectral
sequences. We give topological interpretations, study them in small dimension,
and investigate to what extent one can attach them to projective varieties.

TR was supported by DFG Emmy-Noether-Fellowship RE 3567/1-2. UW was supported
by the NSF and by Simons Foundation Collaboration Grant for Mathematicians
#580839. WZ was supported by the NSF grant DMS#1752081.

###### Contents

  1. 1 Introduction
  2. 2 Čech–de Rham numbers
    1. 2.1 Basic structure results
    2. 2.2 Degeneration
    3. 2.3 Affine complements
      1. 2.3.1 High cohomology groups of the affine complement
      2. 2.3.2 Integrals of Eulerian modules
    4. 2.4 On Veronese maps
  3. 3 Lyubeznik numbers
    1. 3.1 Basic properties
    2. 3.2 Lyubeznik numbers and projective schemes
      1. 3.2.1 Determinantal ideals
      2. 3.2.2 Toric varieties
      3. 3.2.3 Horospherical varieties
    3. 3.3 Lyubeznik numbers in small dimension

##  1\. Introduction

######  Notation 1.1.

Throughout we will use the following conventions: $\mathbb{K}$ will be a field
of characteristic zero,

$I\subseteq R_{n}=\mathbb{K}[x_{1},\ldots,x_{n}],\qquad
X=\operatorname{\textup{Spec}\,}(R_{n})$

an ideal in the polynomial ring in $n$ indeterminates and the associated
affine space. Our default affine variety will be

$Y:={\mathrm{Var}}(I)\subseteq X,\text{ with complement }U=X\smallsetminus Y,$

and if $I$ is homogeneous then

$\tilde{Y}:=\operatorname{\textup{Proj}\,}(R_{n}/I)\subseteq\mathbb{P}^{n-1}_{\mathbb{K}}$

will be the projective scheme to $I$, with complement
$\tilde{U}:=\mathbb{P}U=\mathbb{P}^{n-1}_{\mathbb{K}}\smallsetminus\tilde{Y}$.
The homogeneous irrelevant ideal of $R_{n}$ will be denoted
${\mathfrak{m}}={\langle x_{1},\ldots,x_{n}\rangle}$ and $d$ will stand for
$\dim(Y)$. $\diamond$

Hartshorne’s seminal work [Har75] begins with

> The idea of using differential forms and their integrals to define numerical
> invariants of algebraic varieties goes back to Picard and Lefschetz…

and then outlines the development of this branch of mathematics until the
writing of his article on algebraic de Rham cohomology. While originally the
base field was the complex numbers $\mathbb{C}$, Hartshorne works in greater
generality over fields $\mathbb{K}$ of characteristic zero. It has become
clear since, particularly through the work of Lyubeznik [Lyu93], that
Kashiwara’s framework of $D$-modules is the right set-up for these
investigations. This article is a contribution to this general theme, with the
two main characters defined as follows.

If a variety $Y^{\prime}$ can be embedded into a smooth $\mathbb{K}$-variety
$X^{\prime}$ of $\mathbb{K}$-dimension $n$, one can define the de Rham
homology and cohomology functors of $Y^{\prime}$ as

$H_{q}^{\mathrm{dR}}(Y^{\prime}):=\mathbb{H}^{2n-q}_{Y^{\prime}}(X^{\prime},\Omega^{\bullet}_{X^{\prime}}),\qquad\qquad
H^{q}_{\mathrm{dR}}(Y^{\prime}):=\mathbb{H}^{q}(X^{\prime},{\hat{\Omega}}^{\bullet}_{X^{\prime}}).$

Here, $\mathbb{H}(-)$ denotes hypercohomology functor on complexes of sheaves,
$\Omega^{\bullet}_{X^{\prime}}$ is the de Rham complex (relative to
$\mathbb{K}$) of $X^{\prime}$, and the hat denotes completion along
$Y^{\prime}$. Hartshorne proves that these quantities do not depend on
$X^{\prime}$ or on the chosen embedding of $Y^{\prime}$, and demonstrates many
interesting facts about these two functors.

We focus on de Rham homology for a moment, under the assumption that
$X^{\prime}$ is affine. Then hypercohomology collapses to global sections
since the modules in $\Omega^{\bullet}_{X^{\prime}}$ are coherent, equal to
exterior powers of the free $\mathcal{O}_{X^{\prime}}$-module
$\Omega^{1}_{X^{\prime}}$ of rank $n$ given by the Kähler differentials on
$X^{\prime}$. The set-theoretic sections-with-support functor on a coherent
sheaf agrees with algebraic local cohomology. In particular,
$\mathbb{H}^{2n-q}_{Y^{\prime}}(X^{\prime},\Omega^{i}_{X^{\prime}})$ is just
local cohomology $H^{2n-q}_{Y^{\prime}}(\Omega^{i}_{X^{\prime}})$ of the
module of $i$-forms (identifying sheaves with their global sections).

The sheaf $\omega_{X}:=\Omega^{n}_{X}$ has a natural right module structure
over the ring $\mathcal{D}_{X}$ of $\mathbb{K}$-linear differential operators
on $X$. The global sections of the sheaf of differential operators
$\mathcal{D}_{X}$ on $X$ are the elements of the Weyl algebra

$D_{n}=R_{n}{\langle\partial_{1},\ldots,\partial_{n}\rangle}$

where $\partial_{i}$ stands for the partial differentiation operator
$\frac{\partial}{\partial x_{i}}$. On the other hand, $\Omega^{i}_{X}$ is the
free $\mathcal{O}_{X}$-module of rank ${n\choose i}$ generated by the symbols
${\mathrm{d}}x_{I}={\mathrm{d}}x_{j_{1}}\wedge\cdots\wedge{\mathrm{d}}x_{j_{i}}$
with $I\subseteq 2^{[n]}$ and $|I|=i$, and the global sections of $\omega_{X}$
are the elements of the right $D_{n}$-module $D_{n}/\partial\cdot
D_{n}:=D_{n}/(\partial_{1},\ldots,\partial_{n})D_{n}$.

Let us write $\Omega^{\bullet}_{\mathcal{D},X}$ for the Koszul co-complex on
$\mathcal{D}_{X}$ generated by left multiplication by the derivations
$\partial_{1},\ldots,\partial_{n}$. This is a free resolution of right
$\mathcal{D}_{X}$-modules for $\omega_{X}$ shifted right $n$ steps, and yields
an explicit form of the de Rham cohomology functors

$H^{i}_{\mathrm{dR}}(-):=\mathbb{H}^{i-n}(X,\omega_{X}\otimes_{\mathcal{D}_{X}}^{L}(-))=H^{i}(\Omega^{\bullet}_{\mathcal{D},X}\otimes_{\mathcal{D}_{X}}(-))$

from the category of left $\mathcal{D}_{X}$-modules to the category of
$\mathbb{K}$-vector spaces. Since the constituents of
$\Omega^{\bullet}_{\mathcal{D},X}$ are $\mathcal{D}_{X}$-free and $X$ is
$\mathcal{D}$-affine, for each left $\mathcal{D}_{X}$-module $\mathcal{M}$
with global sections $M$ one has

$H^{i}_{\mathrm{dR}}(\mathcal{M})=H^{i-n}((D_{n}/\partial\cdot
D_{n})\otimes^{L}_{D_{n}}M).$

If $\mathcal{M}$ is holonomic, these vector spaces are $\mathbb{K}$-finite
since they are the cohomology of the $\mathcal{D}$-module theoretic direct
image functor under the map to a point [HTT08].

We return to de Rham homology $\mathbb{H}^{2n-q}_{Y}(X,\Omega^{\bullet}_{X})$
with $X$ equal to affine $n$-space. Since $\Omega^{j}_{X}$ is finite free over
$\mathcal{O}_{X}$, there is a natural identification of
$H^{i}_{Y}(\Omega^{j}_{X})$ with
$\Omega^{j}_{X}\otimes_{\mathcal{O}_{X}}H^{i}_{Y}(\mathcal{O}_{X})$. The
complex

$\ldots\longrightarrow\Omega^{j-1}_{X}\otimes_{\mathcal{O}_{X}}H^{i}_{Y}(\mathcal{O}_{X})\longrightarrow\Omega^{j}_{X}\otimes_{\mathcal{O}_{X}}H^{i}_{Y}(\mathcal{O}_{X})\longrightarrow\Omega^{j+1}_{X}\otimes_{\mathcal{O}_{X}}H^{i}_{Y}(\mathcal{O}_{X})\longrightarrow\ldots$

with differential induced by the usual exterior derivative is quasi-isomorphic
to the complex
$\Omega^{\bullet}_{\mathcal{D},X}\otimes_{\mathcal{D}_{X}}H^{i}_{Y}(\mathcal{O}_{X})$.

Since $X$ is affine, $\Gamma(X,-)$ induces a spectral sequence for
hypercohomology,

(1.0.1) $\displaystyle
H^{p}_{\mathrm{dR}}(H^{q}_{Y}(R_{n}))\Longrightarrow\mathbb{H}^{p+q}_{Y}(\Omega^{\bullet}_{X})=H^{\mathrm{dR}}_{2n-p-q}(Y)$

that has been considered in [Swi17, Lemma 2.16] in the complete local case,
and in [Bri20] in the context we are working in. We note that over the complex
numbers, the abutment is naturally equal to the reduced singular cohomology of
$U:=X\backslash Y$, so there is a spectral sequence

(1.0.2) $\displaystyle
E^{2}_{p,q}=H^{p}_{\mathrm{dR}}(H^{q}_{Y}(R_{n}))\Longrightarrow\tilde{H}^{p+q-1}(U;\mathbb{C})$

to the reduced cohomology of $U$. For $I={\mathfrak{m}}$, the abutment is
$H^{n}_{\mathrm{dR}}(H^{n}_{I}(R_{n}))=\mathbb{C}[2n]$, the reduced cohomology
of the $(2n-1)$-sphere shifted by one. For details see for example [Har75, p.
67] and [LSW16, Thm. 3.1].

The articles [Swi17, Bri20] proceed to show that the $E_{r}$-pages, $r\geq 2$,
of these spectral sequences are isomorphic for all embeddings of $Y$. In
consequence, the terms on pages $r\geq 2$ of (1.0.1) are numerical invariants
of $Y$.

######  Definition 1.2.

Let $Y={\mathrm{Var}}(I)$ be an affine variety embedded in
$X=\operatorname{\textup{Spec}\,}(R_{n})$ defined by the ideal $I\subseteq
R_{n}=\mathbb{K}[x_{1},\ldots,x_{n}]$ over the field $\mathbb{K}$ of
characteristic zero. For $r\geq 2$, the _$(r,p,q)$ -Čech–de Rham number_ of
$Y$ is the dimension

$\rho_{p,q}^{r}(Y):=\dim_{\mathbb{K}}(E^{n-p,n-q}_{r})$

of the corresponding entry in the spectral sequence (1.0.1). If $r=2$ we
denote $\rho_{p,q}^{r}(Y)=H^{n-p}_{\mathrm{dR}}(H^{n-q}_{I}(R_{n}))$ by just
$\rho_{p,q}(Y)$. $\diamond$

Switala defined these for ideals in the power series ring [Swi17, Dfn. 2.23];
they are well-defined by [Swi17, Prop. 2.17] and [Bri20, Thm. 1.1]. The
dimensions $\rho_{p,q}^{r}$ are invariant under field extensions, and one can
compute them algorithmically over any field of definition for $I$, see [OT99,
OT01, Wal00].

A related construction appeared in [Lyu93], where Lyubeznik shows that the
socle dimensions of the $E_{2}$-terms of the Grothendieck spectral sequence

(1.0.3) $\displaystyle
E_{2}^{p,q}=H^{p}_{\mathfrak{m}}(H^{q}_{I}(R_{n}))\Longrightarrow
H^{p+q}_{\mathfrak{m}}(R_{n})$

are independent of the closed embedding of
$Y=\operatorname{\textup{Spec}\,}(R_{n}/I)$ into any affine space
$\mathbb{A}^{n}_{\mathbb{K}}=\operatorname{\textup{Spec}\,}(R_{n})$ and uses
it to define numerical invariants

$\lambda_{p,q}(R_{n}/I):=\dim_{\mathbb{C}}\operatorname{\textup{Hom}}(R/{\mathfrak{m}},H^{p}_{\mathfrak{m}}(H^{n-q}_{I}(R))).$

These numbers, known as _Lyubeznik numbers_ have been investigated for nearly
three decades and are indeed functions of the ring $R/I$ (and do not depend on
the presentation of $R/I$ as a quotient of a polynomial ring). For detailed
information on the history and the _status quo_ we refer to the survey
articles [NnBWZ16, WZ20].

In this article we develop further the theory of the Lyubeznik numbers on one
side, and on the other describe a number of properties that the invariants
introduced by Switala and Bridgland enjoy.

More precisely, in the next section we study the Čech–de Rham numbers for
small dimension of $Y$, and investigate the collapse of the corresponding
spectral sequence. We identify some classes of examples where this collapse
happens on the $E_{2}$-page, and explain why this is so for subspace
arrangements, by stringing together known results of Goresky–MacPherson, and
Àlvarez–García–Zarzuela. We also explore the behavior of the Čech–de Rham
numbers under Veronese maps.

In the third section we discuss Lyubeznik numbers. We elaborate on the results
from [RSW21] by establishing some classes of projective varieties $\tilde{Y}$
with Picard number one that have most Lyubeznik numbers of the affine cone $Y$
independent of the embedding. That includes determinantal varieties, certain
toric varieties, and horospherical varieties. We also prove for certain
projective varieties of dimension four or less that their Lyubeznik numbers
are independent of the embedding.

Some known facts.

Since we will have to refer to them a few times, we state here some results
from the literature.

######  Remark 1.3.

  1. (1)

If $\mathbb{K}$ is of characteristic zero, then local cohomology, algebraic de
Rham cohomology, injective dimension, dimension, socle dimension all behave
well under field extensions. Since all varieties are defined by a finite
number of data, one can restrict all questions we discuss from the given field
$\mathbb{K}$ to a field of definition for $I$, and then extend to
$\mathbb{C}$. In particular, we can assume that $\mathbb{K}=\mathbb{C}$
whenever it is convenient.

  2. (2)

([Lyu93, (4.4.iii)]) Suppose $Y\subseteq X=\mathbb{A}^{n}_{\mathbb{K}}$ is an
affine variety. Then the local cohomology module $H^{i}_{I}(R_{n})$ has
support dimension at most $n-i$, and it vanishes if
$i<c:=\operatorname{\textup{codim}}(Y,X)$. If $Y$ is equi-dimensional and
$i>c$, then $H^{i}_{I}(R_{n})$ has support dimension less than $n-i$.

  3. (3)

([Lyu93, Thm. 2.4]) If $M$ is a holonomic $D$-module, then
$H^{i}_{\mathfrak{m}}(M)$ is a finite sum of copies of the (Artinian,
indecomposable) injective hull $H^{n}_{\mathfrak{m}}(R_{n})$ of
$\mathbb{K}=R_{n}/{\mathfrak{m}}$. More generally, one has for all holonomic
modules that

$\operatorname{\textup{injdim}}_{R}(M)\leq\dim\operatorname{\textup{Supp}}(M).$

Thus, all right derived functors of $R$-modules with derivation level greater
than $n-i$ vanish on $H^{i}_{I}(R)$, and those of derivation level $n-i$
vanish if $I$ is equi-dimensional and $i>c$.

  4. (4)

We will also have to refer to equivariant $\mathcal{D}_{X}$-modules. For
details on equivariance of $\mathcal{D}$-modules, see for example [LW19].

  5. (5)

Let $I\subseteq R=\mathbb{K}[x_{1},\dots,x_{n}]$ be a homogeneous ideal such
that $\dim(R/I)\geq 2$. Assume that $\mathbb{K}$ is separably closed.
Hartshorne proved in [Har68, Theorem 7.5] that if
$\operatorname{\textup{Proj}\,}(R/I)$ is connected then
$H^{n}_{I}(R)=H^{n-1}_{I}(R)=0$, and named this result the Second Vanishing
Theorem. This theorem subsequently has been extended to the local settings as
follows: Let $R$ be either a complete regular local ring of dimension $n$ that
contains a separably closed coefficient field or an unramified complete
regular local ring of dimension $n$ in mixed characteristic with a separably
closed residue field. Let $I\subseteq R$ be an ideal. Then
$H^{n}_{I}(R)=H^{n-1}_{I}(R)=0$ if and only if $\dim(R/I)\geq 2$ and the
punctured spectrum of $R/I$ is connected, [Ogu73, PS73, HL90, Zha21].
$\diamond$

The following is a special case of a more general result comparing direct
image to a point and restriction to a point.

######  Lemma 1.4 ([RW18, Lemma 3.3]).

Suppose $\mathbb{K}=\mathbb{C}$ and $X=\mathbb{C}^{n}$. Assume that
$\mathcal{M}$ is a regular holonomic $\mathcal{D}_{X}$-module and its global
sections form a standard graded $R_{n}$-module. Suppose further that
$\mathcal{M}$ is (strongly) equivariant as a $\mathcal{D}$-module with respect
to the $\mathbb{C}^{*}$-action corresponding to this grading. Then its de Rham
cohomology groups agree with the restriction groups to the origin of the
holonomically dual module. In particular, the dimensions of these groups
satisfy

$\dim_{\mathbb{C}}(H^{i}_{\mathrm{dR}}(\mathcal{M}))=\dim_{\mathbb{C}}(\operatorname{\textup{Hom}}_{R_{n}}(R_{n}/{\mathfrak{m}},H^{n-i}_{\mathfrak{m}}(\mathbb{D}(\mathcal{M}))),$

where $\mathbb{D}$ is the holonomic duality functor. ∎

##  2\. Čech–de Rham numbers

###  2.1. Basic structure results

Basic properties of the de Rham functor imply that $\rho_{p,q}^{r}$ is zero
for $p$ outside the interval $[0,n]$. On the other hand, local cohomology
$H^{j}_{I}(R)$ is nonzero only when $\operatorname{\textup{codim}}(I)\leq
j\leq n$, and so $\rho_{p,q}^{r}$ is zero for $q$ outside the interval
$[0,\dim(Y)]$. Our first statement on these numbers is that they are confined
to a triangular region:

######  Proposition 2.1.

The Čech–de Rham numbers satisfy for all $r\geq 2$ that

$\rho_{p,q}^{r}(Y)=0\qquad\text{ if }p>q.$

Before entering the proof we set up some notation and collect several facts
and from [Dim04, HTT08, KS94] on constructible sheaves and the Riemann–Hilbert
correspondence. All spaces mentioned in the sequel are assumed to be
algebraic.

######  Remark 2.2.

Let $X$ be a smooth algebraic variety.

  1. (1)

For any algebraic map $f$ between algebraic sets we denote, on the level of
constructible sheaves, the usual direct and inverse image functors by $f_{*}$
and $f^{-1}$, and the proper direct and exceptional inverse image functors by
$f_{!}$ and $f^{!}$ respectively. For the sake of notational brevity, we mean
by these symbols always the derived functors on the appropriate derived
categories (so that, for example, we write $j_{*}$ instead of $Rj_{*}$ as a
functor on the bounded derived category of constructible sheaves). This abuse
of notation is common in the relevant literature.

  2. (2)

On the level of $\mathcal{D}$-modules, we will use $f_{+}$ and $f_{!}$ for the
usual and proper direct image functors, and $f^{+}$ and $f^{\dagger}$ for the
usual and exceptional inverse image functors. For reference and comparison,
our $\mathcal{D}$-functors $f_{+},f_{!},f^{+},f^{\dagger}$ are (in this
sequence) denoted by $\int_{f},\int_{f!},f^{\dagger},f^{\star}$ in [HTT08].

  3. (3)

The Riemann–Hilbert correspondence sets up an equivalence between the derived
category of bounded complexes of $\mathcal{D}_{X}$-modules with holonomic
cohomology, and the derived category of bounded complexes of constructible
sheaves $D^{b}_{\mathrm{c.s.}}(X)$. The correspondence is induced by the de
Rham functor
$\Omega^{\bullet}_{X^{\mathit{an}}}\otimes_{\mathcal{D}_{X^{\mathit{an}}}}^{L}(-)$
computed on the analytic space attached to $X$.

Under this correspondence, taking cohomology of a complex of
$\mathcal{D}_{X}$-modules corresponds to an operation on complexes of
constructible sheaves that is denoted ${}^{p}\mathcal{H}$; it is not the same
as taking cohomology of complexes of constructible sheaves. We call _perverse
exact_ any functor on the derived category that commutes with
${}^{p}\mathcal{H}$.

  4. (4)

Suppose $f\colon X\longrightarrow X^{\prime}$ is a morphism of smooth
algebraic varieties. Under the Riemann–Hilbert correspondence, the functors
for $\mathcal{D}$-modules correspond to those on constructible sheaves as
follows:

$\operatorname{\mathit{DR}}_{X^{\prime}}\circ f_{+}\simeq
f_{*}\circ\operatorname{\mathit{DR}}_{X};\qquad\operatorname{\mathit{DR}}_{X^{\prime}}\circ
f_{!}\simeq
f_{!}\circ\operatorname{\mathit{DR}}_{X};\qquad\operatorname{\mathit{DR}}_{X}\circ
f^{+}\simeq
f^{!}\circ\operatorname{\mathit{DR}}_{X^{\prime}};\qquad\operatorname{\mathit{DR}}_{X}\circ
f^{\dagger}\simeq f^{-1}\circ\operatorname{\mathit{DR}}_{X^{\prime}}.$

(The last two identifications are not misprints; for inverse images, the
Riemann–Hilbert correspondence via the de Rham functor aligns a regular
inverse image with an exceptional one).

  5. (5)

Consider an open embedding $j\colon U\hookrightarrow X$ and a closed embedding
$i\colon Z\hookrightarrow X$ where $Z$ is closed (and, _a fortiori_ ,
constructible) and where $U$ is the complement of $Z$ in $X$. We have the
following properties of induced functors for complexes of constructible
sheaves:

     * •

$i_{!}$ is exact and perverse exact;

     * •

$i^{-1}$ is exact but usually not perverse exact;

     * •

$i^{!}$ and $j_{*}$ are usually neither exact nor perverse exact;

     * •

$j^{-1}$ is exact and perverse exact;

     * •

$j_{!}$ is exact but usually not perverse exact.

  6. (6)

In the situation of the previous item, we have the following distinguished
triangles, Verdier dual to one another, in $D^{b}_{\mathrm{c.s.}}(X)$:

$\displaystyle i_{!}i^{!}F^{\bullet}\longrightarrow$ $\displaystyle
F^{\bullet}\longrightarrow j_{*}j^{-1}\overset{+1}{\longrightarrow}\,,$
$\displaystyle j_{!}j^{-1}F^{\bullet}\longrightarrow$ $\displaystyle
F^{\bullet}\longrightarrow
i_{!}i^{-1}F^{\bullet}\overset{+1}{\longrightarrow}\,.$

  7. (7)

We will always denote by $a_{S}$ the map from a space $S$ to a point, which we
occasionally denote with $\mathit{pt}$ and occasionally identify with the
vertex of a cone if a cone is present. $\diamond$

We now enter the

###### Proof of Proposition 2.1.

It suffices to consider $r=2$. We will use the Riemann–Hilbert correspondence
to translate
$\rho_{p,q}=\dim_{\mathbb{C}}(H^{n-p}_{{\mathrm{dR}}}(H^{n-q}_{I}(R_{n})))$
into the language of constructible sheaves. The de Rham functor takes the
local cohomology $H^{n-q}_{I}(\mathcal{O}_{X})$ to
${{}^{p}}\mathcal{H}^{n-q}h_{!}h^{!}\mathbb{C}_{X}[n]\simeq
h_{!}({{}^{p}}\mathcal{H}^{-q}\omega_{Y})$ where $h:Y\rightarrow
X=\mathbb{A}^{n}_{\mathbb{C}}$ is the canonical embedding, $\mathbb{C}_{X}[n]$
is the constant sheaf on $X$ shifted to the left by $n$ and
$\omega_{Y}=\mathbb{D}\mathbb{C}_{Y}$ is the (topological) dualizing complex
$R\operatorname{\textup{Hom}}_{{\mathrm{c.s.}}}(\mathbb{C}_{Y},\mathbb{C}_{Y})$
for constructible sheaves on $Y$. (We use $\mathbb{D}$ also to denote Verdier
duality, the operation corresponding to holonomic duality under the
Riemann–Hilbert correspondence).

By Remark 1.3,
$\dim(\operatorname{\textup{Supp}}(h_{!}({{}^{p}}\mathcal{H}^{-q}\omega_{Y})))=\dim(\operatorname{\textup{Supp}}(H^{n-q}_{I}(R)))\leq
q$. Set
$Y_{q}:=\operatorname{\textup{Supp}}({{}^{p}}\mathcal{H}^{-q}\omega_{Y})$ and
denote by

$i_{q}\colon Y_{q}\rightarrow Y\qquad\text{and}\qquad j_{q}\colon
Y\smallsetminus Y_{q}\rightarrow Y$

the embeddings of $Y_{q}$ and its complement into $Y$ and denote by
$\tilde{i}_{q}:Y_{q}\longrightarrow X$ resp. $\tilde{j}_{q}:X\smallsetminus
Y_{q}\longrightarrow X$ the corresponding embeddings of $Y_{q}$ and its
complement into $X$.

On the level of $\mathcal{D}_{X}$-modules with support in $Y$, this gives an
exact triangle

$R\Gamma_{Y_{q}}\longrightarrow\operatorname{\textup{id}}\longrightarrow(\tilde{j}_{q})_{+}(\tilde{j}_{q})^{\dagger}\stackrel{{\scriptstyle+1}}{{\longrightarrow}}$

that corresponds via Riemann–Hilbert to

$(\tilde{i}_{q})_{!}(\tilde{i}_{q})^{!}\longrightarrow\operatorname{\textup{id}}\longrightarrow(\tilde{j}_{q})_{*}(\tilde{j}_{q})^{-1}\stackrel{{\scriptstyle+1}}{{\longrightarrow}}$

for constructible sheaves.

Since the support of ${{}^{p}}\mathcal{H}^{-q}\omega_{Y}$ is $Y_{q}$,
$(j_{q})_{*}j_{q}^{-1}({{}^{p}}\mathcal{H}^{-q}\omega_{Y})=0$ and so
$(i_{q})_{!}(i_{q})^{!}({{}^{p}}\mathcal{H}^{-q}\omega_{Y})\simeq{{}^{p}}\mathcal{H}^{-q}\omega_{Y}$.
In particular
$\mathcal{G}_{q}:=(i_{q})^{!}({{}^{p}}\mathcal{H}^{-q}\omega_{Y})$ is a
perverse sheaf on $Y_{q}$ satisfying
$(i_{q})_{!}\mathcal{G}_{q}={{}^{p}}\mathcal{H}^{-q}\omega_{Y}$. As
$\tilde{i}_{q}$ is proper, we find that
$(\tilde{i}_{q})_{*}(\mathcal{G}_{q})=(\tilde{i}_{q})_{!}(\mathcal{G}_{q})={{}^{p}}\mathcal{H}^{-q}\omega_{Y}$
and it follows that the hypercohomology of $\mathcal{G}_{q}$ agrees with the
hypercohomology of ${{}^{p}}\mathcal{H}^{-q}\omega_{Y}$.

But the hypercohomology $\mathbb{H}^{k}(\mathcal{F})$ vanishes for
$k\not\in[-q,0]$ and every perverse sheaf $\mathcal{F}$ on $Y_{q}$ (see e.g.
[Dim04, Corollary 5.2.18] and [Dim04, Proposition 5.2.20]. The claim follows
now from

$\rho_{p,q}^{2}=\dim(H^{n-p}_{{\mathrm{dR}}}(H^{n-q}_{I}(R_{n})))=\dim(\mathbb{H}^{-p}({{}^{p}}\mathcal{H}^{-q}\omega_{Y}))=\dim(\mathbb{H}^{-p}(\mathcal{G}_{q})).$

∎

######  Remark 2.3.

If $I$ is homogeneous, a more elementary argument can be made. Indeed, the
local cohomology module $H^{p}_{I}(R_{n})$ is Eulerian (see Definition 2.11
below). This means that the Lie algebra action induced from the
differentiating the $\mathbb{C}^{*}$-action corresponding to the grading
agrees with the action of the Lie algebra via the morphism from the universal
enveloping algebra to the Weyl algebra. In other words, $H^{p}_{I}(R_{n})$ is
equivariant and [RW18, Lem. 3.3] applies in the form of Lemma 1.4.

Holonomic duality does not affect the support of the underlying module and
preserves the category of regular holonomic $\mathcal{D}$-modules. Thus, for
graded $I$ we have that $H^{n-p}_{\mathrm{dR}}(H^{n-q}_{I}(R_{n}))=0$ if and
only if $H^{p}_{\mathfrak{m}}(\mathbb{D}(H^{n-q}_{I}(R_{n}))=0$. This latter
vanishing holds whenever
$\dim(\operatorname{\textup{Supp}}(H^{n-q}_{I}(R_{n})))<p$ since
$H^{p}_{\mathfrak{m}}(\mathcal{M})$ is zero whenever
$p>\dim(\operatorname{\textup{Supp}}(\mathcal{M}))$, [Har67]. But by Remark
1.3, $\dim(\operatorname{\textup{Supp}}(H^{n-q}_{I}(R_{n})))\leq q$.
$\diamond$

If one pictures the $\rho^{r}_{p,q}$ as a table, it thus takes the following
general form, assuming that $Y$ is embedded into
$\mathbb{A}^{n}_{\mathbb{K}}$, cut out by the ideal $I\subseteq R$ of
dimension $d$:

$P^{r}(Y)=((\rho^{r}_{p,q})):=\begin{pmatrix}\rho^{r}_{0,0}&\cdots&\cdots&\rho_{0,d}^{r}\\\
0&\ddots&&\vdots\\\ \vdots&\ddots&\ddots&\vdots\\\
0&\cdots&0&\rho^{r}_{d,d}\end{pmatrix}$

Here, $p$ is the row index counting downward, $q$ the column index counting
towards the right, and the arrows of the Čech–de Rham spectral sequence point
North to Northeast.

###  2.2. Degeneration

Switala raised in [Swi17, Question 8.2] the following question for a complete
local ring $A$ with coefficient field $\mathbb{K}$ of characteristic zero:

> “Does the local Hodge–de Rham homology spectral sequence degenerate at
> $E_{2}$ ?”

One can ask a similar question for the affine scenario. We discuss interesting
classes where this question has a positive answer.

######  Example 2.4.

Suppose $Y={\mathrm{Var}}(I)$ is a complex subspace arrangement. Let
${\mathsf{P}}_{Y}$ be its _intersection lattice_ , the collection of all
possible intersections of the components of $Y$, ordered by inclusion. (This
differs from standard notation in arrangement theory, where the order is the
reverse). We agree that ${\mathsf{P}}_{Y}$ has a unique maximal element
corresponding to the ambient space, but it may have several minimal elements
as we do not insist that $I$ be homogeneous (so, the arrangement may not be
central).

It is well-known that the cohomology of the complement
$\mathbb{C}^{n}\smallsetminus Y$ is determined by the combinatorics of
${\mathsf{P}}$: building on work of Brieskorn, Orlik and Solomon [OS80] showed
that the cohomology algebra of this complement is given by a purely
combinatorial algebra constructed from the matroid of the arrangement.

Goresky and MacPherson [GM88, III, Thm. 1.3] proved that the Betti numbers of
the complement can be computed as a sum of non-negative integers, one for each
element of ${\mathsf{P}}_{Y}$. Here, the integers for each flat
$p\in{\mathsf{P}}$ are computed as Betti numbers of the simplicial complex
$K(>p)$. (While Goresky and MacPherson phrase this in terms of relative
homology for the pair $(K(\geq p),K(>p))$, the space $K(\geq p)$ is
contractible and one can convert into an absolute homology without harm).

Àlvarez, García and Zarzuela established the degeneration on page two of a
certain spectral sequence

(2.2.1) $\displaystyle
E^{-i,j}_{2}={\varprojlim_{p\in{\mathsf{P}}_{Y}}}^{(i)}H^{j}_{I_{p}}(R_{n})$
$\displaystyle\Longrightarrow$ $\displaystyle H^{j-i}_{I}(R_{n})$

for the local cohomology groups $H^{\bullet}_{I}(R_{n})$, the inverse limits
being taken over the poset ${\mathsf{P}}_{Y}$ viewed as a category with a
morphism for each containment. In [AMGLZA03, Thm. 1.2], the structure of the
derived inverse limits is explained as direct sums of modules
$H^{j}_{I_{p}}(R)$ with $\operatorname{\textup{codim}}(I_{p})=j$ and
multiplicity given by the topological Betti numbers of $K(>p)$. In [AMGLZA03,
Cor. 1.3], this is used to give a formula for the cohomology groups of the
complement of $Y$, by translating the Goresky–McPherson formula.

The affine complement of an affine space is homotopy equivalent to a sphere,
hence applying the de Rham functor to a module of the form $H^{j}_{I_{p}}(R)$
gives exactly one (reduced) cohomology group. Thus, the entries of the
$E_{2}$-page of the Čech–de Rham spectral sequence (1.0.2) correspond exactly
to the composition factors of $H^{j-i}_{I}(R)$ in the spectral sequence
(2.2.1) from [AMGLZA03] on one side, and to the direct summands for
$H^{\bullet}(\mathbb{C}^{n}\smallsetminus Y)$ in [GM88] on the other. It
follows that for complex subspace arrangements $Y$ the Čech–de Rham spectral
sequence collapses on the $E_{2}$-page. $\diamond$

In small dimensions we show that Switala’s question has a positive answer as
well.

######  Proposition 2.5.

If $I$ is homogeneous and $\dim({\mathrm{Var}}(I))\leq 3$ then the Čech–de
Rham spectral sequence degenerates at $E_{2}$.

###### Proof.

Let $Y$ be of dimension 2 or less. If follows from Proposition 2.1 that no
nonzero differential can exist in the spectral sequence.

Let now $\dim(Y)=3$. Then Proposition 2.1 implies that then there might be at
most one nonzero differential,

(2.2.2) $\displaystyle d_{2}\colon
H^{n-2}_{\mathrm{dR}}(H^{n-2}_{I}(R_{n}))\longrightarrow
H^{n}_{\mathrm{dR}}(H^{n-3}_{I}(R_{n})),$

linking $\rho_{2,2}$ and $\rho_{0,3}$.

Assume for the time being that $Y$ is purely 3-dimensional. Remark 1.3 says
that $\dim\operatorname{\textup{Supp}}(H^{n-i}_{I}(R_{n}))<i$ for $i<3$. In
particular, by Lemma 1.4, $\dim H^{n-2}_{\mathrm{dR}}(H^{n-2}_{I}(R_{n}))$
equals the socle dimension of
$H^{2}_{\mathfrak{m}}(\mathbb{D}H^{n-2}_{I}(R_{n}))=0$. Thus, the degeneration
of the spectral sequence is forced.

Now relax the equi-dimensionality condition and let $Y_{3}$ and $Y^{\prime}$
be the 3-dimensional and smaller dimensional components of $Y$ respectively.
Then $Y_{3}\cap Y^{\prime}$ is of dimension 1 or less, and the Mayer–Vietoris
sequence implies that $H^{n-3}_{Y_{3}}(R_{n})\oplus
H^{n-3}_{Y^{\prime}}(R_{n})=H^{n-3}_{Y}(R_{n})$ and that there is a short
exact sequence

$0\longrightarrow H^{n-2}_{Y_{3}}(R_{n})\oplus
H^{n-2}_{Y^{\prime}}(R_{n})\longrightarrow H^{n-2}_{Y}(R_{n})\longrightarrow
C\longrightarrow 0$

where $C$ is a (graded) submodule of $H^{n-1}_{Y_{3}\cap Y^{\prime}}(R_{n})$.
In particular, the dimension of the support of $C$ is one or less by Remark
1.3 and so $H^{\leq n-2}_{\mathrm{dR}}(C)$ is zero, being dual to the socle of
$H^{\geq 2}_{\mathfrak{m}}(C)=0$.

Applying the de Rham functor, the resulting long exact sequence shows that
$H^{n-2}_{\mathrm{dR}}(H^{n-2}_{Y_{3}}(R_{n})\oplus
H^{n-2}_{Y^{\prime}}(R_{n}))$ equals
$H^{n-2}_{\mathrm{dR}}(H^{n-2}_{Y}(R_{n}))$. Then the map (2.2.2) is the
direct sum of the corresponding $d_{2}$-morphism for $Y_{3}$ and for
$Y^{\prime}$ separately. But it is zero on
$H^{n-2}_{\mathrm{dR}}(H^{n-2}_{Y_{3}}(R_{n}))$ since the source of $d_{2}$ is
zero in that case, and it is zero on
$H^{n-2}_{\mathrm{dR}}(H^{n-2}_{Y^{\prime}}(R_{n}))$ since the target is zero
in that case. ∎

###  2.3. Affine complements

In the next two subsections we investigate to what extent the cohomology of
the affine complement, or its table of Čech–de Rham numbers, of a homogeneous
variety $Y$ is determined by the associated projective variety $\tilde{Y}$. We
start with looking at the top cohomology group of the affine complement, and
then investigate the affine complement under Veronese maps. In the process we
review some algorithmic ideas that lead to a condition on the de Rham classes
of graded $\mathcal{D}$-modules on affine space.

So, throughout, $\tilde{Y}$ is a projective variety and
$Y\subseteq\mathbb{C}^{n}$ is a cone for $\tilde{Y}$.

####  2.3.1. High cohomology groups of the affine complement

######  Remark 2.6.

Let $\tilde{Y}$ be a projective variety with cone
$Y=\operatorname{\textup{Spec}\,}(R_{n}/I)$. The following facts are due to
Ogus [Ogu73] Let

$f_{Y}:=\min(k\in\mathbb{N}|H^{\ell}_{I}(R)\text{ is Artinian for all
}\ell>k)$

and

$v_{Y}:=\min(k\in\mathbb{N}|H^{\ell}_{I}(R)\text{ is zero for all }\ell>k).$

  1. (1)

The number $n-f_{Y}$ is intrinsic to $\tilde{Y}$, it does not depend on the
choice of the cone $Y$,[Ogu73, Thm. 4.1].

  2. (2)

The number $n-v_{Y}$ is intrinsic to $\tilde{Y}$, it does not depend on the
choice of the cone $Y$,[Ogu73, Thm. 4.4] and the remark following it.

In particular,

$\rho^{r}_{p,q}=0\quad\left\\{\begin{array}[]{ll}\text{ if }&q<n-\nu_{Y},\\\
\text{ or }&p>0\text{ and }q<n-f_{Y}.\end{array}\right.$

$\diamond$

We show next that in fact the top de Rham cohomology group of the affine cone
complement is usually determined by $\tilde{Y}$.

######  Lemma 2.7.

Let $X=\mathbb{A}^{n}_{\mathbb{C}}$ and suppose
$\tilde{Y}\subseteq\mathbb{P}X=\mathbb{P}^{n-1}_{\mathbb{C}}$ is defined by
the homogeneous ideal $I\subseteq R_{n}:=\Gamma(X,\mathcal{O}_{X})$. Let
$Y={\mathrm{Var}}(I)\subseteq X$ and assume that $Y$ has codimension at least
two. Then the index and the dimension of the top non-vanishing de Rham
cohomology group of $U:=X\smallsetminus Y$ is encoded on $\tilde{Y}$.

###### Proof.

We recall Alexander duality, compare [Ive86, V.6.6]: if $\mathbb{P}$ is a
$\mathbb{C}$-orientable manifold and $\tilde{Y}$ a closed subset then the
topological local cohomology group $H^{i}_{\tilde{Y}}(\mathbb{P};\mathbb{C})$
is canonically identified with the $\mathbb{C}$-dual of the cohomology with
compact support
$H^{2\dim_{\mathbb{C}}\mathbb{P}-i}_{c}(\tilde{Y};\mathbb{C})$. If $\tilde{Y}$
is, in addition, compact, the latter is just
$H^{2\dim_{\mathbb{C}}\mathbb{P}-i}(\tilde{Y};\mathbb{C})$.

On the other hand, [Ive86, II.9.2] states the existence of a long exact
sequence

(2.3.1) $\displaystyle H^{i}_{\tilde{Y}}(P;\mathcal{F})\longrightarrow
H^{i}(P;\mathcal{F})\longrightarrow
H^{i}(P\smallsetminus\tilde{Y};\mathcal{F})\stackrel{{\scriptstyle+1}}{{\longrightarrow}}$

where $\mathcal{F}$ is a sheaf of Abelian groups on $P$ and we ease notation
by ignoring the pull-backs of $\mathcal{F}$ to $\tilde{Y}$ and its complement
respectively. Notice that one can get this long exact sequence by applying
hypercohomology to the first triangle in Remark 2.2 (4). We use these with
$\mathbb{P}=\mathbb{P}X$ and $\tilde{Y}$ as above.

Via Poincaré duality, the map
$H^{i}_{\tilde{Y}}(\mathbb{P}X;\mathbb{C})\longrightarrow
H^{i}(\mathbb{P}X;\mathbb{C})$ becomes
$H^{2n-2-i}(\tilde{Y};\mathbb{C})^{\vee}\longrightarrow
H^{2n-2-i}(\mathbb{P}X;\mathbb{C})^{\vee}$. This is the dual of
$H^{2n-2-i}(\mathbb{P}X;\mathbb{C})\longrightarrow
H^{2n-2-i}(\tilde{Y};\mathbb{C})$ induced by restriction from $\mathbb{P}X$ to
$\tilde{Y}$. The restriction $H^{i}(\mathbb{P}X;\mathbb{C})\longrightarrow
H^{i}(\tilde{Y};\mathbb{C})$ is injective 111The cohomology fundamental class
of $Y$ in $H^{2\dim_{\mathbb{C}}(\tilde{Y})}(\mathbb{P}X;\mathbb{C})$
evaluates on the homology class of a generic
$\mathbb{P}^{n-1-\dim_{\mathbb{C}}(\tilde{Y})}\subseteq\mathbb{P}X$ to the
0-cycle given by the intersection of $\tilde{Y}$ with that generic subspace.
But this intersection is the degree of $\tilde{Y}$, hence positive. Thus the
restriction of the class represented by this subspace on $\mathbb{P}X$, a
generator of $H^{2\dim_{\mathbb{C}}(\tilde{Y})}(\mathbb{P}X;\mathbb{C})$, to
$\tilde{Y}$ is nonzero. But cohomology of projective space is a polynomial
algebra in the hyperplane section, and if the
$\dim_{\mathbb{C}}(\tilde{Y})$-power of the hyperplane restricts to a nonzero
class on $\tilde{Y}$ then so do all smaller powers. for $i\leq 2\dim\tilde{Y}$
and necessarily zero for $i>2\dim_{\mathbb{C}}(\tilde{Y})$ since $\tilde{Y}$
is a CW-complex of dimension $2\dim_{\mathbb{C}}(\tilde{Y})$. Thus, one can
determine from the topological Betti numbers of $\tilde{Y}$ alone the sizes of
the kernels of the left-most morphisms in display (2.3.1). This in turn
determines the sizes of the cohomology groups of
$\mathbb{P}U:=\mathbb{P}X\smallsetminus\tilde{Y}$.

As $\operatorname{\textup{codim}}(Y)\geq 2$, $U$ is simply connected by
[God71, Thm. 2.3]. Thus, the $\mathbb{C}^{*}$-fiber bundle
$U\longrightarrow\mathbb{P}U$ has a Leray spectral sequence

$H^{i}(\mathbb{P}U;H^{j}(\mathbb{C}^{*};\mathbb{C}))\Longrightarrow
H^{i+j}(U;\mathbb{C})$

in which the coefficients on the left are global (in a trivial vector bundle).
Let $m$ be the largest index with $H^{m}(\mathbb{P}U;\mathbb{C})\neq 0$. Since
all differentials out of and into
$H^{m}(\mathbb{P}U;H^{1}(\mathbb{C}^{*};\mathbb{C}))\neq 0$ are zero, $m+1$
must be the largest index with $H^{m+1}(U;\mathbb{C})\neq 0$ and
$\dim_{\mathbb{C}}H^{m}(\mathbb{P}U;\mathbb{C})=\dim_{\mathbb{C}}H^{m+1}(U;\mathbb{C})$.
∎

######  Corollary 2.8.

Let $Y$ be an affine variety defined by the homogeneous ideal $I\subseteq
R_{n}=\mathbb{C}[x_{1},\ldots,x_{n}]$. If the local cohomology group
$H^{\ell}_{I}(R)$ is Artinian and $H^{>\ell}_{I}(R_{n})=0$ then the socle
dimension $s$ of $H^{\ell}_{I}(R)$ is a function of the projective variety
$\tilde{Y}=\mathbb{P}(Y)$ and does not depend on the choice of the cone $Y$.

###### Proof.

Let $X=\operatorname{\textup{Spec}\,}(R)$ and set $U=X\smallsetminus Y$. By
[LSW16, Thm. 3.1], $s=\dim_{\mathbb{C}}H^{n+\ell-1}(U;\mathbb{C})$, and $U$
has no higher non-vanishing cohomology groups. Now use the previous lemma. ∎

######  Remark 2.9.

Even if $H^{\ell}_{I}(R)$ is not Artinian, the dimension
$\dim_{\mathbb{C}}H^{n}_{\mathrm{dR}}(H^{\ell}_{I}(R_{n}))$ is encoded by
$\tilde{Y}$. $\diamond$

####  2.3.2. Integrals of Eulerian modules

We investigate next to what extent the $\rho_{p,q}^{r}$, or the abutment terms
$H^{\mathrm{dR}}_{\bullet}(Y)$ of the Čech–de Rham spectral sequence are
independent of the cone $Y$ (_i.e._ , the line bundle $\mathcal{L}$ on
$\tilde{Y}$ that induces the cone). In the following we show that replacing
$\mathcal{L}$ by a power of itself does not change the
$H^{\mathrm{dR}}_{\bullet}(Y)$.

For this we give an account on the main results on algorithmic computation of
the integral of a $D_{n}$-module along $\partial_{1},\ldots,\partial_{n}$. See
[OT99, OT01, Wal00] for details, and a generalization to the case when $M$ is
a bounded complex of finitely generated modules that has holonomic cohomology.

We define a grading
$\operatorname{\textup{gr}}^{i}_{\tilde{V}}(D_{n}):=\\{P\in
D_{n}|\deg(P)=i\\}$ on $D_{n}$ by setting

$\deg(x_{j})=1=-\deg(\partial_{j})$

for all $1\leq j\leq n$. With it we define a filtration on $D_{n}$ by

${\tilde{V}}^{k}(D_{n})=\sum_{i\leq
k}\operatorname{\textup{gr}}^{i}_{{\tilde{V}}}(D_{n}).$

Let $M$ be a $D_{n}$-module, finitely generated by elements
$m_{1},\ldots,m_{r}$, and choose integers $s_{1},\ldots,s_{r}$. Then define a
filtration on $M$ by setting

${\tilde{V}}^{k}(M)=\sum_{i=1}^{r}{\tilde{V}}^{k-s_{i}}(D_{n})\cdot m_{i}.$

Denote the operator $-\sum_{j=1}^{n}\partial_{j}\cdot x_{j}$ by $\tilde{E}$.

It is a result of Kashiwara [Kas78] that when $M$ is holonomic there is a _$b$
-function for integration_ $\tilde{b}_{M}(s)$. This is a univariate polynomial
that satisfies

(2.3.2)
$\displaystyle\tilde{b}_{M}(\tilde{E}+n+k)\cdot{\tilde{V}}^{k}(M)\subseteq{\tilde{V}}^{k-1}(M)$

for all $k\in\mathbb{Z}$. We describe now ideas that lead to a proof for
Proposition 2.12 below.

As before, let $\omega_{n}$ be the right $D_{n}$-module $(D_{n}/\partial\cdot
D_{n})$ where $\partial=\\{\partial_{1},\ldots,\partial_{n}\\}$. This is a
free rank one $R_{n}$-module, and can be naturally identified with the
$D_{n}$-module $\operatorname{Ext}^{n}_{D_{n}}(R_{n},D_{n})$, and with the
global sections of the right $\mathcal{D}_{X}$-module of top differential
forms
$\mathcal{O}_{X}\cdot{\mathrm{d}}x_{1}\wedge\ldots\wedge{\mathrm{d}}x_{n}$,
[HTT08]. Give it a ${\tilde{V}}$-filtration by placing the generator
$1+\partial\cdot D_{n}$ into ${\tilde{V}}$-level $n$.

The $D$-module theoretic direct image functor $\pi_{+}$ for the projection map
$\pi\colon\mathbb{C}^{n}\longrightarrow\mathbb{C}^{0}$ can on global sections
be identified with $\omega_{n}\otimes_{D_{n}}^{L}(-)$ shifted by $n$,
computing the Tor-functors against $\omega_{n}$. This derived tensor product
can be viewed as the tensor product of $\omega_{n}$ with a free
$D_{n}$-resolution $F^{\bullet}$ of the input module $M$, or of a free
resolution $K^{\bullet}$ of $\omega$ with $M$, or of the tensor product of
$K^{\bullet}$ with $F^{\bullet}$. There are natural morphisms from the last
scenario to the two former ones that induce isomorphisms on cohomology.

One major difficulty in identifying $\pi_{+}(M)$ is that its homology consists
of finite-dimensional vector spaces with no further module structure, while
the modules that appear in the complex are infinite-dimensional vectors spaces
with no further module structure.

A free resolution $F^{\bullet}$ of $M$ is _${\tilde{V}}$ -strict_ if each
$F^{i}$ is equipped with a ${\tilde{V}}$-filtration ${\tilde{V}}(F^{i})$ such
that every differential $\delta^{i}\colon F^{i}\longrightarrow F^{i+1}$
satisfies
$\delta^{i}({\tilde{V}}^{k}(F^{i}))\subseteq{\tilde{V}}^{k}(F^{k+1})$ and
moreover
$\delta^{i}(F^{i})\cap{\tilde{V}}^{k}(F^{i+1})=\delta^{i}({\tilde{V}}^{k}(F^{i}))$.
It is a theorem of algorithmic algebraic analysis that finitely generated
$V$-filtered $D_{n}$-modules do allow ${\tilde{V}}$-strict resolutions of
finite length. The ${\tilde{V}}$-filtration on $F^{\bullet}$ induces a
quotient filtration on $\omega\otimes_{D_{n}}F^{\bullet}$. This filtered
complex may not be strict anymore, but still the morphisms will respect the
filtration. The ${\tilde{V}}$-filtration on
$\omega_{n}\otimes_{D_{n}}F^{\bullet}$ is bounded below while on $F^{\bullet}$
it is not. Moreover, $\operatorname{\textup{gr}}^{k}_{\tilde{V}}(F^{i})$ is
infinite dimensional over $\mathbb{C}$, while each
$\operatorname{\textup{gr}}^{k}_{\tilde{V}}(\omega_{n}\otimes_{D_{n}}F^{i})$
is $\mathbb{C}$-finite. Nonetheless, the $\mathbb{C}$-dimension of each
$\omega_{n}\otimes_{D_{n}}F^{i}$ is still infinite.

Let $\ell$ be the largest and $s$ the smallest integral root of the
$b$-function $\tilde{b}_{M}(s)$.

######  Theorem 2.10 (Integration Theorem [OT99, OT01]).

With notation as introduced above, the morphisms

$\omega_{n}\otimes_{D_{n}}F^{\bullet}\hookleftarrow{\tilde{V}}^{\ell}(\omega_{n}\otimes_{D_{n}}F^{\bullet})\twoheadrightarrow{\tilde{V}}^{\ell}(\omega_{n}\otimes_{D_{n}}F^{\bullet})/{\tilde{V}}^{s-1}(\omega_{n}\otimes_{D_{n}}F^{\bullet})$

are quasi-isomorphisms.

In other words, every cohomology class of
$\operatorname{Tor}_{\bullet}^{D_{n}}(\omega_{n},M)$ has a representative
inside ${\tilde{V}}^{\ell}(\omega_{n}\otimes_{D_{n}}F^{\bullet})$, and the
complex ${\tilde{V}}^{s-1}(\omega_{n}\otimes_{D_{n}}F^{\bullet})$ is exact.

Note that the subquotient complex
${\tilde{V}}^{\ell}(\omega_{n}\otimes_{D_{n}}F^{\bullet})/{\tilde{V}}^{s-1}(\omega_{n}\otimes_{D_{n}}F^{\bullet})$
is, in contrast to $\omega_{n}\otimes F^{\bullet}$, $\mathbb{C}$-finite,
reducing the computation of $\pi_{+}(M)$ to finite-dimensional linear algebra
in this subquotient complex.

One can now just as well resolve $\omega_{n}$ and $M$, or just $\omega_{n}$,
and obtain other complexes that represent $\omega_{n}\otimes^{L}_{D_{n}}M$. A
natural resolution for $\omega_{n}$ is the cohomological Koszul complex
$K^{\bullet}$ on the left-multiplications on $D_{n}$ by the various
$\partial_{j}$. (So, $K^{\bullet}$ is the complex of global sections of
$\Omega^{\bullet}_{\mathcal{D},X}$). The module $K^{\ell}$ has a natural
generating set given by the size-$\ell$-subsets of $1,\ldots,n$. We place
these generators in ${\tilde{V}}$-level $\ell$ and extend ${\tilde{V}}$ to
each $K^{\ell}$ by $D_{n}$-linearity. Since $\partial_{i}$ is in
${\tilde{V}}$-level $-1$, this produces a ${\tilde{V}}$-strict resolution of
$\omega_{n}$. Having resolutions $K^{\bullet},F^{\bullet}$ with
${\tilde{V}}$-filtration, there is an induced ${\tilde{V}}$-filtration on
$K^{\bullet}\otimes_{D_{n}}F^{\bullet}$.

The complex $K^{\bullet}\otimes_{D_{n}}M$ is sometimes called the (affine,
global) _de Rham complex_ of $M$. If $M$ is a space of functions on which one
can differentiate, multiplication by $\partial_{i}$ in $K^{\bullet}$
corresponds to differentiation by $x_{i}$ in the usual de Rham complex.

Suppose now that $M=\bigoplus_{\ell\in\mathbb{Z}}M_{\ell}$ is a graded module
over the graded ring $D_{n}$, with homogeneous generators $m_{1},\ldots,m_{r}$
of degrees $s_{1},\ldots,s_{r}$. Choosing the degrees of the generators as
shifts (_i.e._ , $s_{i}=\deg(m_{i})$) for the ${\tilde{V}}$-filtration on $M$
one obtains a direct sum of the graded components of $M$,

(2.3.3)
$\displaystyle{\tilde{V}}^{k}(M)=\sum_{i=1}^{r}{\tilde{V}}^{k-s_{i}}(D_{n})\cdot
m_{i}=\bigoplus_{\ell\leq k}M_{\ell}.$

Since the twisted Euler operator $\tilde{E}=-\sum_{j=1}^{n}\partial_{j}x_{j}$
is ${\tilde{V}}$-homogeneous of degree zero, the defining equation (2.3.2)
becomes

$\tilde{b}_{M}(\tilde{E}+n+k)\cdot M_{k}=0$

for all $k\in\mathbb{Z}$.

For ${\tilde{V}}$-graded $M$ one can arrange the resolution $F^{\bullet}$ to
respect this grading, and $K^{\bullet}$ is graded in any case. If now
$\eta_{F}$ is a cohomology class generator in
$H^{i}(\omega_{n}\otimes_{D_{n}}F^{\bullet})$, one can lift it into
$K^{n}\otimes_{D_{n}}F^{i}$ and then chase it into a class $\eta_{K}$ of
$K^{\bullet}\otimes_{D_{n}}M$, since Tor is a balanced functor. The grading of
the resolutions involved implies that the ${\tilde{V}}$-level of this class in
$K^{\bullet}\otimes_{D_{n}}M$ is the same as the ${\tilde{V}}$-level of
$\eta_{F}$ in $\omega_{n}\otimes_{D_{n}}F^{\bullet}$.

We recall the notion of an Eulerian $D_{n}$-module.

######  Definition 2.11 ([MZ14]).

The graded $D_{n}$-module $M=\bigoplus_{i\in\mathbb{Z}}M_{i}$ is _Eulerian_ if
for every homogeneous $m\in M_{i}$ one has
$(\sum_{j=1}^{n}x_{j}\partial_{j})m=i\cdot m$.

In terms of $\tilde{E}$ this is equivalent to $(\tilde{E}+n+\deg(m))m=0$.
$\diamond$

Eulerian $D_{n}$-modules are a very special case of Brylinski’s _monodromic_
modules, which are those on which the Euler operator has a minimal polynomial.
They include (iterated) local cohomology modules
$H^{i_{1}}_{I_{1}}(\ldots(H^{i_{k}}_{I_{k}}(R_{n})\ldots)$ for homogeneous
ideals $I_{1},\ldots,I_{k}$.

######  Proposition 2.12.

Let $M$ be a finitely generated Eulerian $D_{n}$-module. Then every nonzero
cohomology class of $\omega_{n}\otimes^{L}_{D_{n}}M$ has degree zero.

###### Proof.

Since the module is Eulerian, we have $(\tilde{E}+n+\deg(m))m=0$ for every
homogeneous $m\in M$. We put the ${\tilde{V}}$-filtration on $M$ that is
induced by a finite set of homogeneous generators as in (2.3.3), with shifts
$s_{i}=\deg(m_{I})$. Then, a $b$-function for integration is given by
$\tilde{b}(s)=s$. The conclusion is immediate from the Integration Theorem
2.10. ∎

######  Remark 2.13.

Let us call _quasi-Eulerian_ a graded monodromic $D_{n}$-module $M$. Then one
can easily generalize Proposition 2.12 to: if $M$ is quasi-Eulerian then the
degree of every cohomology class of $\omega_{n}\otimes^{L}_{D_{n}}M$ must be
an integral root of the minimal polynomial of $\tilde{E}$ on $M$.

There is a version of the Integration Theorem for complexes of holonomic
modules (more generally, for complexes that have a $b$-function for
integration), see [Wal00]. This allows a further generalization to finite
graded complexes with quasi-Eulerian cohomology. $\diamond$

We now consider the Eulerian $D_{n}$-module that arises as the localization
$M=R_{n}[1/f]$ of $R_{n}$ at a homogeneous polynomial $f$. It is clear that
this is an Eulerian module since the Euler operator $E$ acts on a rational
homogeneous function of degree $k$ by multiplication with $k$. Thus,
$K^{\bullet}\otimes_{D_{n}}M$ is ${\tilde{V}}$-graded and every class in
$\omega_{n}\otimes^{L}_{D_{n}}M$ has native degree zero.

If one reads elements of $K^{\ell}\otimes M$ as differential $\ell$-forms on
$M$, this implies that the cohomology of $K^{\bullet}\otimes_{D_{n}}M$ is
spanned as vector space by differential forms of degree zero: forms of the
type

$\sum_{|I|=\ell\atop
I\subseteq\\{1,\ldots,n\\}}\frac{g_{I}{\mathrm{d}}x_{I}}{f^{k_{I}}}$

where ${\mathrm{d}}x_{I}=\wedge_{i\in I}{\mathrm{d}}x_{i}$, where $g_{I}$ is a
homogeneous element of $R_{n}$, and where
$\deg(g_{I})+\ell=k_{I}\cdot\deg(f)$. Similarly, integrating a graded complex
$M^{\bullet}$ with Eulerian cohomology modules yields a de Rham complex of
$M^{\bullet}$ with cohomology groups concentrated in degree zero.

######  Corollary 2.14.

If $I$ is a homogeneous ideal then the de Rham cohomology of the affine
complement $U(I)=X\smallsetminus{\mathrm{Var}}(I)$ of the affine variety
${\mathrm{Var}}(I)\subseteq X:=\mathbb{C}^{n}$ is generated by chains of
differential forms of degree zero. Moreover, the de Rham cohomology groups
$H^{i}_{\mathrm{dR}}(H^{j}_{I}(R))$ all are concentrated in degree zero. ∎

###### Proof.

The Grothendieck comparison theorem asserts that the cohomology of
$K^{\bullet}\otimes_{D_{n}}\check{C}^{\bullet}$ is the de Rham cohomology of
$U(I)$. The rest follows from Proposition 2.12. ∎

######  Remark 2.15.

Since multiplication by $\mathbb{C}\ni\lambda\neq 0$ is an isomorphism on
$U(f)$, the de Rham cohomology of $U(f)$ of a divisor is spanned by
homogeneous differential forms (homothety eigenvectors) for all homogeneous
$f\in R_{n}$. Alex Dimca pointed out that path-connectedness of
$\mathbb{C}^{*}$ implies that this multiplication is in fact homotopy
equivalent to the identity, and thus does not change the class. Hence, the
cohomology of $U(f)$ must be eigenvectors to eigenvalue 1, and thus of degree
zero. $\diamond$

###  2.4. On Veronese maps

Throughout this subsection, $2\leq d,n\in\mathbb{N}$. Let

$v_{n}^{d}\colon
X=\mathbb{A}^{n}_{\mathbb{C}}\longrightarrow\mathbb{A}^{N}_{\mathbb{C}}=:W$

be the $d$-th Veronese morphism on the affine level, so $N={n+d-1\choose
n-1}$. If $n,d$ are understood, we abbreviate $v_{n}^{d}$ to just $v$. We set
$X^{\prime}:=v(X)\subseteq W$, $W^{\circ}=W\smallsetminus\\{0\\}$,
$X^{\circ}:=X\smallsetminus\\{0\\}$ and
${X^{\prime}}^{\circ}:=X^{\prime}\smallsetminus\\{0\\}$.

Let $R_{n}=\mathbb{C}[x_{1},\ldots,x_{n}]=\mathcal{O}_{X}(X)$ and
$R_{N}=\mathbb{C}[\\{y_{S}\mid
S\in\mathbb{N}^{n},|S|=d\\}]=\mathcal{O}_{W}(W)$. Let $I\subseteq R_{n}$ be a
homogeneous ideal and $Y$ the associated variety. Denote $U$ the complement
$X\smallsetminus Y$, and let $Y^{\prime},U^{\prime}$ the images of $Y,U$ under
$v$. Let $U_{W}$ be the complement $W\smallsetminus v(Y)$. We wish to compare
here the cohomology of the affine complements of $Y$ and $Y^{\prime}$.

Note that $v^{\\#}\colon R_{N}\longrightarrow R_{n}$ sends $y_{S}\mapsto
x^{S}$ in multi-index notation. The $d$-th roots of unity
${\makebox[7.04979pt]{$\mu\mu$}}_{d}$ act diagonally on $X$, as well as on
every other variety of a homogeneous ideal of $R_{n}$, by multiplication on
each $x_{i}$. Moreover, $v$ is the orbit map to this action, followed by
inclusion into $W$. The image of $v$ has a unique isolated singularity at the
origin, and $v$ is a $d:1$ covering of ${X^{\prime}}^{\circ}$ by $X^{\circ}$.

Note that ${\makebox[7.04979pt]{$\mu\mu$}}_{d}$ is the covering group of the
map $U\longrightarrow U^{\prime}$, and its order $d$ is nonzero in
$\mathbb{C}$. Under these circumstances, $H^{\bullet}(U^{\prime};\mathbb{C})$
is the group of ${\makebox[7.04979pt]{$\mu\mu$}}_{d}$-invariants in
$H^{\bullet}(U;\mathbb{C})$. Using the de Rham manifestation of
$H^{\bullet}(U;\mathbb{C})$, in which we showed that every class has a
representative that is of degree zero, the entire space
$H^{\bullet}(U;\mathbb{C})$ is
${\makebox[7.04979pt]{$\mu\mu$}}_{d}$-invariant, so that

$H^{\bullet}_{\mathrm{dR}}(U;\mathbb{C})=H^{\bullet}_{\mathrm{dR}}(U^{\prime};\mathbb{C}).$

In what follows, we replace de Rham cohomology by singular cohomology, since
we will have need to step outside the category of smooth algebraic varieties.
Known comparison theorems over $\mathbb{C}$ assure functorial isomorphisms
between these cohomology theories whenever both exist.

It will turn out to be useful to know the cohomology of $W\smallsetminus
X^{\prime}=W^{\circ}\smallsetminus{X^{\prime}}^{\circ}$. Note that
${X^{\prime}}^{\circ}$ has the homology of the homotopy $(2n-1)$-sphere
$X^{\circ}$ and is a closed submanifold of the $2N$-dimensional manifold
$W^{\circ}$, the latter being homotopy equivalent to the $(2N-1)$-sphere
$\SS^{2N-1}$. Alexander duality gives an isomorphism
$H^{i}_{{X^{\prime}}^{\circ}}(W^{\circ};\mathbb{C})\simeq\operatorname{\textup{Hom}}_{\mathbb{C}}(H^{2N-i}_{c}({X^{\prime}}^{\circ};\mathbb{C}),\mathbb{C})$
with the dual of compactly supported cohomology, [Ive86, Alexander Duality
V.6.6]. But then ${X^{\prime}}^{\circ}$ being a $2n$-dimensional real manifold
yields by Poincaré duality that
$\operatorname{\textup{Hom}}_{\mathbb{C}}(H^{2N-i}_{c}({X^{\prime}}^{\circ};\mathbb{C}),\mathbb{C})\simeq
H^{2n-2N+i}({X^{\prime}}^{\circ};\mathbb{C})$, [BT82, I.(5.4)]. The latter is
$\mathbb{C}$ for $i=2N-1$ and $i=2N-2n$, and zero otherwise. In the long exact
sequence

$\cdots\longrightarrow
H^{i}_{{X^{\prime}}^{\circ}}(W^{\circ};\mathbb{C})\longrightarrow
H^{i}(W^{\circ};\mathbb{C})\longrightarrow
H^{i}(W^{\circ}\smallsetminus{X^{\prime}}^{\circ};\mathbb{C})\stackrel{{\scriptstyle+1}}{{\longrightarrow}},$

we have $H^{i}_{{X^{\prime}}^{\circ}}(W^{\circ};\mathbb{C})\neq 0$ only when
$i=2N-1,2(N-n)$ and $H^{i}(W^{\circ};\mathbb{C})\neq 0$ only if $i=2N-1,0$.
The map
$\mathbb{C}=H^{2N-1}_{{X^{\prime}}^{\circ}}(W^{\circ};\mathbb{C})\longrightarrow
H^{2N-1}(W^{\circ};\mathbb{C})=\mathbb{C}$ is surjective (hence bijective)
since $W^{\circ}\smallsetminus{X^{\prime}}^{\circ}$ is homotopy equivalent to
an open subset of a $(2N-1)$-sphere) and so
$H^{2N-1}(W^{\circ}\smallsetminus{X^{\prime}}^{\circ};\mathbb{C})=0$. It
follows that

(2.4.3) $\displaystyle H^{i}(W\smallsetminus{X^{\prime}};\mathbb{C})$
$\displaystyle=$ $\displaystyle
H^{i}(W^{\circ}\smallsetminus{X^{\prime}}^{\circ};\mathbb{C})=\left\\{\begin{array}[]{ccc}\mathbb{C}&\text{
if }&i=0,2(N-n)-1;\\\ 0&&\text{else.}\end{array}\right.$

Next we compute the cohomology of $U_{W}=W\smallsetminus
Y^{\prime},Y^{\prime}=v(Y)$ where $Y={\mathrm{Var}}(I)$ for some homogeneous
ideal $I\subseteq R_{n}$. Since $U^{\prime}=v(U)$ is an embedded submanifold
of $U_{W}$ with complex codimension $N-n$, we can consider the tubular
neighborhood $T^{\prime}$ of $U^{\prime}$ that arises via the tubular
neighborhood theorem as the total space of the normal bundle of $U^{\prime}$
in $U_{W}$. Then

$U_{W}=W\smallsetminus Y^{\prime}=(W\smallsetminus X^{\prime})\cup
U^{\prime}=(W\smallsetminus X^{\prime})\cup T^{\prime},$

with intersection $(W\smallsetminus X^{\prime})\cap
T^{\prime}={T^{\prime}}^{\circ}$.

As $U^{\prime},U_{W}$ are complex manifolds, the removal of the zero section
$U^{\prime}$ from $T^{\prime}$ leaves a space ${T^{\prime}}^{\circ}$ homotopic
to an oriented sphere bundle
$\SS^{q}\hookrightarrow{T^{\prime}}^{\circ}\twoheadrightarrow U^{\prime}$
where

$q=2(N-n)-1.$

The $q$-sphere bundle $T^{\circ}$ yields a Gysin sequence

$\ldots\longrightarrow
H^{i}({T^{\prime}}^{\circ};\mathbb{C})\stackrel{{\scriptstyle\pi_{*}}}{{\longrightarrow}}H^{i-q}(U^{\prime};\mathbb{C})\stackrel{{\scriptstyle
e\wedge}}{{\longrightarrow}}H^{i+1}(U^{\prime};\mathbb{C})\stackrel{{\scriptstyle\pi^{*}}}{{\longrightarrow}}H^{i+1}({T^{\prime}}^{\circ};\mathbb{C})\longrightarrow\ldots$

Here, $\pi\colon{T^{\prime}}^{\circ}\longrightarrow U^{\prime}$ is the
fibration map, $\pi^{*}$ is the pullback under this map, and $e$ is the Euler
class of the bundle ${T^{\prime}}^{\circ}$ when restricted from relative
cohomology to absolute cohomology on $T$. The map $\pi_{*}$ is special to the
situation of bundles with fibers homotopic to compact manifolds, and is
induced by integration along the fibers in the following sense. For any
oriented $\mathbb{R}^{k}$-bundle $E\longrightarrow B$ with
$E^{\circ}=E\smallsetminus B$ there is a fundamental class $u\in
H^{k}(E,E^{\circ};\mathbb{Z})$ that restricts in each fiber to the canonical
class in $H^{k}(\mathbb{R}^{k},\mathbb{R}^{k}\setminus\\{0\\};\mathbb{Z})$;
this canonical class is the given orientation on the bundle (an orientation is
a global section of the orientation bundle with fiber
$H^{k}(\mathbb{R}^{k},\mathbb{R}^{k}\setminus\\{0\\};\mathbb{Z})$). The
existence of the fundamental class is the content of the Thom isomorphism
theorem for oriented vector bundles, and the cup product with $u$ sets up an
isomorphism $u\cup\colon H^{j}(E;\mathbb{Z})\longrightarrow
H^{j+k}(E,E^{\circ};\mathbb{Z})$. The cap product with the Poincaré dual of
$u$ induces an isomorphism $H_{j}(E,E^{\circ};\mathbb{Z})\longrightarrow
H_{j-k}(E;\mathbb{Z})$, the “integration along the fibers” above (compare
[MS74, Ch. 9-12]). The image of $u$ in $H^{k}(E;\mathbb{Z})$ is the Euler
class (by definition). If the fiber dimension $k$ is large,
$H^{k}(E;\mathbb{Z})=H^{k-1}(E;\mathbb{Z})=0$. In that case, the Euler class
of the bundle must be zero and then $u$ corresponds to the class in
$H^{k-1}(E^{\circ};\mathbb{Z})$ with the property that it restricts in each
fiber to the canonical generator of
$H^{k-1}(\mathbb{R}^{k}\smallsetminus\\{0\\};\mathbb{Z})$.

Our Gysin sequence above arises from the long exact sequence to the pair
$(T^{\prime},{T^{\prime}}^{\circ})$ with replacements coming from the Thom
isomorphism and the fact that $U^{\prime},T^{\prime}$ are homotopic.

Since $\dim_{\mathbb{C}}(U^{\prime})=n$, $H^{i}(U^{\prime};\mathbb{C})=0$ if
$i\geq 2n$. On the other hand, the Euler class is of homological degree
$q+1=2(N-n)$. Thus, if $2(N-n)\geq 2n$ then either the source or the target of
the Euler map $H^{i-q}(U^{\prime};\mathbb{C})\stackrel{{\scriptstyle
e\wedge}}{{\longrightarrow}}H^{i+1}(U^{\prime};\mathbb{C})$ is zero for every
$i$. But $N={n+d-1\choose n-1}\geq 2n$ for $n,d\geq 2$ unless $d=n=2$, and
usually much larger. Thus the Gysin sequence splits into isomorphisms

(2.4.4) $\displaystyle H^{i}({T^{\prime}}^{\circ};\mathbb{C})$
$\displaystyle\stackrel{{\scriptstyle\pi_{*}}}{{\longrightarrow}}H^{i-q}(U^{\prime};\mathbb{C})=H^{i}(T^{\prime};\mathbb{C})$
$\displaystyle\text{ if }i\geq 2n;$ (2.4.5) $\displaystyle
H^{i}(T^{\prime};\mathbb{C})=H^{i}(U^{\prime};\mathbb{C})$
$\displaystyle\stackrel{{\scriptstyle\pi^{*}}}{{\longrightarrow}}H^{i}({T^{\prime};\mathbb{C}}^{\circ})$
$\displaystyle\text{ if }i<2n.$

Note that the composition $H^{i}(T^{\prime};\mathbb{C})\longrightarrow
H^{i}({T^{\prime}}^{\circ};\mathbb{C})\longrightarrow
H^{i}(U^{\prime};\mathbb{C})$ is an isomorphism since
$U^{\prime}\hookrightarrow T^{\prime}$ is a homotopy equivalence; so the left
map is an isomorphism if and only if the right one is. Now consider the
Mayer–Vietoris sequence to the pair $(W\smallsetminus X^{\prime})\cup
T^{\prime}=U_{W}$ with ${T^{\prime}}^{\circ}=(W\smallsetminus X^{\prime})\cap
T^{\prime}$:

$\cdots\longrightarrow H^{i}((W\smallsetminus X^{\prime})\cup
T^{\prime};\mathbb{C})\longrightarrow H^{i}(W\smallsetminus
X^{\prime};\mathbb{C})\oplus H^{i}(T^{\prime};\mathbb{C})\longrightarrow
H^{i}({T^{\prime}}^{\circ};\mathbb{C})\longrightarrow\cdots$

Here, each (component of a) map is the natural restriction, possibly with a
$(-1)$ factor.

If $i<2n$, the map $H^{i}(T^{\prime};\mathbb{C})\longrightarrow
H^{i}({T^{\prime}}^{\circ};\mathbb{C})$ in the Mayer–Vietoris sequence is
therefore the identity by (2.4.5). It follows that in this range,
$H^{i}((W\smallsetminus X^{\prime})\cup T^{\prime};\mathbb{C})\longrightarrow
H^{i}(W\smallsetminus X^{\prime};\mathbb{C})$ is an isomorphism as well. But
in that range, by (2.4.3), only $H^{0}(W\smallsetminus X^{\prime};\mathbb{C})$
is nonzero and so $H^{i}((W\smallsetminus X^{\prime})\cup
T^{\prime};\mathbb{C})$ is zero for $0<i<2n$.

If $2n-1\leq i<q$, then $H^{i}((W\smallsetminus X^{\prime})\cup
T^{\prime};\mathbb{C})$ vanishes since $H^{i}(W\smallsetminus
X^{\prime};\mathbb{C})=H^{i}(T^{\prime};\mathbb{C})=H^{i}({T^{\prime}}^{\circ};\mathbb{C})=0$.

Let us look at the situation when $i=q$:

$\displaystyle\underbrace{H^{q-1}({T^{\prime}}^{\circ};\mathbb{C})}_{=H^{-1}(U^{\prime};\mathbb{C})=0}\longrightarrow
H^{q}((W\smallsetminus X^{\prime})\cup
T^{\prime};\mathbb{C})\longrightarrow\underbrace{H^{q}(W\smallsetminus
X^{\prime};\mathbb{C})}_{=\mathbb{C}}\oplus\underbrace{H^{q}(T^{\prime};\mathbb{C})}_{=0}\longrightarrow\qquad$
$\displaystyle\underbrace{H^{q}({T^{\prime}}^{\circ};\mathbb{C})}_{=H^{0}(U^{\prime};\mathbb{C})=\mathbb{C}}\longrightarrow
H^{q+1}((W\smallsetminus X^{\prime})\cup
T^{\prime};\mathbb{C})\longrightarrow{\underbrace{H^{q+1}(W\smallsetminus
X^{\prime};\mathbb{C})}_{=0}\oplus\underbrace{H^{q+1}(T^{\prime};\mathbb{C})}_{=0}}.$

If one restricts the morphism $H^{q}(W\smallsetminus
X^{\prime};\mathbb{C})\longrightarrow H^{q}({T^{\prime}}^{\circ};\mathbb{C})$
to the intersection with a small ball around a generic point of $Y^{\prime}$,
both spaces become homotopic to $\SS^{q}$ and so the morphism
$H^{q}(W\smallsetminus X^{\prime};\mathbb{C})\longrightarrow
H^{q}({T^{\prime}}^{\circ};\mathbb{C})$ restricts to an isomorphism
$\mathbb{C}\longrightarrow\mathbb{C}$. But since $H^{q}(W\smallsetminus
X^{\prime};\mathbb{C})$ and $H^{q}({T^{\prime}}^{\circ};\mathbb{C})$ are also
equal to $\mathbb{C}$, the morphism $H^{q}(W\smallsetminus
X^{\prime};\mathbb{C})\longrightarrow H^{q}({T^{\prime}}^{\circ};\mathbb{C})$
is an isomorphism. Thus,
$H^{q}(U_{W};\mathbb{C})=H^{q+1}(U_{W};\mathbb{C})=0$.

If $i>q$, $H^{i}(T^{\prime};\mathbb{C})=H^{i}(W\smallsetminus
X^{\prime};\mathbb{C})=0$. Thus,
$H^{i-q}(U^{\prime};\mathbb{C})=H^{i}({T^{\prime}}^{\circ};\mathbb{C})=H^{i+1}((W\smallsetminus
X^{\prime})\cup T^{\prime};\mathbb{C})$.

We have proved

######  Proposition 2.16.

We use notation as defined at the start of Subsection 2.4. Let
$T^{\prime}\longrightarrow U^{\prime}$ be the normal bundle of $U^{\prime}$ in
$W^{\circ}$. With $U_{W}:=W\setminus Y^{\prime}=(W\smallsetminus
X^{\prime})\cup T^{\prime}$, and $q={n+d-1\choose n-1}>2n$ we have on the
level of reduced cohomology for every $i\in\mathbb{Z}$ the isomorphisms

---
$\textstyle{H^{i}(U^{\prime};\mathbb{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\pi^{*}}}$$\scriptstyle{\simeq}$$\textstyle{H^{i}(T^{\prime};\mathbb{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e\cup}$$\scriptstyle{\simeq}$$\scriptstyle{e_{0}\cup}$$\scriptstyle{\simeq}$$\textstyle{H^{i+q}({T^{\prime}}^{\circ};\mathbb{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta^{*}}$$\scriptstyle{\simeq}$$\scriptstyle{\delta^{*}}$$\scriptstyle{\simeq}$$\textstyle{H^{i+q+1}(U_{W};\mathbb{C})}$$\textstyle{H^{i+q+1}(T^{\prime},{T^{\prime}}^{\circ};\mathbb{C})}$

Here $e\in H^{q+1}(T^{\prime},{T^{\prime}}^{\circ};\mathbb{Z})$ is the Euler
class of the bundle, $e_{0}\in H^{q}({T^{\prime}}^{\circ};\mathbb{Z})$ is its
preimage, the vertical $\delta^{*}$ is the connecting morphism for the pair
$(T^{\prime},{T^{\prime}}^{\circ})$, and the horizontal $\delta^{*}$ is the
connecting morphism for the Mayer–Vietoris spectral sequence for the cover
$U_{W}=(W\smallsetminus X^{\prime})\cup T^{\prime}$.

In particular, the singular reduced cohomology groups of the complements of
the cones $Y$ and $Y^{\prime}$ over $\tilde{Y}$ are the same up to a
cohomological shift by $q+1=2(N-n)$.

######  Corollary 2.17.

If $Y\subseteq X=\mathbb{C}^{n}$ is homogeneous and of equi-dimension three,
then the Čech–de Rham numbers $\rho_{p,q}^{r}(Y)$ are invariant under Veronese
maps of $Y$.

###### Proof.

With $Y^{\prime}\subseteq W$ and notation as in Subsection 2.4, the Čech–de
Rham spectral sequence degenerates for dimensional reasons by Proposition 2.5.
According to Proposition 2.16, the two complements have the same reduced
cohomology up to a shift by the relative dimension. Hence, up to that same
shift, the two Čech–de Rham spectral sequences have the same abutment. The
degeneration shows that the abutment determines the $\rho^{2}_{p,q}$, except
for the numbers $\rho^{2}_{2,2}+\rho^{2}_{3,0}=\dim H^{n-3}_{\mathrm{dR}}(U)$.
However, $\rho^{2}_{2,2}$ is the dimension of
$H^{n-2}_{\mathrm{dR}}(H^{n-2}_{I}(R))$ and thus equals the socle dimension of
$H^{2}_{\mathfrak{m}}(\mathbb{D}H^{n-2}_{I}(R))$ by Lemma 1.4. But equi-
dimensionality and Remark 1.3 show that
$H^{2}_{\mathfrak{m}}(\mathbb{D}H^{n-2}_{I}(R))=0$. This settles the case
$r=2$. But no higher nonzero differentials can exist by degeneration. ∎

It turns out that, similarly to the corresponding result on Lyubeznik numbers
in [RSW21], the Čech–de Rham numbers of level two do not change under Veronese
maps of projective varieties.

######  Theorem 2.18.

Let $\tilde{Y}$ be a projective variety and suppose $Y$ is a cone for
$\tilde{Y}$, embedded into an affine space $X$. Then the Čech–de Rham numbers
$\rho^{2}_{k,\ell}$ do for $k\geq 2$ not depend on $Y$ but are a function of
$\tilde{Y}$ alone.

The overall plan of the proof is quite similar to the proof in [RSW21] that
the $\lambda_{k,\ell}$ are (largely) unchanged under Veronese maps. We start
with translating the $\rho_{k,\ell}$ into objects of constructible sheaves
involving the Verdier dual $\omega_{Y}$ of the constant sheaf on $Y$. After
some rewriting we use Lemma 1.4 to exchange a direct image functor to a point
for the pullback to the origin and then to use an adjunction triangle to
reformulate them in terms of $Y^{\circ}$. We finally lift to $\tilde{Y}$ on
which one uses an interpretation in terms of Chern classes.

During the proof we shall use the following diagram of maps

---
$\textstyle{\\{0\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{Y}}$$\textstyle{\\{0\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{Y}}$$\scriptstyle{h}$$\textstyle{\mathbb{C}^{n}=:X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{X}}$$\textstyle{Y_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{Y}}$$\scriptstyle{\pi}$$\scriptstyle{h_{0}}$$\textstyle{\mathbb{C}^{n}\setminus\\{0\\}=:X^{\circ}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\scriptstyle{p}$$\textstyle{\tilde{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{\mathbb{P}^{n-1}}$

###### Proof.

The Čech–deRham numbers of level 2 are given by

$\rho_{k,\ell}:=\dim
H^{n-k}_{{\mathrm{dR}}}H^{n-\ell}_{I}R=\dim{{}^{p}}\mathcal{H}^{-k}a_{*}({{}^{p}}\mathcal{H}^{n-\ell}h_{!}h^{!}\underline{\mathbb{C}}_{X}[n]).$

(The shift from $-n+k$ on the left to $-k$ on the right occurs since the de
Rham functor used in the Riemann–Hilbert correspondence arises from the
“natural” algebraic de Rham functor—which goes along with the tensor product
with $\omega$—by analytification _and_ a shift by $n$).

We have

$\displaystyle{{}^{p}}\mathcal{H}^{-k}a_{*}({{}^{p}}\mathcal{H}^{n-\ell}h_{!}h^{!}\underline{\mathbb{C}}_{X}[n])$
$\displaystyle\stackrel{{\scriptstyle({\rm a})}}{{\simeq}}$
$\displaystyle{{}^{p}}\mathcal{H}^{-k}a_{*}({{}^{p}}\mathcal{H}^{n-\ell}h_{!}h^{!}a_{X}^{!}\mathbb{C}_{\mathit{pt}}[-n])$
$\displaystyle\stackrel{{\scriptstyle({\rm b})}}{{\simeq}}$
$\displaystyle{{}^{p}}\mathcal{H}^{-k}a_{*}({{}^{p}}\mathcal{H}^{n-\ell}h_{!}\omega_{Y}[-n])$
$\displaystyle\stackrel{{\scriptstyle({\rm c})}}{{\simeq}}$
$\displaystyle{{}^{p}}\mathcal{H}^{-k}a_{*}h_{!}({{}^{p}}\mathcal{H}^{n-\ell}\omega_{Y}[-n])$
$\displaystyle\stackrel{{\scriptstyle({\rm d})}}{{\simeq}}$
$\displaystyle{{}^{p}}\mathcal{H}^{-k}a_{*}h_{*}({{}^{p}}\mathcal{H}^{n-\ell}\omega_{Y}[-n])$
$\displaystyle{\simeq}$
$\displaystyle{{}^{p}}\mathcal{H}^{-k}(a_{Y})_{*}({{}^{p}}\mathcal{H}^{n-\ell}\omega_{Y}[-n])$
$\displaystyle{\simeq}$
$\displaystyle{{}^{p}}\mathcal{H}^{-k}(a_{Y})_{*}({{}^{p}}\mathcal{H}^{-\ell}\omega_{Y})$
$\displaystyle\stackrel{{\scriptstyle({\rm e})}}{{\simeq}}$
$\displaystyle{{}^{p}}\mathcal{H}^{-k}(i_{Y})^{-1}({{}^{p}}\mathcal{H}^{-\ell}\omega_{Y})$

The justfications are as follows: (a) holds since the real dimension of $X$ is
$2n$; (b) follows from the definition of $\omega_{Y}$; (c) holds since $h$ is
a closed embedding and hence $h_{!}$ is perverse exact; (d) comes from
$h_{*}=h_{!}$ for closed embeddings; (e) is Lemma 3.3 in [RW18].

We have the following triangle from the inclusion of the origin into $Y$:

$j_{Y!}j_{Y}^{-1}({{}^{p}}\mathcal{H}^{-\ell}\omega_{Y})\longrightarrow{{}^{p}}\mathcal{H}^{-\ell}\omega_{Y}\longrightarrow
i_{Y!}i_{Y}^{-1}({{}^{p}}\mathcal{H}^{-\ell}\omega_{Y})\overset{+1}{\longrightarrow}$

and it induces the following long exact sequence

$\textstyle{\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{}^{p}}\mathcal{H}^{-3}i_{Y!}(i_{Y})^{-1}({{}^{p}}\mathcal{H}^{-\ell}\omega_{Y})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{}^{p}}\mathcal{H}^{-2}j_{Y!}(j_{Y})^{-1}({{}^{p}}\mathcal{H}^{-\ell}\omega_{Y})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{}^{p}}\mathcal{H}^{-2}i_{Y!}(i_{Y})^{-1}({{}^{p}}\mathcal{H}^{-\ell}\omega_{Y})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{}^{p}}\mathcal{H}^{-1}j_{Y!}(j_{Y})^{-1}({{}^{p}}\mathcal{H}^{-\ell}\omega_{Y})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{}^{p}}\mathcal{H}^{-1}i_{Y!}(i_{Y})^{-1}({{}^{p}}\mathcal{H}^{-\ell}\omega_{Y})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{}^{p}}\mathcal{H}^{0}j_{Y!}(j_{Y})^{-1}({{}^{p}}\mathcal{H}^{-\ell}\omega_{Y})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{}^{p}}\mathcal{H}^{-\ell}\omega_{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{}^{p}}\mathcal{H}^{0}i_{Y!}(i_{Y})^{-1}({{}^{p}}\mathcal{H}^{-\ell}\omega_{Y})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

We now use that $k\geq 2$, which yields the following isomorphisms from the
long exact sequence above:

$\displaystyle{{}^{p}}\mathcal{H}^{-k}i_{Y!}(i_{Y})^{-1}({{}^{p}}\mathcal{H}^{-\ell}\omega_{Y})$
$\displaystyle\simeq$
$\displaystyle{{}^{p}}\mathcal{H}^{-k+1}j_{Y!}(j_{Y})^{-1}({{}^{p}}\mathcal{H}^{-\ell}\omega_{Y})$
$\displaystyle\stackrel{{\scriptstyle({\rm f})}}{{\simeq}}$
$\displaystyle{{}^{p}}\mathcal{H}^{-k+1}j_{Y!}({{}^{p}}\mathcal{H}^{-\ell}\omega_{Y^{\circ}})$
$\displaystyle\stackrel{{\scriptstyle({\rm g})}}{{\simeq}}$
$\displaystyle{{}^{p}}\mathcal{H}^{-k+1}j_{Y!}({{}^{p}}\mathcal{H}^{-\ell}\pi^{!}\omega_{\tilde{Y}})$
$\displaystyle{\simeq}$
$\displaystyle{{}^{p}}\mathcal{H}^{-k+1}j_{Y!}({{}^{p}}\mathcal{H}^{-\ell+1}\pi^{!}[-1]\omega_{\tilde{Y}})$
$\displaystyle\stackrel{{\scriptstyle({\rm h})}}{{\simeq}}$
$\displaystyle{{}^{p}}\mathcal{H}^{-k}j_{Y!}\pi^{!}({{}^{p}}\mathcal{H}^{-\ell+1}\omega_{\tilde{Y}})$

with justifications as follows: (f) since $j_{Y}$ is open and so
$(j_{Y})^{-1}$ is perverse exact; (g) is dual to the fact that
$\mathbb{C}_{Y^{\circ}}=\pi^{-1}\mathbb{C}_{\tilde{Y}}$; (h) is because $\pi$
is smooth so that $\pi^{!}[-1]$ is perverse exact.

We have then

$\displaystyle{{}^{p}}\mathcal{H}^{i}a_{Y!}{{}^{p}}\mathcal{H}^{-k}(j_{Y!}\pi^{!}({{}^{p}}\mathcal{H}^{-\ell+1}\omega_{\tilde{Y}}))\simeq{{}^{p}}\mathcal{H}^{i}a_{Y!}({{}^{p}}\mathcal{H}^{-k}(i_{Y!}(i_{Y})^{-1}({{}^{p}}\mathcal{H}^{-\ell}\omega_{Y})))=0\qquad\text{for}\quad
i\neq 0$

since
${{}^{p}}\mathcal{H}^{-n+k}i_{Y!}(i_{Y})^{-1}({{}^{p}}\mathcal{H}^{-\ell}\omega_{Y})$
is at most supported on a point. A spectral sequence argument shows therefore
that

(2.4.8)
$\displaystyle{{}^{p}}\mathcal{H}^{0}a_{Y!}{{}^{p}}\mathcal{H}^{-k}j_{Y!}\pi^{!}({{}^{p}}\mathcal{H}^{-\ell+1}\omega_{\tilde{Y}})\simeq{{}^{p}}\mathcal{H}^{-k}a_{Y!}j_{Y!}\pi^{!}({{}^{p}}\mathcal{H}^{-\ell+1}\omega_{\tilde{Y}}).$

Summarizing we have for $k\geq 2$ that

$\displaystyle{{}^{p}}\mathcal{H}^{-k}a_{*}({{}^{p}}\mathcal{H}^{n-\ell}h_{!}h^{!}\underline{\mathbb{C}}_{V}[n])$
$\displaystyle\stackrel{{\scriptstyle({\rm i})}}{{\simeq}}$
$\displaystyle{{}^{p}}\mathcal{H}^{-k}(i_{Y})^{-1}({{}^{p}}\mathcal{H}^{-\ell}\omega_{Y})$
$\displaystyle\stackrel{{\scriptstyle({\rm j})}}{{\simeq}}$
$\displaystyle{{}^{p}}\mathcal{H}^{0}a_{Y!}i_{Y!}({{}^{p}}\mathcal{H}^{-k}(i_{Y})^{-1}({{}^{p}}\mathcal{H}^{-\ell}\omega_{Y}))$
$\displaystyle\stackrel{{\scriptstyle({\rm k})}}{{\simeq}}$
$\displaystyle{{}^{p}}\mathcal{H}^{0}a_{Y!}({{}^{p}}\mathcal{H}^{-k}i_{Y!}(i_{Y})^{-1}({{}^{p}}\mathcal{H}^{-\ell}\omega_{Y}))$
(2.4.10) $\displaystyle\stackrel{{\scriptstyle({\rm l})}}{{\simeq}}$
$\displaystyle{{}^{p}}\mathcal{H}^{-k}a_{Y!}j_{Y!}\pi^{!}({{}^{p}}\mathcal{H}^{-\ell+1}\omega_{\tilde{Y}}),$

where (i) follows from display (2.4); (j) follows since $a_{Y}\circ i_{Y}$ is
the identity on $\\{0\\}$; (k) is since $i_{Y!}$ is perverse exact as $i_{Y}$
is closed; (l) comes from displays (2.4) and (2.4.8).

Now let $\mathcal{L}$ be the quasi-coherent pullback of
$\mathcal{O}_{\mathbb{P}^{n-1}}(1)$ via $g$. By abuse of notation we denote
the total space of the corresponding line bundle by the same letter. Notice
that $Y^{\circ}\simeq\mathcal{L}\smallsetminus\\{\text{zero section}\\}$.
Consider the following diagram

$\textstyle{\\{0\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{Y}}$$\textstyle{\tilde{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{i}}$$\textstyle{Y}$$\textstyle{\mathcal{L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\scriptstyle{q}$$\textstyle{\tilde{Y}}$$\textstyle{Y^{\circ}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{Y}}$$\textstyle{Y^{\circ}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{j}}$$\scriptstyle{\pi}$

in which $q$ is the bundle map, $\tilde{i}$ the embedding of the zero section,
and $u$ is the contraction of the zero section. We have

$\displaystyle{{}^{p}}\mathcal{H}^{-k}a_{Y!}j_{Y!}\pi^{!}({{}^{p}}\mathcal{H}^{-\ell+1}\omega_{\tilde{Y}})$
$\displaystyle\simeq{{}^{p}}\mathcal{H}^{-k}a_{Y!}j_{Y!}\pi^{-1}[2]({{}^{p}}\mathcal{H}^{-\ell+1}\omega_{\tilde{Y}})$
$\displaystyle\simeq{{}^{p}}\mathcal{H}^{-k+2}a_{Y!}j_{Y!}\pi^{-1}({{}^{p}}\mathcal{H}^{-\ell+1}\omega_{\tilde{Y}})$
$\displaystyle\simeq{{}^{p}}\mathcal{H}^{-k+2}a_{Y!}j_{Y!}(\widetilde{j})^{-1}q^{-1}({{}^{p}}\mathcal{H}^{-\ell+1}\omega_{\tilde{Y}})$
$\displaystyle\simeq{{}^{p}}\mathcal{H}^{-k+2}a_{X!}q_{!}\
\widetilde{j}_{!}(\widetilde{j})^{-1}q^{-1}({{}^{p}}\mathcal{H}^{-\ell+1}\omega_{\tilde{Y}})$
$\displaystyle\simeq{{}^{p}}\mathcal{H}^{-k+2}a_{X!}\pi_{!}\pi^{-1}({{}^{p}}\mathcal{H}^{-\ell+1}\omega_{\tilde{Y}})$
$\displaystyle\simeq\mathbb{H}^{-k+2}_{c}(X,\pi_{!}\pi^{-1}({{}^{p}}\mathcal{H}^{-\ell+1}\omega_{\tilde{Y}}))$

Here the first isomorphism comes from the fact that $\pi^{-1}[1]=\pi^{!}[-1]$
is perverse exact, and the last one because cohomology of compact supports is
the cohomology of the exceptional direct image functor.

From the closed embedding of $\tilde{Y}$ into $\mathcal{L}$ arises a triangle

(2.4.11)
$\widetilde{j}_{!}(\widetilde{j})^{-1}q^{-1}({{}^{p}}\mathcal{H}^{-\ell+1}\omega_{\tilde{Y}})\longrightarrow
q^{-1}({{}^{p}}\mathcal{H}^{-\ell+1}\omega_{\tilde{Y}})\longrightarrow\tilde{i}_{!}(\tilde{i})^{-1}q^{-1}({{}^{p}}\mathcal{H}^{-\ell+1}\omega_{\tilde{Y}})\overset{+1}{\longrightarrow}$

Applying $q_{!}$ we get

$\pi_{!}\pi^{-1}\mathcal{G}_{-\ell+1}\longrightarrow
q_{!}q^{-1}\mathcal{G}_{-\ell+1}\longrightarrow\mathcal{G}_{-\ell+1}\overset{+1}{\longrightarrow}$

where we have set
$\mathcal{G}_{-\ell}:={{}^{p}}\mathcal{H}^{-\ell}\omega_{\tilde{Y}}$ and used
$\pi=q\circ\tilde{j}$. We have $\mathcal{G}\simeq q_{!}q^{!}\mathcal{G}\simeq
q_{!}q^{-1}\mathcal{G}[2]$ for any $\mathcal{G}\in\textup{Perv}(\tilde{Y})$
since $q$ is smooth of relative dimension $1$. This gives the triangle

$\pi_{!}\pi^{-1}\mathcal{G}_{-\ell+1}\longrightarrow\mathcal{G}_{-\ell+1}[-2]\longrightarrow\mathcal{G}_{-\ell+1}\overset{+1}{\longrightarrow}$

As in [RSW21, (1.3.1)], this triangle is dual to a triangle
$\mathcal{F}\longrightarrow\mathcal{F}[2]\longrightarrow
p_{*}\pi^{!}\mathcal{F}\overset{+1}{\longrightarrow}$ where the first map is
induced by

$e\otimes
1:\mathbb{C}_{\tilde{Y}}\otimes\mathcal{F}\longrightarrow\mathbb{C}_{\tilde{Y}}[2]\otimes\mathcal{F},$

with
$e\in\operatorname{\textup{Hom}}_{D^{b}_{\mathrm{c.s.}}({\tilde{Y}})}(\mathbb{C}_{\tilde{Y}},\mathbb{C}_{\tilde{Y}}[2])\simeq\operatorname{\textup{Hom}}_{D^{b}_{\mathrm{c.s.}}(\mathit{pt})}(\mathbb{C},R\Gamma({\tilde{Y}};\mathbb{C}_{\tilde{Y}}[2]))\simeq
H^{2}({\tilde{Y}};\mathbb{C})$ is the image of the Euler class of the vector
bundle $\mathcal{L}$.

We get a long exact sequence

$\longrightarrow\mathbb{H}^{-k+2}_{c}({\tilde{Y}},\pi_{!}\pi^{-1}\mathcal{G}_{-\ell+1})\longrightarrow\mathbb{H}^{-k}_{c}({\tilde{Y}},\mathcal{G}_{-\ell+1})(-1)\longrightarrow\mathbb{H}^{-k+2}_{c}({\tilde{Y}},\mathcal{G}_{-\ell+1})\longrightarrow\mathbb{H}^{-k+3}_{c}({\tilde{Y}},\pi_{!}\pi^{-1}\mathcal{G}_{-\ell+1})\longrightarrow$

In particular we get short exact sequences

$0\longrightarrow\mathbb{H}^{-k+1}_{c}({\tilde{Y}},\mathcal{G}_{-\ell+1})_{\mathcal{L}}\longrightarrow\mathbb{H}^{-k+2}_{c}({\tilde{Y}},\pi_{!}\pi^{-1}({{}^{p}}\mathcal{H}^{-\ell+1}\omega_{\tilde{Y}}))\longrightarrow\mathbb{H}^{-k}_{c}({\tilde{Y}},\mathcal{G}_{-\ell+1})^{\mathcal{L}}\longrightarrow
0$

where

$\displaystyle\mathbb{H}^{-k+1}_{c}({\tilde{Y}},\mathcal{G}_{-\ell+1})_{\mathcal{L}}$
$\displaystyle:=$
$\displaystyle\textup{coker}\left(\mathbb{H}^{-k-1}_{c}({\tilde{Y}},\mathcal{G}_{-\ell+1})(-1)\longrightarrow\mathbb{H}^{-k+1}_{c}({\tilde{Y}},\mathcal{G}_{-\ell+1})\right)$
$\displaystyle\mathbb{H}^{-k}_{c}({\tilde{Y}},\mathcal{G}_{-\ell+1})^{\mathcal{L}}$
$\displaystyle:=$
$\displaystyle\ker\left(\mathbb{H}^{-k}_{c}({\tilde{Y}},\mathcal{G}_{-\ell+1})(-1)\longrightarrow\mathbb{H}^{-k+2}_{c}({\tilde{Y}},\mathcal{G}_{-\ell+1})\right)$

Putting everything together we get that

$\rho_{k,\ell}=\dim\mathbb{H}^{-k+2}_{c}({\tilde{Y}},\pi_{!}\pi^{-1}({{}^{p}}\mathcal{H}^{-\ell+1}\omega_{\tilde{Y}}))=\dim\mathbb{H}^{-k+1}_{c}({\tilde{Y}},\mathcal{G}_{-\ell+1})_{\mathcal{L}}+\dim\mathbb{H}^{-k}_{c}({\tilde{Y}},\mathcal{G}_{-\ell+1})^{\mathcal{L}}$

is unchanged under Veronese maps for $k\geq 2$, since the Euler class of a
bundle power is a multiple of the original Euler class (and so over
$\mathbb{C}$ kernels and cokernels are preserved).

∎

##  3\. Lyubeznik numbers

In this section we study the Lyubeznik numbers and their spectral sequence.
After surveying some known facts we discuss to what extent a projective
variety determines the Lyubeznik numbers of its cone(s). We look first
specifically at varieties of Picard number 1, listing some examples and open
questions. After that we discuss cases where in small dimension the Lyubeznik
tables of all cones agree.

###  3.1. Basic properties

We should begin with drawing some parallels to the case of the Čech–de Rham
numbers. Quite immediately, being defined as the socle dimensions of the
$E_{2}$-terms in the Grothendieck spectral sequence (1.0.3), the Lyubeznik
numbers vanish for $q\not\in[\operatorname{\textup{codim}}(I),n]$ and for
$p\not\in[0,n]$. In fact, similarly to the $\rho^{r}_{p,q}$, Lyubeznik numbers
fit into a triangular region

$\Lambda(Y):=\begin{pmatrix}\lambda_{0,0}&\cdots&\cdots&\lambda_{0,d}\\\
0&\ddots&&\vdots\\\ \vdots&\ddots&\ddots&\vdots\\\
0&\cdots&0&\lambda_{d,d}\end{pmatrix}.$

In this picture, the differentials of the Grothendieck spectral sequence point
South to Southeast.

######  Remark 3.1.

The fact that the abutment is $H^{n}_{\mathfrak{m}}(R)$ implies that the
entries

$\lambda_{0,d}=\lambda_{1,d}=0$

always vanish unless the dimension of $I$ is less than two, in which cases the
Lyubeznik tables are $(1)$ and $\begin{pmatrix}\cdot&\cdot\\\
\cdot&1\end{pmatrix}$ respectively. $\diamond$

The number $\lambda_{d,d}$ is never zero by [Lyu93] and related to
connectedness issues. For example, if $\dim(Y)=2$ then
$\Lambda=\begin{pmatrix}\cdot&a-1&\cdot\\\ \cdot&\cdot&\cdot\\\
\cdot&\cdot&a\end{pmatrix}$ where $a$ is the number of connected components of
the punctured spectrum of the ring defining the purely 2-dimensional part of
$I$, [Wal01, Kaw02]. By [Zha07], $\lambda_{d,d}$ is the number of connected
components of the Hochster–Huneke graph of the completed strict Henselization
of $R/I$.

It was first observed in [GLS98] that the Lyubeznik numbers encode interesting
topological information also in higher dimension. However, it is often not
easy to decode this information. Garcia and Sabbah concentrate on the case of
an isolated singularity and find that the topology of the singularity link
carries all information on $\Lambda$. Other relations to connectedness
dimensions are discussed in the survey [NnBWZ16].

A new angle was introduced in [RSW21] by applying the theory of perverse
sheaves and mixed Hodge modules to the problem. Indeed, over $\mathbb{C}$, the
local cohomology groups $H^{\bullet}_{I}(R)$ are, as the pushforward of the
structure sheaf on $U(I)$ to $\mathbb{C}^{n}$, all equipped with a natural
mixed Hodge module structure. The corresponding perverse sheaves carry
information on the intersection homology of $U(I)$. Using this connection it
is proved in [RSW21] that if $I$ is homogeneous then the $\lambda_{p,q}(R/I)$
can be recovered from the kernel and cokernel of certain maps of sheaves on
the corresponding projective scheme, at least as long as $p>1$.

###  3.2. Lyubeznik numbers and projective schemes

Suppose $\tilde{Y}$ is a projective variety in
$\mathbb{P}^{n-1}_{\mathbb{K}}$, with defining ideal $I\subseteq
R_{n}=\mathbb{K}[x_{1},\ldots,x_{n}]$. Different embeddings of $\tilde{Y}$
give rise to different ideals in different polynomial rings, and thus
potentially to different sets of Lyubeznik numbers. That this is indeed a
possibility was shown to be the case in [RSW21, Sections 2.2, 2.3] where a
projective variety with two embeddings is constructed that produce (partially)
different $\lambda_{p,q}$; see also [Wan20]. On the other hand, if $\tilde{Y}$
is smooth or an $\mathbb{Q}$-homology manifold or analytically locally a set-
theoretic complete intersection, then all cones for $\tilde{Y}$ yield the same
Lyubeznik numbers [GLS98, Swi15, RSW21].

That examples of cones over $\tilde{Y}$ with varying Lyubeznik numbers exist
over $\mathbb{C}$ is rather surprising at first, since similar examples cannot
exist in any positive characteristic. Indeed, it is shown in [Zha11] that
Lyubeznik numbers in finite characteristic can be seen as eigenvalues of
certain operators on sheaves that are intrinsic to the projective variety
associated to $I$.

An interesting feature of [RSW21] is the realization that the non-vanishing
Lyubeznik numbers of a homogeneous ideal can, for $p>1$, be viewed as a
measure for the failure of a certain morphism of perverse sheaves to be an
isomorphism. All known examples of projective varieties with possibly varying
Lyubeznik numbers of their cones come from varieties with Picard number at
least two. An application of [RSW21, Prop. 3] is that most Lyubeznik numbers
are unchanged under Veronese maps.

######  Lemma 3.2.

Let $Y$ be a cone over the projective variety
$\tilde{Y}\subseteq\mathbb{P}^{n-1}_{\mathbb{C}}$. Let $v_{n}^{d}$ be the
$d$-th Veronese applied to the cone $Y$, and write $Y^{\prime}=v_{n}^{d}(Y)$
for the new cone. Then for $p\geq 2$, the Lyubeznik numbers $\lambda_{p,q}(Y)$
and $\lambda_{p,q}(Y^{\prime})$ agree.

In particular, if the Picard number $\tilde{Y}$ equals one, then the Lyubeznik
numbers $\lambda_{\geq 2,q}(Y)$ to cones over $\tilde{Y}$ are independent of
the cone.

###### Proof.

Let $\iota_{1},\iota_{2}$ be two embeddings of $\tilde{Y}$ into projective
spaces $\mathbb{P}^{n-1}_{\mathbb{K}},\mathbb{P}^{m-1}_{\mathbb{K}}$ and
denote $Y_{1}\subseteq X_{1},Y_{2}\subseteq X_{2}$ the two cones over
$\tilde{Y}$, sitting in the respective affine spaces that belong to the two
embeddings. Let $\mathcal{L}_{1},\mathcal{L}_{2}$ be the associated line
bundles on $\tilde{Y}$ obtained as pullbacks of
$\mathcal{O}_{\mathbb{P}^{n-1}_{\mathbb{K}}}(1)$ and
$\mathcal{O}_{\mathbb{P}^{m-1}_{\mathbb{K}}}(1)$ respectively. Then by [RSW21,
Prop. 1,2,3], the Lyubeznik numbers $\lambda_{p,q}$ of $\tilde{Y}$ that belong
to $Y_{i}$ and have $p\geq 2$ are determined by the (co)kernel sizes of the
Chern classes of $\mathcal{L}_{i}$ on certain cohomology groups of $\tilde{Y}$
with rational coefficients. These cohomology groups themselves (see [RSW21,
Prop. 2]) do not depend on the bundles $\mathcal{L}_{i}$.

If $\iota_{2}$ is the $d$-fold Veronese applied to $\iota_{1}$ then the
(first) Chern class of $\mathcal{L}_{2}$ is $d$ times that of $L_{1}$. In
particular, their kernels and cokernels on $\mathbb{Q}$-spaces are identical
and the first claim follows.

Now suppose that the target of the natural map

$\phi\colon\operatorname{\textup{Pic}}(\tilde{Y})\longrightarrow\operatorname{\textup{Pic}}(\tilde{Y})\otimes_{\mathbb{Z}}\mathbb{Q}$

is $\mathbb{Q}$. If $\iota_{1},\iota_{2}$ are both projective embeddings of
$\tilde{Y}$, ampleness implies that $q(\mathcal{L}_{i})>0$. Then if
$\phi(\mathcal{L}_{1})=q_{1}$ and $\phi(\mathcal{L}_{2})=q_{2}$, both positive
rational numbers, we have for $k\gg 0$ with $kq_{1},kq_{2}\in\mathbb{N}$ that
$\mathcal{L}_{1}^{k|q_{2}|}=\mathcal{L}_{2}^{k|q_{1}|}$. Then by the first
part of the proof, $\iota_{1}$, $\iota_{1}^{kq_{2}}$, $\iota_{2}^{kq_{1}}$ and
$\iota_{2}$ all yield the same Lyubeznik numbers $\lambda_{p,q}$ for $p\geq 2$
(where we write $\iota^{\ell}$ for the $\ell$-th Veronese of the embedding
$\iota$). ∎

For $p<2$, we do not know how to compare the $\lambda_{p,q}$ of different
cones.

######  Problem 3.3.

Is it true that if the Picard number of $\tilde{Y}$ is one, then all Lyubeznik
numbers of all cones $Y$ of $\tilde{Y}$ agree? $\diamond$

Here are three interesting sets of varieties to which this lemma applies.

####  3.2.1. Determinantal ideals

######  Proposition 3.4.

The Lyubeznik numbers $\lambda_{p,q}$ with $p\geq 2$ of (the cones over) the
projective determinantal varieties $\tilde{Y}_{m,n,t}$ cut out by the $t\times
t$ minors of an $m\times n$ matrix of indeterminates are unique.

###### Proof.

Let $A_{m,n,t}$ be the ring obtained as quotient of the polynomial ring
$\mathbb{K}[x_{i,j}|1\leq i\leq m,1\leq j\leq n]$ by the $t$-minors of the
matrix $x:=((x_{i,j}))$.

The case $t=1$ is trivial. If $t=2$, the associated projective variety is the
product of two projective spaces, and in particular smooth. By [GLS98], or
[Swi15], the Lyubeznik numbers of $\tilde{Y}_{m,n,2}$ are independent of the
embedding.

Now consider the case $t>2$. By [BV88, Cor. 8.4], the divisor class group of
$A_{m,n,t}$ is $\mathbb{Z}$, a generator being the ideal $\bar{I}_{m,n,t-1}$
of $A_{m,n,t}$ generated by the $(t-1)$-minors of the first $t-1$ rows (or
columns) of $x$.

Since determinantal varieties are normal, they satisfy condition $(*)$ in
[Har77, Page 130]. By [Har77, Exercise II.6.3], there is a short exact
sequence
$0\longrightarrow\mathbb{Z}\longrightarrow\operatorname{\textup{Cl}}(\tilde{Y})\longrightarrow\operatorname{\textup{Cl}}(Y)\longrightarrow
0$, $Y$ the cone over $\tilde{Y}$, where the last map factors through the
class group $\operatorname{\textup{Cl}}(Y\smallsetminus P)$ of the complement
of the origin in $Y$. For $t\geq 2$ this implies that
$\operatorname{\textup{Cl}}(\tilde{Y}_{m,n,t})=\mathbb{Z}\oplus\mathbb{Z}$. In
this sequence, $1\in\mathbb{Z}$ is sent to the generic hyperplane section of
$\tilde{Y}$. In order to determine the Picard group of $\tilde{Y}_{m,n,t}$ we
need by [Har77, Prop. II.6.15] to determine the Cartier classes of
$\operatorname{\textup{Cl}}(\tilde{Y}_{m,n,t})$. From the preceding, this
amounts to checking which multiples of $\bar{I}_{m,n,t-1}$ are Cartier on the
punctured spectrum of $A_{m,n,t}$. One sees easily that for $t=2$,
$\bar{I}_{m,n,t-1}$ is Cartier on the punctured spectrum. For $t>2$ only its
trivial power is Cartier: by the coordinate change expounded in [LSW16],
powers of $\bar{I}_{m,n,t-1}$ are locally principal on the open set
$U_{x_{1,1}}$ if and only if corresponding powers of $\bar{I}_{m-1,n-1,t-2}$
are locally principal everywhere on $Y_{m-1,n-1,t-1}$; for $t=3$ this is
clearly not so. Hence the Picard group of $\tilde{Y}_{m,n,t}$ is $\mathbb{Z}$
for $t>2$. Now use Lemma 3.2.

∎

######  Remark 3.5.

In particular, the Lyubeznik numbers $\lambda_{p,q}$ of determinantal
varieties computed by Lörincz and Raicu in [LR20] for the standard embedding
equal those of any embedding, at least for $p\geq 2$. $\diamond$

######  Remark 3.6.

Suppose $G$ is a semisimple linear algebraic group, $P$ a parabolic subgroup
and $w$ an element of the Weyl group of $G$. The Schubert variety
$X_{P}(w):=BwP/P$ sits inside the homogeneous space $G/P$, and every line
bundle on $X_{P}(w)$ is the restriction of a line bundle on $G/P$, [Mat88]. In
particular, the Picard group of $X_{P}(w)$ is generated by the Schubert
divisors (the Schubert varieties inside $X_{P}(w)$ of codimension one),
[Bri05, Prop. 2.2.8]. $\diamond$

######  Problem 3.7.

Compute the Lyubeznik numbers of Schubert varieties in their Schubert divisor
embeddings. $\diamond$

####  3.2.2. Toric varieties

Suppose $\tilde{Y}$ is the toric variety attached to a complete fan $\Delta$
that is projective. If $\Delta$ is smooth, or at least simplicial, then the
Picard group of $\tilde{Y}$ is a free Abelian group generated by the torus
invariant (Cartier) divisors corresponding to the $n$ rays of $\Delta$,
[CLS11, Thm. 4.2.1]. The ambient lattice imposes $d:=\dim(\tilde{Y})$ many
independent relations on these divisors, so that
$\operatorname{\textup{Pic}}(\tilde{Y})=\mathbb{Z}^{n-d}$. In order for this
number to be 1, there is very little choice for $\Delta$; it forces
$\tilde{Y}$ to be a weighted projective space. These are $\mathbb{Q}$-homology
manifolds and thus yield the same Lyubeznik numbers under all embeddings by
[RSW21].

However, singular fans fail the Picard rank formula above and can have Picard
group $\mathbb{Z}$ with greater variety. The Picard group is free if the fan
is full-dimensional by [CLS11, Thm. 4.2.5], and equals the inverse limit of
the quotient lattices $M/M(\sigma)$, taken modulo $M$ by [CLS11, Thms.
4.2.1,4.2.9].

######  Example 3.8.

If $\Delta$ is a complete rational fan in $\mathbb{Z}^{3}$, one can use the
description of the Picard group via support functions to show that if $\Delta$
has at most one simplicial cone, then the Picard group of the associated toric
variety is rank one. For example, the fan over the sides of a cube leads to a
projective three-fold with Picard number one. The generating support function
takes the value zero on one square and one on the opposing square (see [Ful93,
Exa. 1.5.(3)]). Our next result shows that all projective toric threefolds
have their Lyubeznik table independent of the embedding.

######  Theorem 3.9.

Let $\tilde{Y}$ be the projective variety to a complete projective fan in
$\mathbb{Z}^{3}$ with Picard number $\tilde{p}+1$. Then for any cone $Y$ over
$\tilde{Y}$ its Lyubeznik numbers take the form

$\Lambda(Y)=\begin{pmatrix}\cdot&\cdot&\cdot&\tilde{p}&\cdot\\\
\cdot&\cdot&\cdot&\cdot&\cdot\\\ \cdot&\cdot&\cdot&\cdot&\tilde{p}\\\
\cdot&\cdot&\cdot&\cdot&\cdot\\\ \cdot&\cdot&\cdot&\cdot&1\end{pmatrix}$

We postpone the proof until the end of the final section.

######  Problem 3.10.

Express Lyubeznik numbers of projective toric varieties (of Picard rank 1 or
otherwise) in terms of fan data (and, possibly, polytopes). $\diamond$

####  3.2.3. Horospherical varieties

Horospherical varieties are complex normal algebraic varieties on which a
connected complex reductive algebraic group $G$ acts with an open orbit that
is isomorphic to a torus bundle over a flag variety; the dimension of this
torus is referred to as the rank of the variety. In particular, toric and flag
varieties are examples of horospherical varieties.

Any flag variety $G/P$ with $P$ a parabolic subgroup of $G$ is smooth and
projective. Their Lyubeznik numbers are hence all topological, by [Swi15]. By
[Pas09, Thm. 0.1], a smooth projective horospherical variety of Picard number
1 must either be a homogeneous space or have horospherical rank one.

There are many singular horospherical varieties of Picard number one. For
example, let $G$ be a simple linear algebraic group and choose two dominant
weights $\chi_{1}$ and $\chi_{2}$ that cannot be written as the sum of a
common dominant weight with another dominant weight. Writing $V(\chi)$ for the
simple $G$-module of weight $\chi$, let $\tilde{Y}$ be the closure of the
$G$-orbit of the sum of two highest weight vectors in
$\mathbb{P}(V(\chi_{1})\oplus V(\chi_{2}))$. It is a projective variety of
horospherical rank one, it has Picard number one and is smooth only in very
few cases, namely when $\chi_{1}$ and $\chi_{2}$ are fundamental weights
$\varpi_{\alpha}$ and $\varpi_{\beta}$ and $(G,\alpha,\beta)$ is in the list
of [Pas09, Thm. 1.7]. It has three G-orbits (one open and two closed), the
singularities if they exist, are on the closed orbit(s). By taking a longer
list of weights $\chi_{1},\ldots,\chi_{n}$ one can produce (usually singular)
varieties of horospherical rank $n-1$.

In all these Picard rank 1 cases, the Lyubeznik numbers $\lambda_{p,q}$ with
$p>1$ of the cone of $\tilde{Y}$ are embedding independent, and can hence be
computed from the embedding that arises from the definition.

######  Problem 3.11.

Compute Lyubeznik numbers of horospherical varieties of Picard number one for
the standard families. $\diamond$

###  3.3. Lyubeznik numbers in small dimension

We consider here to what extent the Lyubeznik numbers of varieties of small
dimension are functions of the associated projective variety only. Some
independence is known quite generally.

######  Remark 3.12.

  1. (1)

If $\tilde{Y}$ is a projective scheme of dimension at most $1$ over any field
(not necessarily connected or equi-dimensional) then $R/I$ is two-dimensional
for any embedding, and so $\Lambda$ is independent of embeddings by [Wal01,
Kaw02]

  2. (2)

Set $d-1=\dim(\tilde{Y})$. Then $\lambda_{d,d}$ is independent of embeddings
unconditionally by [Zha07]. $\diamond$

We begin with some preparations involving Hartshorne’s (local) algebraic de
Rham cohomology.

######  Theorem 3.13.

Let $\tilde{Y}\subseteq\mathbb{P}^{n-1}_{\mathbb{K}}$ be a projective variety
over a field $\mathbb{K}$ of characteristic 0, let
$Y\subseteq\mathbb{A}^{n}_{\mathbb{K}}$ be the affine cone of $\tilde{Y}$, and
let $P$ be its vertex. Let $H^{j}_{P}(Y)$ denote the local de Rham cohomology
of $Y$ supported in $\\{P\\}$. Assume that the Picard group of $\tilde{Y}$ has
rank 1. Then $\dim_{k}(H^{j}_{P}(Y))$ depends only on $\tilde{Y}$, but not on
the embedding $\tilde{Y}\subseteq\mathbb{P}^{n}_{\mathbb{K}}$. More precisely,
if $\tilde{Y}\subseteq\mathbb{P}^{n^{\prime}-1}_{\mathbb{K}}$ is another
embedding of $\tilde{Y}$ into a projective space and $Y^{\prime}$ is its
affine cone with the vertex $P^{\prime}$ , then
$\dim_{\mathbb{K}}(H^{j}_{P}(Y))=\dim_{\mathbb{K}}(H^{j}_{P^{\prime}}(Y^{\prime}))$.

To prove Theorem 3.13, we need the following result of Hartshorne.

######  Theorem 3.14 (Proposition III.3.2 in [Har75]).

Let $\tilde{Y},Y,P$ be the same as in Theorem 3.13. Then $H^{0}_{P}(Y)=0$ and
there are two exact sequences:

$0\longrightarrow\mathbb{K}\longrightarrow
H^{0}_{{\mathrm{dR}}}(\tilde{Y})\longrightarrow H^{1}_{P}(Y)\longrightarrow 0$

and

$0\longrightarrow H^{1}_{{\mathrm{dR}}}(\tilde{Y})\longrightarrow
H^{2}_{P}(Y)\longrightarrow H^{0}_{{\mathrm{dR}}}(\tilde{Y})\longrightarrow
H^{2}_{{\mathrm{dR}}}(\tilde{Y})\longrightarrow
H^{3}_{P}(\tilde{Y})\longrightarrow
H^{1}_{{\mathrm{dR}}}(\tilde{Y})\longrightarrow
H^{3}_{{\mathrm{dR}}}(\tilde{Y})\longrightarrow\cdots$

where the maps $H^{i}_{{\mathrm{dR}}}(\tilde{Y})\longrightarrow
H^{i+2}_{{\mathrm{dR}}}(\tilde{Y})$ are given by the cup product with the
Chern class $\xi\in H^{2}_{{\mathrm{dR}}}(\tilde{Y})$ of the hyperplane
section (_i.e._ , the first Chern class of $\mathcal{O}_{\tilde{Y}}(1)$).∎

###### Proof of Theorem 3.13.

The case when $j\leq 1$ is clear from the long exact sequence above.

Since the Picard group of $\tilde{Y}$ has rank 1, any two very ample line
bundles on $\tilde{Y}$ have a common power. It is thus sufficient to consider
the case where the two ample line bundles in question are $\mathcal{L}$ and
$\mathcal{L}^{m}$.

Let $\xi(\mathcal{L})\in H^{2}_{dR}(\tilde{Y})$ be the first Chern class of
$\mathcal{L}$, represented by a generic hyperplane section with the embedding
given by $\mathcal{L}$. Then we have $\xi(\mathcal{L}^{m})=m\xi(\mathcal{L})$.
Since the cup product is linear and
$\operatorname{\textup{char}}(\mathbb{K})=0$, the maps
$H^{i}_{{\mathrm{dR}}}(\tilde{Y})\xrightarrow{\cup\xi}H^{i+2}_{{\mathrm{dR}}}(\tilde{Y})$
and $H^{i}_{{\mathrm{dR}}}(\tilde{Y})\xrightarrow{\cup
m\xi}H^{i+2}_{{\mathrm{dR}}}(\tilde{Y})$ have the same rank. Therefore
$\dim_{\mathbb{K}}(\ker(H^{i}_{dR}(\tilde{Y})\longrightarrow
H^{i+2}_{{\mathrm{dR}}}(\tilde{Y})))$ and
$\dim_{\mathbb{K}}(\operatorname{\textup{coker}}(H^{i}_{dR}(\tilde{Y})\longrightarrow
H^{i+2}_{{\mathrm{dR}}}(\tilde{Y})))$ depend only on $\tilde{Y}$, but not on
the choice of the embedding (or equivalently, not on the choice of ample line
bundles $\mathcal{L}$). When $j\geq 2$ we have

$\dim_{\mathbb{K}}(H^{j}_{P}(Y)=\dim_{\mathbb{K}}(\ker(H^{j-2}_{{\mathrm{dR}}}(\tilde{Y})\longrightarrow
H^{j}_{{\mathrm{dR}}}(\tilde{Y}))))+\dim_{\mathbb{K}}(\operatorname{\textup{coker}}(H^{j-3}_{{\mathrm{dR}}}(\tilde{Y})\longrightarrow
H^{j-1}_{{\mathrm{dR}}}(\tilde{Y})))),$

hence the conclusion holds for $\dim_{k}(H^{j}_{P}(Y))$ when $j\geq 2$. ∎

######  Corollary 3.15.

Assume that the Picard group of $\tilde{Y}$ has rank 1. Then $\lambda_{p,q}$
is independent of embeddings for all $q<n-f_{Y}$, with $f_{Y}$ as in Remark
2.6.

###### Proof.

Assume $q<n-f_{Y}$. Since
$\operatorname{\textup{Supp}}(H^{n-q}_{I}(R_{n}))\subseteq\\{{\mathfrak{m}}\\}$,
[Lyu93] shows that $H^{n-q}_{I}(R_{n})\cong
H^{n}_{{\mathfrak{m}}}(R_{n})^{\lambda_{0,q}}$ and
$H^{p}_{{\mathfrak{m}}}(H^{n-q}_{I}(R_{n}))=0$ for $p\geq 1$. Hence
$\lambda_{p,q}=0$ for all $p\geq 1$ (and $q<n-f_{Y}$).

Let $D(-)$ denote the Matlis dual. Then
$D(H^{\ell}_{I}(R_{n}))\cong\hat{R_{n}}^{\lambda_{0,n-\ell}}$ whenever
$H^{\ell}_{I}(R_{n})$ is Artinian. On the other hand, [Ogu73, Proposition
2.2,Theorem 2.3] shows that, for $q<n-f_{Y}$,

$D(H^{n-q}_{I}(R_{n}))\cong
H^{q}_{P}(\hat{X},\mathcal{O}_{\hat{X}})\cong\hat{R_{n}}\otimes H^{q}_{P}(Y)$

where $Y$ denotes the affine cone of $\tilde{Y}$ with vertex $P$ and $\hat{X}$
denotes the formal completion of $\operatorname{\textup{Spec}\,}(\hat{R_{n}})$
along the subscheme defined by $I$. This shows that
$\dim_{\mathbb{K}}(H^{q}_{P}(Y))=\lambda_{0,q}$. Hence $\lambda_{0,q}$ depends
only on $\tilde{Y}$ by Theorem 3.13. ∎

######  Remark 3.16.

An alternative way to look at Corollary 3.15 arises through Proposition 2.16:
for $q>f_{Y}$, the multiplicities of $H^{n}_{\mathfrak{m}}(R_{n})$ in
$H^{j}_{I}(R_{n})$ are exactly the Betti numbers $H^{n-1+j}(U)$ where $U$ is
the affine complement of $Y$, because of the spectral sequence (1.0.2). By
Proposition 2.16 these do not change under Veronese maps. $\diamond$

We now consider the effect of Serre’s conditions $(S_{t})$ in $R_{n}/I$ on the
Lyubeznik numbers.

######  Remark 3.17.

Assume that “$\tilde{Y}$ satisfies $(S_{t})$ locally everywhere”, by which we
mean that each local ring $\mathcal{O}_{\tilde{Y},\tilde{\mathfrak{y}}}$ of
the projective scheme $\tilde{Y}=\operatorname{\textup{Proj}\,}(R_{n}/I)$
satisfies Serre condition $(S_{t})$.

Let $Y$ be the cone $\operatorname{\textup{Spec}\,}(R_{n}/I)$ as always and
$P$ the vertex; then the punctured cone $Y^{\circ}=Y\smallsetminus P$ is a
bundle over $\tilde{Y}$. It follows that every local ring of $Y^{\circ}$ also
is $(S_{t})$. So for each non-maximal prime ideal ${\mathfrak{p}}$ of $R_{n}$
such that $\dim((R_{n}/I)_{\mathfrak{p}})\geq t$, one has
$\operatorname{\textup{depth}}((R_{n}/I)_{\mathfrak{p}})\geq t$.

In general, if
$(A,{\mathfrak{n}})\longrightarrow(A^{\prime},{\mathfrak{n}}^{\prime})$ is a
faithfully flat morphism, then

$\operatorname{\textup{depth}}(A^{\prime})=\operatorname{\textup{depth}}(A)+\operatorname{\textup{depth}}(A^{\prime}/{\mathfrak{n}}A).$

If $A^{\prime}$ is the strict Henselization $A^{sh}$ or the completion
$\hat{A}$ of $A$, then $A^{\prime}$ is faithfully flat over $A$. Therefore,

$\operatorname{\textup{depth}}\left(\left(\left((R_{n}/I)_{\mathfrak{p}}\right)^{\widehat{~{}}}\right)^{sh}\right)^{\widehat{~{}}}\geq
t.$

$\diamond$

######  Lemma 3.18.

If $\tilde{Y}$ is equi-dimensional and locally everywhere $(S_{2})$ then the
off-diagonal entries $\lambda_{i-1,i}$ vanish for $1<i<d:=\dim(\tilde{Y})+1$,
and $H^{n-1}_{I}(R_{n})$ is Artinian and injective.

###### Proof.

By Remark 3.17, for each non-maximal prime ideal ${\mathfrak{p}}$ of $R_{n}$
with $\dim((R_{n}/I)_{\mathfrak{p}})\geq 2$, we have
$\operatorname{\textup{depth}}\left(\left(((R_{n}/I)_{\mathfrak{p}})^{\widehat{~{}}}\right)^{sh}\right)^{\widehat{~{}}}\geq
2$. Hence the punctured spectrum of this ring is connected by [Har62]. The
Second Vanishing Theorem implies that
$H^{>\operatorname{\textup{codim}}(P)-2}_{I}(R_{n})_{\mathfrak{p}}=0$ for each
prime ideal $P$ such that $\dim((R_{n}/I)_{\mathfrak{p}})\geq 2$. Therefore
the support dimension of $H^{i}_{I}(R_{n})$ with $n-1>i>n-d$ is at most equal
to $n-i-2$ and so $H^{i-1}_{\mathfrak{m}}H^{n-i}_{I}(R)=0$ by Grothendieck’s
vanishing theorem. For $H^{n-1}_{I}(R_{n})$, localization shows in conjunction
with the Hartshorne–Lichtenbaum theorem that its support is at best at $P$. By
Lyubzenik’s work, it is hence Artinian and injective. ∎

For the next three result we will use the following reduction.

######  Lemma 3.19.

Let $\tilde{Y}$ be an equi-dimensional projective variety of dimension at
least two. If the Lyubeznik numbers for the cones over all connected
components of $\tilde{Y}$ are independent of the choice of the cone then the
same is true for $\tilde{Y}$ itself.

###### Proof.

Let $Y^{\prime},Y^{\prime\prime}$ be two cones for $\tilde{Y}$ and let
$\tilde{Y}=\tilde{Y}_{1}\sqcup\tilde{Y}_{2}$ be a disconnection. The resulting
cones $Y^{\prime}_{1},Y^{\prime\prime}_{1}$ and
$Y^{\prime}_{2},Y^{\prime\prime}_{2}$ satisfy: $Y^{\prime}_{1}\cap
Y^{\prime}_{2}$ and $Y^{\prime\prime}_{1}\cap Y^{\prime\prime}_{2}$ both equal
the origin. Let $I^{\prime},I^{\prime\prime}$ be the defining ideals for
$Y^{\prime},Y^{\prime\prime}$ and denote the defining ideals of
$Y^{\prime}_{1},Y^{\prime}_{2},Y^{\prime\prime}_{1},Y^{\prime\prime}_{2}$ by
$I^{\prime}_{1},I^{\prime}_{2}\subseteq R_{n^{\prime}}$ and
$I^{\prime\prime}_{1},I^{\prime\prime}_{2}\subseteq R_{n^{\prime\prime}}$
respectively. All these ideals have dimension three or more.

Then
$H^{q}_{I^{\prime}}(R_{n^{\prime}})=H^{q}_{I^{\prime}_{1}}(R_{n^{\prime}})\oplus
H^{q}_{I^{\prime}_{2}}(R_{n^{\prime}})$ and
$H^{q}_{I^{\prime\prime}}(R_{n^{\prime\prime}})=H^{q}_{I^{\prime\prime}_{1}}(R_{n^{\prime\prime}})\oplus
H^{q}_{I^{\prime\prime}_{2}}(R_{n^{\prime\prime}})$ for all $q<n-1$ as follows
from the Mayer–Vietoris sequence.

It follows that, apart from $q=n,n-1$, the Lyubeznik numbers satisfy
$\lambda_{p,q}(Y^{\prime})=\lambda_{p,q}(Y^{\prime}_{1})+\lambda_{p,q}(Y^{\prime}_{2})$
and
$\lambda_{p,q}(Y^{\prime\prime})=\lambda_{p,q}(Y^{\prime\prime}_{1})+\lambda_{p,q}(Y^{\prime\prime}_{2})$.
By the presumed embedding independence of $\Lambda(Y_{1})$ and
$\Lambda(Y_{2})$, the same follows for $\Lambda(Y)$, except for columns
$n,n-1$.

In column $n$ all entries in all cases are zero by the Hartshorne–Lichtenbaum
theorem. So is the diagonal entry $\lambda_{1,1}$ for all three ideals by
equi-dimensionality. Thus,
$\lambda_{0,1}(Y_{i})=\lambda_{0,1}(Y^{\prime}_{i})+\lambda_{0,1}(Y^{\prime\prime}_{i})+1$
for $i=1,2$ as follows from the Grothendieck spectral sequence (which implies
that the alternating sum of all $\lambda_{p.q}$ is 1). Therefore, all
Lyubeznik numbers of $\tilde{Y}$ are embedding independent. ∎

######  Theorem 3.20.

Let $\tilde{Y}$ be an equi-dimensional projective scheme of dimension two,
which

  1. (1)

either satisfies locally everywhere Serre’s condition $S_{2}$,

  2. (2)

or has Picard number one.

Then the Lyubeznik numbers of all affine cones $Y$ over $\tilde{Y}$ agree.

###### Proof.

Let $Y$ be any cone over $\tilde{Y}$. It is a scheme of pure dimension 3, and
thus by Remark 1.3, the Lyubeznik table of $Y$ is

$\begin{pmatrix}\cdot&\lambda_{0,1}&\lambda_{0,2}&\cdot\\\
\cdot&\cdot&\lambda_{1,2}&\cdot\\\ \cdot&\cdot&\cdot&\lambda_{2,3}\\\
\cdot&\cdot&\cdot&\lambda_{3,3}\end{pmatrix}.$

By Lemma 3.19, we can assume that $\tilde{Y}$ is connected. That assures that
$\lambda_{0,1}$ is zero by the Second Vanishing Theorem [Har68, Theorem 7.5].

If $\tilde{Y}$ is $(S_{2})$ locally everywhere then by Lemma 3.18,
$H^{n-2}_{I}(R_{n})$ has support dimension zero and is the top local
cohomology module, and so $\lambda_{1,2}=0$. It follows from [Lyu93] that
$H^{n-2}_{I}(R_{n})$ is injective. By Corollary 2.8, the socle dimension
$\lambda_{0,2}$ of this module is determined by the topology of $\tilde{Y}$.
Finally, the convergence of the spectral sequence to
$H^{n}_{\mathfrak{m}}(R_{n})$ implies that $\lambda_{2,3}=\lambda_{0,2}$.

Suppose now that $\tilde{Y}$ has Picard number one. Then by Lemma 3.2, the
$\lambda_{i,j}$ with $i>1$ are a function of $\tilde{Y}$ alone. The only
possibly nonzero differentials are:

  * •

on page two the morphism $E^{0,n-2}_{2}\longrightarrow E^{2,n-3}_{2}$ and
$E^{1,n-2}_{2}\longrightarrow E^{3,n-3}_{2}$;

  * •

on page three the morphism $E^{0,n-1}_{3}\longrightarrow E^{3,n-3}_{3}$.

Convergence of the spectral sequence forces $E^{0,n-2}_{2}\longrightarrow
E^{2,n-3}_{2}$ to be an isomorphism222This isomorphism property holds for any
ideal $I$ of dimension greater than two., and the maps
$E^{1,n-2}_{2}\longrightarrow E^{3,n-3}_{2}$ and $E^{0,n-1}_{3}\longrightarrow
E^{3,n-3}_{3}$ to be injective. Moreover, the cokernel of
$E^{0,n-1}_{3}\longrightarrow E^{3,n-3}_{3}$ must be one copy of
$H^{n}_{\mathfrak{m}}(R_{n})$.

Since all modules in $E^{p,q}_{\geq 2}$ are injective, socle dimensions are
additive in short exact sequences. Thus, $\lambda_{0,2}=\lambda_{2,3}$, and
$\lambda_{3,3}=\lambda_{1,2}+\lambda_{0,1}+1=\lambda_{1,2}+1$. This settles
the claim for $\lambda_{0,2}$. But $\lambda_{3,3}$ is a function of
$\tilde{Y}$ by [Zha07], and it follows that $\lambda_{1,2}$ is a function of
$\tilde{Y}$ as well. ∎

######  Theorem 3.21.

Let $\tilde{Y}$ be a projective complex scheme that is of equi-dimension
three. Assume that every local ring $\mathcal{O}_{\tilde{Y},\tilde{y}}$
satisfies $(S_{2})$ and that the Picard group of $\tilde{Y}$ has rank 1. Then
$\Lambda(Y)$ is independent of the choice of the cone $Y$ for $\tilde{Y}$.

###### Proof.

By Lemma 3.19 we can assume that $\tilde{Y}$ is connected. This forces
$\lambda_{0,1}(Y)=0$ for any cone $Y$ of $\tilde{Y}$ by the Second Vanishing
Theorem [Har68, Theorem 7.5].

Using the equi-dimensionality and the $(S_{2})$-property, the Lyubeznik table
is by Remark 1.3 and Lemma 3.18 equal to

$\Lambda=\begin{pmatrix}\cdot&\cdot&\lambda_{0,2}&\lambda_{0,3}&\cdot\\\
\cdot&\cdot&\cdot&\lambda_{1,3}&\cdot\\\
\cdot&\cdot&\cdot&\cdot&\lambda_{2,4}\\\
\cdot&\cdot&\cdot&\cdot&\lambda_{3,4}\\\
\cdot&\cdot&\cdot&\cdot&\lambda_{4,4}\end{pmatrix},$

Moreover, $H^{n-2}_{I}(R_{n})$ is supported only in the origin, hence
injective. By Corollary 2.8, its socle dimension is the dimension of the top
de Rham group of the affine cone complement. By Proposition 2.16, this
dimension is well-defined. Thus, $\lambda_{0,2}$ is a function of $\tilde{Y}$
alone, reflecting the de Rham group
$H^{2n-2}(\mathbb{A}^{n}_{\mathbb{K}}\smallsetminus Y)$ independent of the
choice of the cone.

Convergence of the spectral sequence forces, similarly to the proof of Theorem
3.20, that $\lambda_{3,4}=\lambda_{1,3}+\lambda_{0,2}$ and that
$\lambda_{2,4}=\lambda_{0,3}$. By the Picard number condition, $\lambda_{2,4}$
is the same for every cone, and hence so is $\lambda_{0,3}$. Since
$\lambda_{0,2}$ is a function of $\tilde{Y}$, and since $\lambda_{\geq 2,*}$
is independent of the embedding by the Picard number condition, the same is
true for $\lambda_{1,3}$. ∎

######  Theorem 3.22.

Let $\tilde{Y}$ be a projective complex scheme of equi-dimension four. Assume
that $\tilde{Y}$ is locally everywhere $(S_{3})$, and that the Picard group of
$\tilde{Y}$ has rank 1. Then $\Lambda(Y)$ is independent of the choice of the
cone $Y$ for $\tilde{Y}$.

###### Proof.

By Lemma 3.19 we can assume that $\tilde{Y}$ is connected. This forces
$\lambda_{0,1}(Y)=0$ for any cone $Y$ of $\tilde{Y}$ by the Second Vanishing
Theorem [Har68, Theorem 7.5].

Write $\tilde{Y}=\operatorname{\textup{Proj}\,}(R/I)$ where
$R=\mathbb{C}[x_{1},\dots,x_{n}]$. Since $(S_{3})$ implies $(S_{2})$, Remark
1.3 and Lemma 3.18 assure that the Lyubeznik table of $R/I$ is

$\Lambda=\begin{pmatrix}\cdot&\cdot&\lambda_{0,2}&\lambda_{0,3}&\lambda_{0,4}&\cdot\\\
\cdot&\cdot&\cdot&\lambda_{1,3}&\lambda_{1,4}&\cdot\\\
\cdot&\cdot&\cdot&\cdot&\lambda_{2,4}&\lambda_{2,5}\\\
\cdot&\cdot&\cdot&\cdot&\cdot&\lambda_{3,5}\\\
\cdot&\cdot&\cdot&\cdot&\cdot&\lambda_{4,5}\\\
\cdot&\cdot&\cdot&\cdot&\cdot&\lambda_{5,5}\end{pmatrix}.$

Now take a prime ${\mathfrak{p}}$ of height $n-2$ that contains $I$. Then
$\operatorname{\textup{depth}}((R_{n}/I)_{\mathfrak{p}})=3$ and so by [DT16,
Corollary 2.8], $(H^{(n-2)-3+1}_{I}(R_{n}))_{\mathfrak{p}}=0$. Thus,
$\dim(H^{n-4}_{I}(R_{n}))\leq 1$ and $\lambda_{2,4}=0$.

Localizing at primes of height $n-1$ yields, with the result of Dao and Takagi
[DT16, Corollary 2.8], that $H^{n-2}_{I}(R_{n})$ and $H^{n-3}_{I}(R_{n})$ are
Artinian. It follows that $\lambda_{1,3}=0$, and $f_{Y}\leq n-3$. By Corollary
3.15, since the Picard number is one, $\lambda_{0,2}$ and $\lambda_{0,3}$ are
independent of the embedding choice.

Convergence of the spectral sequence to $H^{n}_{\mathfrak{m}}(R)$ forces that

$\lambda_{0,4}=\lambda_{2,5}\qquad{\rm
and}\qquad\lambda_{1,4}=\lambda_{3,5}-\lambda_{0,3}\qquad{\rm(and}\qquad\lambda_{0,2}=\lambda_{4,5}).$

As the Picard number is one, the $\lambda_{i,j}$ are independent of embeddings
for all $i\geq 2$ and all $j$. This then fixes all $\lambda_{p,q}$. ∎

###### Proof of Theorem 3.9.

Toric projective varieties are connected and locally the spectra of semigroup
rings to saturated semigroups. They are hence normal, and by Hochster’s
theorem Cohen–Macaulay, [Hoc72]. The coordinate ring $R_{n}/I$ of the cone $Y$
thus has a Lyubeznik table as in the proof of Theorem 3.21. Moreover,
$H^{n-2}_{I}(R_{n})$ is Artinian.

Additional vanishings are due to the $(S_{3})$-condition. As in the proof of
Theorem 3.22, localization at a prime of $R_{n}$ of height $n-1$ shows with
[DT16, Thm. 2.8] that the support of $H^{n-3}_{I}(R_{n})$ is zero-dimensional,
hence $\lambda_{1,3}=0$.

At this point, let us assume that $\tilde{Y}$ is not a hypersurface, and hence
of codimension two or more.

If $\lambda_{0,2}$ is nonzero, it is therefore the dimension of
$H^{n}_{\mathrm{dR}}(H^{n-2}_{I}(R_{n}))=H^{2n-3}(U;\mathbb{C})$ where $U$ is
the affine complement of $Y$. By the spectral sequence argument in the proof
of Lemma 2.7, it also equals the dimension of the top cohomology group
$H^{2n-4}(\mathbb{P}U;\mathbb{C})$ of the projective complement $\mathbb{P}U$.

The long exact sequence (2.3.1) takes the form

$H^{2n-4}_{\tilde{Y}}(\mathbb{P}^{n-1};\mathbb{C})\longrightarrow
H^{2n-4}(\mathbb{P}^{n-1};\mathbb{C})\longrightarrow
H^{2n-4}(\mathbb{P}U;\mathbb{C})\longrightarrow
H^{2n-3}_{\tilde{Y}}(\mathbb{P}^{n-1};\mathbb{C})\longrightarrow\cdots$

and by [Ive86, V.6.6] $H^{2n-3}_{\tilde{Y}}(\mathbb{P}^{n-1};\mathbb{C})$ is
dual to $H^{1}_{c}(\tilde{Y};\mathbb{C})=H^{1}(\tilde{Y};\mathbb{C})$. But
projective toric varieties (or more generally toric varieties to a fan with a
full-dimensional cone) are simply connected by [Ful93, 3.2]. So
$H^{1}(\tilde{Y};\mathbb{C})$ and
$H^{2n-3}_{\tilde{Y}}(\mathbb{P}^{n-1};\mathbb{C})$ are zero.

The morphism $H^{2n-4}_{\tilde{Y}}(\mathbb{P}^{n-1};\mathbb{C})\longrightarrow
H^{2n-4}(\mathbb{P}^{n-1};\mathbb{C})$ is via Alexander and Poincaré duality
dual to the (injective) restriction morphism
$H^{2}(\mathbb{P}^{n-1};\mathbb{C})\longrightarrow
H^{2}(\tilde{Y};\mathbb{C})$, hence itself surjective. It follows that
$H^{2n-4}(\mathbb{P}U;\mathbb{C})=H^{2n-3}(U;\mathbb{C})=0$, and hence
$\lambda_{0,2}$ and $H^{n-2}_{I}(R_{n})$ are both zero.

It now follows that actually the Artinian module $H^{n-3}_{I}(R_{n})$ is the
top local cohomology group of $I$, and $H^{2n-5}(\mathbb{P}U;\mathbb{C})$ is
the top cohomology group of $\mathbb{P}U$. Repeating the above computations,
we now have a long exact sequence

$0=H^{2n-5}(\mathbb{P}^{n-1};\mathbb{C})\longrightarrow
H^{2n-5}(\mathbb{P}U;\mathbb{C})\longrightarrow
H^{2n-4}_{\tilde{Y}}(\mathbb{P}^{n-1};\mathbb{C})\longrightarrow
H^{2n-4}(\mathbb{P}^{n-1};\mathbb{C})\longrightarrow\cdots$

in which the arrow
$H^{2n-4}_{\tilde{Y}}(\mathbb{P}^{n-1};\mathbb{C})\longrightarrow
H^{2n-4}(\mathbb{P}^{n-1};\mathbb{C})$ is dual to the (injective) morphism
$H^{2}(\mathbb{P}^{n-1};\mathbb{C})\longrightarrow
H^{2}(\tilde{Y};\mathbb{C})$, and where
$H^{2n-5}(\mathbb{P}U;\mathbb{C})=H^{2n-4}(U;\mathbb{C})$ is a vector space of
dimension $\lambda_{0,3}$.

For projective toric varieties (and more generally, when all cones of the fan
are top-dimensional), the Picard group of $\tilde{Y}$ is isomorphic to
$H^{2}(\tilde{Y};\mathbb{C})$, [CLS11, Thm. 12.3.2]. The long exact sequence
above thus shows the equation $\lambda_{0,3}=\tilde{p}$.

Finally, by convergence of the spectral sequence, $\lambda_{4,4}=1$ and
$\lambda_{3,4}=\lambda_{0,2}=0$ and $\lambda_{2,4}=\lambda_{0,3}$.

This settles the problem for all embeddings in which $\tilde{Y}$ is not a
hypersurface. If in some embedding $\tilde{Y}$ happens to be a hypersurface,
necessarily in $\mathbb{P}^{4}_{\mathbb{C}}$, its Lyubeznik table is trivial
for this embedding, simply for lack of higher local cohomology. On the other
hand, [Gro05, Exp. XII, Cor 3.7] asserts that the Picard group of $\tilde{Y}$
is then cyclic, equal to that of $\mathbb{P}^{4}_{\mathbb{C}}$. Thus,
$\tilde{p}$ is zero and we see that all Lyubeznik tables of $\tilde{Y}$ agree.
∎

## Acknowledgements

We would like to thank Winfried Bruns, Boris Pasquier and Alex Dimca for
helpful conversations.

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