# Weighted Ehrhart theory via equivariant toric geometry

Laurenţiu Maxim  L. Maxim : Department of Mathematics, University of
Wisconsin-Madison, 480 Lincoln Drive, Madison WI 53706-1388, USA,
and
Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 70700
Bucharest, ROMANIA<EMAIL_ADDRESS>and  Jörg Schürmann  J. Schürmann :
Mathematische Institut, Universität Münster, Einsteinstr. 62, 48149 Münster,
Germany<EMAIL_ADDRESS>
###### Abstract.

We give a $K$-theoretic and geometric interpretation for a generalized
weighted Ehrhart theory of a full-dimensional lattice polytope $P$, depending
on a given homogeneous polynomial function $\varphi$ on $P$, and with Laurent
polynomial weights $f_{Q}(y)\in{\mathbb{Z}}[y^{\pm 1}]$ associated to the
faces $Q\preceq P$ of the polytope. For this purpose, we calculate equivariant
$K$-theoretic Hodge-Chern classes of an equivariant mixed Hodge module
${\mathcal{M}}$ on the toric variety $X_{P}$ associated to $P$ (defined via an
equivariant embedding of $X_{P}$ into an ambient smooth variety). For any
integer $\ell$, we introduce a corresponding equivariant Hodge
$\chi_{y}$-polynomial $\chi_{y}(X_{P},\ell D_{P};[{\mathcal{M}}])$, with
$D_{P}$ the corresponding ample Cartier divisor on $X_{P}$ (defined by the
facet presentation of $P$). Motivic properties of the Hodge-Chern classes are
used to express this equivariant Hodge polynomial in terms of weighted
character sums fitting with a generalized weighted Ehrhart theory. The
equivariant Hodge polynomials are shown to satisfy a reciprocity and purity
formula fitting with the duality for equivariant mixed Hodge modules, and
implying the corresponding properties for the generalized weighted Ehrhart
polynomials. In the special case of the equivariant intersection cohomology
mixed Hodge module, with the weight function corresponding to Stanley’s
$g$-function of the polar polytope of $P$, we recover in geometric terms a
recent combinatorial formula of Beck–Gunnells–Materov.

###### Key words and phrases:

Toric varieties, lattice polytopes, Ehrhart polynomial, reciprocity, purity,
equivariant Hodge-Chern classes, equivariant mixed Hodge module, equivariant
Serre duality

###### 2020 Mathematics Subject Classification:

14M25, 14C17, 14C40, 19L47, 52B20, 55N91

##  1\. Introduction

We provide a geometric (equivariant) perspective for the generalized weighted
Ehrhart theory developed in [3] for a full-dimensional lattice polytope $P$
with a homogeneous polynomial function $\varphi$ defined on it. This is
achieved by attaching Laurent polynomial weights
$f_{Q}(y)\in{\mathbb{Z}}[y^{\pm 1}]$ of geometric origin to the faces
$\emptyset\neq Q\preceq P$ of the polytope (with the partial order given by
face inclusion). The Brion-Vergne [8] combinatorial approach to reciprocity
can be linearly extended to this generalized weighted Ehrhart theory.
Motivation for using such weights comes from our prior work [9] on equivariant
Hodge-Chern and Hirzebruch characteristic classes of toric varieties
(associated to such polytopes), as well as from the use of Stanley’s
$g$-functions in the generalized Ehrhart reciprocity theorem of [3].

###  1.1. Combinatorial and geometrical posets and their identification

We review here some basic notions explaining the relation between lattice
polytopes and toric geometry; for more details see [11, 17].

Let $M\cong{\mathbb{Z}}^{n}$ be a lattice and $P\subset
M_{{\mathbb{R}}}:=M\otimes{\mathbb{R}}\cong{\mathbb{R}}^{n}$ be a full-
dimensional lattice polytope. To $P$ one associates a fan
$\Sigma=\Sigma_{P}\subset N_{\mathbb{R}}:=N\otimes{\mathbb{R}}$, with $N$ the
dual lattice of $M$, called the inner normal fan of $P$. To the fan
$\Sigma_{P}$ one further associates a projective toric variety $X=X_{P}$ with
torus ${\mathbb{T}}$ (whose character lattice is $M$) and an ample Cartier
divisor $D=D_{P}$. There is an order-reversing one-to-one correspondence
between the faces $\emptyset\neq Q\preceq P$ of $P$ and the cones $\sigma_{Q}$
of $\Sigma_{P}$, with $\dim_{\mathbb{R}}(Q)=n-\dim_{\mathbb{R}}(\sigma_{Q})$.
There is an order-reversing one-to-one correspondence between the cones
$\sigma_{Q}$ of $\Sigma_{P}$ and the ${\mathbb{T}}$-orbits $O_{Q}$ of the
toric variety $X_{P}$, with
$\dim_{\mathbb{C}}(O_{Q})=n-\dim_{\mathbb{R}}(\sigma_{Q})$. Then the
identification between the faces $\emptyset\neq Q\preceq P$ of $P$ and the
torus orbits $O_{Q}$ of the toric variety $X_{P}$ is order-preserving, with
$\dim_{\mathbb{R}}(Q)=\dim_{\mathbb{C}}(O_{Q})$.

If, in addition, $P$ contains the origin in its interior, then its polar
polytope $P^{\circ}\subset N_{\mathbb{R}}$ is defined as in [17, Section 1.5],
as a full-dimensional lattice polytope with respect to the lattice $N$, and
containing the origin in the interior. By taking cones at the origin of
$N_{\mathbb{R}}$ over the proper faces of $P^{\circ}$, with the empty set
$\emptyset$ corresponding to the origin, one gets the same lattice fan
$\Sigma_{P}$ (hence the same toric variety $X_{P}$). This correspondence
between the proper faces $\emptyset\preceq Q^{\circ}\prec P^{\circ}$ of
$P^{\circ}$ and the cones of $\Sigma_{P}$ is order-preserving, increasing real
dimension by $1$. (Here $\dim_{\mathbb{R}}\emptyset=-1$ by convention.) This
induces an order-reversing one-to-one correspondence between the faces $Q$ of
$P$, and the faces $Q^{\circ}$ of the polar polytope $P^{\circ}$, switching
the roles of polytopes and emptysets (seen as faces). Moreover, for a proper
face $\emptyset\neq Q\prec P$ of $P$, one has
$\dim_{\mathbb{R}}(Q)+\dim_{\mathbb{R}}(Q^{\circ})=n-1$.

The original polytope $P\subset M_{\mathbb{R}}$ is classically used for
counting lattice points (and the corresponding Ehrhart theory), fitting with
sections of the ample line bundle ${\mathcal{O}}_{X_{P}}(D_{P})$ on $X_{P}$
[12, 17, 11]. The torus orbits of $X_{P}$ give a natural
${\mathbb{T}}$-invariant Whitney stratification, which is particularly useful
in our geometric interpretation of weighted lattice points counting via the
theory of Hirzebruch homology classes $T_{y*}$ of [6] and the corresponding
Hirzebruch-Riemann-Roch theorem; see [21, 22]. The polar polytope $P^{\circ}$
appears in Stanley’s work [29] for recursively defining his $g$-polynomials
$g_{Q^{\circ}}(t)\in{\mathbb{Z}}[t]$, with $g(\emptyset)=1$, for the Eulerian
graded poset given by the faces $\emptyset\preceq Q^{\circ}\preceq P^{\circ}$.
It is well-known [29, 5, 13, 16, 27] that these $g$-polynomials are related to
the intersection cohomology complex of $X_{P}$. In the use of the
$g$-polynomials, we will (implicitly) assume that $P$ contains the origin in
its interior (so that the polar polytope is well-defined).

###  1.2. Generalized weighted Ehrhart theory

Let $\varphi\colon M_{\mathbb{R}}\cong{\mathbb{R}}^{n}\to{\mathbb{C}}$ be a
homogeneous polynomial function. We assign to the posets mentioned in the
previous subsection Laurent polynomial weights via a weight vector
$f=\\{f_{Q}\\}$, with $f_{Q}(y)\in{\mathbb{Z}}[y^{\pm 1}]$ indexed here by the
non-empty faces $\emptyset\neq Q\preceq P$ of $P$. Motivated by [3], we
assemble these weights, the combinatorics of $P$, and the polynomial function
$\varphi$ into the generalized weighted Ehrhart polynomial
$E^{\varphi}_{P,f}(\ell,y)$, resp., $\widetilde{E}^{\varphi}_{P,f}(\ell,y)$,
as follows.

######  Definition 1.1.

For $\ell\in{\mathbb{Z}}_{>0}$, define the generalized weighted Ehrhart
polynomial of $P$, $f$ and $\varphi$ by

(1.1) $\begin{split}E^{\varphi}_{P,f}(\ell,y)&:=\sum_{Q\preceq
P}f_{Q}(y)\cdot(1+y)^{\dim(Q)+\deg(\varphi)}\cdot\sum_{m\in{\rm
Relint}({{\ell}}Q)\cap M}\varphi(m)\\\
&=:(1+y)^{\deg(\varphi)}\cdot\widetilde{E}^{\varphi}_{P,f}(\ell,y),\end{split}$

with ${\rm Relint}({{\ell}}Q)$ denoting the relative interior of the face
${\ell}Q$ of the dilated polytope $\ell P$.

The renormalization $\widetilde{E}^{\varphi}_{P,f}(\ell,y)$ is more natural to
work with from the geometric perspective considered in this paper. The Brion-
Vergne [8] combinatorial approach to reciprocity can be linearly extended to
this generalized weighted Ehrhart theory, so that $E^{\varphi}_{P,f}(\ell,y)$,
resp., $\widetilde{E}^{\varphi}_{P,f}(\ell,y)$, have the following properties:

  1. (1)

$E^{\varphi}_{P,f}(\ell,y)$ is obtained by evaluating a polynomial
$E^{\varphi}_{P,f}(z,y)$ at $z=\ell\in{\mathbb{Z}}_{>0}$. Similarly for
$\widetilde{E}^{\varphi}_{P,f}(\ell,y)$.

  2. (2)

(Constant term) For $\ell=0$,

(1.2) $E^{\varphi}_{P,f}(0,y)=\sum_{Q\preceq
P}f_{Q}(y)\cdot(-1-y)^{\dim(Q)+\deg(\varphi)}\cdot\varphi(0),$

that is, evaluating $\sum_{m\in{\rm Relint}({{\ell}}Q)\cap M}\varphi(m)$ at
$\ell=0$ as $(-1)^{\dim(Q)+\deg(\varphi)}\cdot\varphi(0)$ (see [8, Prop.4.1]).
Similarly,

(1.3) $\widetilde{E}^{\varphi}_{P,f}(0,y)=\sum_{Q\preceq
P}f_{Q}(y)\cdot(-1-y)^{\dim(Q)}\cdot(-1)^{\deg(\varphi)}\cdot\varphi(0).$

  3. (3)

(Reciprocity formula) For $\ell\in{\mathbb{Z}}_{>0}$, using [8, Prop.4.1] for
each face $Q$ of $P$, one gets

(1.4) $E^{\varphi}_{P,f}(-\ell,y)=\sum_{Q\preceq
P}f_{Q}(y)\cdot(-1-y)^{\dim(Q)+\deg(\varphi)}\cdot\sum_{m\in{{\ell}}Q\cap
M}\varphi(m).$

Similarly,

(1.5) $\widetilde{E}^{\varphi}_{P,f}(-\ell,y)=\sum_{Q\preceq
P}f_{Q}(y)\cdot(-1-y)^{\dim(Q)}\cdot(-1)^{\deg(\varphi)}\cdot\sum_{m\in{{\ell}}Q\cap
M}\varphi(m).$

Assume now that the origin is an interior point of $P$, so that we can
consider the weight vector given by Stanley’s $g$-polynomials

(1.6) $f_{Q}(y)=g_{Q^{\circ}}(-y)=:\widetilde{g}_{Q}(-y)$

for the polar polytope of $P$, with
$g_{\emptyset}(-y)=\widetilde{g}_{P}(-y)=1$. In this setup, it was shown in
[3, Thm.1.3, Thm.2.6] that the following purity property holds:

(1.7) $E^{\varphi}_{P,f}(-\ell,y)=(-y)^{\dim(P)+\deg(\varphi)}\cdot
E^{\varphi}_{P,f}(\ell,1/y).$

This property can be reformulated as

(1.8)
$\widetilde{E}^{\varphi}_{P,f}(-\ell,y)=(-y)^{\dim(P)}\cdot(-1)^{\deg(\varphi)}\cdot\widetilde{E}^{\varphi}_{P,f}(\ell,{1}/{y}).$

Let us note here that our definition (1.1) for $E^{\varphi}_{P,f}(\ell,y)$
equals $G_{\varphi}(\ell,y)$ from [3, formula (14)]. In the case when $P$ is a
simple polytope, the polar polytope $P^{\circ}$ is simplicial, so that
$g_{Q^{\circ}}(-y)=1$, for all faces $Q$ of $P$. In this case, as explained in
[3], the equality (1.7) implies the Dehn-Sommerville relations for $P$.

If $\varphi=1$ in (1.1), we recover the weighted Ehrhart polynomials studied
in [22] via the theory of mixed Hodge modules on the toric variety $X_{P}$.

###  1.3. Geometric viewpoint on generalized weighted Ehrhart theory

In this paper, we give a geometric interpretation and proofs of the above
properties for any homogeneous polynomial function $\varphi$ and any weight
vector $f$ via the theory of equivariant mixed Hodge modules on a projective
toric variety and their $K$-theoretic equivariant characteristic classes.
Moreover, using the equivariant intersection Hodge module ${IC}^{H}_{X_{P}}$
and its purity, we give a geometric proof of formula (1.7).

####  1.3.1. Geometric realization of weight vectors and duality

As already indicated in [22], we identify the weight vectors $\\{f_{Q}\\}$,
seen as ${\mathbb{Z}}[y^{\pm 1}]$-valued functions on the set of faces $Q$ of
$P$, with the ${\mathbb{T}}$-invariant ${\mathbb{Z}}[y^{\pm 1}]$-valued
constructible functions on $X=X_{P}$, via

$\\{f_{Q}\\}\mapsto\sum_{Q\preceq P}1_{O_{Q}}\cdot f_{Q}\in
F^{{\mathbb{T}}}(X)[y^{\pm 1}].$

Note that $F^{{\mathbb{T}}}(X)[y^{\pm 1}]$ is a ${\mathbb{Z}}[y^{\pm
1}]$-module freely generated by the ${\mathbb{T}}$-invariant constructible
functions $1_{O_{Q}}\in F^{{\mathbb{T}}}(X)\subset F^{{\mathbb{T}}}(X)[y^{\pm
1}].$ In what follows, we adapt the formulations from [22] in motivic and
Hodge-theoretic terms to the equivariant context.

There is a tautological (injective) homomorphism

$\tau^{\mathbb{T}}:F^{{\mathbb{T}}}(X)[y^{\pm 1}]\longrightarrow
K^{\mathbb{T}}_{0}(var/X)[{\mathbb{L}}^{-1}]$

to the equivariant Grothendieck group of ${\mathbb{T}}$-varieties over $X$,
localized at the Lefschetz motive ${\mathbb{L}}=[{\mathbb{C}}\to pt]$ (with
the trivial ${\mathbb{T}}$-action on ${\mathbb{C}}$), given by

$1_{O_{Q}}\cdot f_{Q}(y)\mapsto[O_{Q}\hookrightarrow X]\cdot
f_{Q}(-{\mathbb{L}}).$

Here, $K^{\mathbb{T}}_{0}(var/X)$ is a $K^{\mathbb{T}}_{0}(var/pt)$-module via
the identification $X\times pt\simeq X$.

In order to identify the equivariant intersection cohomology complex and
duality we need to go one step further, and work with
${\mathbb{T}}$-equivariant mixed Hodge modules. For technical reasons, here we
need to choose a ${\mathbb{T}}$-equivariant closed embedding
$i:X\hookrightarrow Z$ into an ambient smooth ${\mathbb{T}}$-variety $Z$, and
work with the Grothendieck group $K_{0}({\rm MHM}^{\mathbb{T}}_{X}(Z))$ of
equivariant mixed Hodge modules on $Z$ with support on $X$. There is a
canonical homomorphism

$\chi^{\mathbb{T}}_{Hdg}:K^{\mathbb{T}}_{0}(var/X)[{\mathbb{L}}^{-1}]\to
K_{0}({\rm MHM}^{\mathbb{T}}_{X}(Z))$

such that

$\chi^{\mathbb{T}}_{Hdg}([O_{Q}\hookrightarrow X]\cdot
f_{Q}(-{\mathbb{L}}))=[(j_{Q})_{!}{\mathbb{Q}}^{H}_{O_{Q}}]\cdot f_{Q}(-u),$

with $j_{Q}:O_{Q}\hookrightarrow X\hookrightarrow Z$ the orbit inclusion in
the ambient smooth ${\mathbb{T}}$-variety $Z$, and ${\mathbb{Q}}^{H}_{O_{Q}}$
denoting the equivariant constant Hodge module on the orbit $O_{Q}$. Here,

$u=[{\mathbb{Q}}(-1)]\in K_{0}({\rm MHM}^{\mathbb{T}}(pt))\simeq K_{0}({\rm
MHM}(pt))\simeq K_{0}({\rm MHS}^{p}),$

and the module structure on $K_{0}({\rm MHM}^{\mathbb{T}}_{X}(Z))$ is induced
from the exterior product for the identification $Z\simeq pt\times Z$. Also,
${\rm MHS}^{p}$ is the abelian category of graded polarizable
${\mathbb{Q}}$-mixed Hodge structures, with ${\mathbb{Q}}(-1)$ the
corresponding Tate Hodge structure of weight $2$.

In our geometric applications, we recover the corresponding weight vector
$\\{f_{Q}\\}$ via

$f_{Q}(y):=\chi_{y}(i_{x_{Q}}^{*}[{\mathcal{M}}])$

as the Hodge $\chi_{y}$-polynomial of the stalk class
$i_{x_{Q}}^{*}[{\mathcal{M}}]$, with $[{\mathcal{M}}]\in K_{0}({\rm
MHM}^{\mathbb{T}}_{X}(Z))$, and $i_{x_{Q}}:\\{x_{Q}\\}\hookrightarrow Z$ the
inclusion (in the ambient smooth ${\mathbb{T}}$-variety) of a point $x_{Q}$
chosen in the orbit $O_{Q}$. Then our aim is to show that many invariants of
$[{\mathcal{M}}]$ depend only of this choice of a weight function.

Specific to the toric context is the fact that the stabilizers of the torus
action are connected, so that, using [31, Lem.1.2], any class
$[{\mathcal{M}}]\in K_{0}({\rm MHM}^{\mathbb{T}}_{X}(Z))$ can be decomposed as
(see Corollary 3.2)

(1.9) $\begin{split}[{\mathcal{M}}]&=\sum_{Q\preceq
P}\,\big{[}(j_{Q})_{!}{\mathbb{Q}}_{O_{Q}}^{H}\big{]}\cdot
i_{x_{Q}}^{*}[{\mathcal{M}}],\end{split}$

and the stalk classes do not see the ${\mathbb{T}}$-action, i.e.,

$i_{x_{Q}}^{*}[{\mathcal{M}}]\in K_{0}({\rm MHM}^{\mathbb{T}}(pt))\simeq
K_{0}({\rm MHM}(pt))\simeq K_{0}({\rm MHS}^{p}).$

Since the composed map

${\mathbb{Z}}[u^{\pm 1}]\longrightarrow K_{0}({\rm
MHS}^{p})\overset{\chi_{y}}{\longrightarrow}{\mathbb{Z}}[y^{\pm 1}]$

is the isomorphism given by $u\mapsto-y$, we get the injectivity of the
composition

$F^{{\mathbb{T}}}(X)[y^{\pm 1}]\longrightarrow
K^{\mathbb{T}}_{0}(var/X)[{\mathbb{L}}^{-1}]\longrightarrow K_{0}({\rm
MHM}^{\mathbb{T}}_{X}(Z)).$

In this way we also see that any weight vector $f$ is realized by the stalk
class of a certain equivariant mixed Hodge module complex via
$f_{Q}(y):=\chi_{y}(i_{x_{Q}}^{*}[{\mathcal{M}}])$.

Let $IC^{H}_{X}\in{\rm MHM}^{\mathbb{T}}_{X}(Z)$ be the equivariant
intersection cohomology Hodge module of $X$, and
${IC^{\prime}}^{H}_{X}:=IC^{H}_{X}[-n]$ with
$n=\dim_{\mathbb{C}}(X)=\dim_{\mathbb{R}}(P)$. Since ${IC}^{H}_{X}$ is pure of
weight $n$, we get that
${\mathbb{D}}^{\mathbb{T}}_{Z}{IC^{\prime}}^{H}_{X}\simeq{IC^{\prime}}^{H}_{X}[2n](n)$,
with ${\mathbb{D}}_{Z}^{\mathbb{T}}$ the equivariant duality functor of
equivariant mixed Hodge modules on the ambient smooth ${\mathbb{T}}$-variety
$Z$. This implies the following key purity property:

(1.10)
${\mathbb{D}}^{\mathbb{T}}_{Z}[{IC^{\prime}}^{H}_{X}]=u^{-n}\cdot[{IC^{\prime}}^{H}_{X}]\in
K_{0}({\rm MHM}^{\mathbb{T}}_{X}(Z)).$

As already explained in the non-equivariant context of [22], the weight vector
associated to $[{IC^{\prime}}^{H}_{X}]$ is the collection of Stanley’s
$g$-functions $\widetilde{g}:=\\{\widetilde{g}_{Q}(-y)\\}$, i.e.,
$\widetilde{g}_{Q}(-y)=\chi_{y}(i_{x_{Q}}^{*}[{IC^{\prime}}^{H}_{X}])$ (cf.
also [5, 16, 13, 27]).

####  1.3.2. Twisted equivariant Hodge-Chern classes and generalized weighted
Ehrhart theory

We can now explain our geometric interpretation of the generalized weighted
Ehrhart theory introduced in Subsection 1.2, in terms of suitable twists (by
the line bundle of the ample Cartier divisor $D_{P}$ of $P$) of the
equivariant Hodge-Chern class transformation

$\hbox{\rm DR}^{\mathbb{T}}_{y}:K_{0}({\rm
MHM}^{\mathbb{T}}_{X}(Z))\longrightarrow K^{\mathbb{T}}_{0,X}(Z)[y^{\pm
1}]\simeq K^{\mathbb{T}}_{0}(X)[y^{\pm 1}],$

generalizing the corresponding non-equivariant Hodge-Chern classes of [6, 28].
This equivariant Hodge-Chern transformation is defined in terms of the
equivariant filtered De Rham complex of ${\mathcal{M}}\in{\rm
MHM}^{\mathbb{T}}_{X}(Z)$, which equivariantly can only be defined on the
ambient smooth ${\mathbb{T}}$-variety $Z$. Here,
$K^{\mathbb{T}}_{0,X}(Z)\simeq K^{\mathbb{T}}_{0}(X)$ is the Grothendieck
group of ${\mathbb{T}}$-equivariant coherent sheaves on ($Z$ with support in)
$X$.

The equivariant Hodge-Chern class transformation $\hbox{\rm
DR}^{\mathbb{T}}_{y}$ commutes with cross product $\boxtimes$ for the
identification $Z\times pt\simeq Z$. Moreover, it is functorial for
${\mathbb{T}}$-equivariant morphisms $f\colon(Z,X)\to(Z^{\prime},X^{\prime})$
of pairs, with $Z,Z^{\prime}$ smooth complex algebraic
${\mathbb{T}}$-varieties, and $f:Z\to Z^{\prime}$ restricting to a proper
morphism $f\colon X\to X^{\prime}$ of closed ${\mathbb{T}}$-invariant
subvarieties. In this paper, the functoriality is used for the closed
inclusions $k_{Q}:V_{Q}\hookrightarrow Z$ of the closure of the
${\mathbb{T}}$-orbits $O_{Q}$ into the ambient smooth ${\mathbb{T}}$-variety
$Z$, and their compositions with toric resolutions of these orbit closures, as
well as for taking degrees of these classes (i.e., pushing down to a point
space). These properties are then used to prove the following basic identity
(see (5.7)):

(1.11) $\hbox{\rm
DR}^{\mathbb{T}}_{y}(\big{[}(j_{Q})_{!}{\mathbb{Q}}^{H}_{O_{Q}}\big{]})=(1+y)^{\dim(O_{Q})}\cdot(k_{Q})_{*}[\omega_{V_{Q}}]_{\mathbb{T}},$

with $\omega_{V_{Q}}$ the ${\mathbb{T}}$-equivariant canonical (and dualizing)
sheaf of the orbit closure $V_{Q}$.

As explained in [14, Sect.3], the equivariant Hodge-Chern class transformation
$\hbox{\rm DR}^{\mathbb{T}}_{y}$ commutes with equivariant duality for
Grothendieck classes of equivariant mixed Hodge modules on $Z$ with support in
$X$, which are in the image of $\chi^{\mathbb{T}}_{Hdg}$, i.e., on such
elements one has

(1.12) $\hbox{\rm
DR}^{\mathbb{T}}_{y}\circ{\mathbb{D}}^{\mathbb{T}}_{Z}={\mathbb{D}}^{\mathbb{T}}\circ\hbox{\rm
DR}^{\mathbb{T}}_{y},$

Here, ${\mathbb{D}}^{\mathbb{T}}_{Z}$ is induced from the exact duality
functor of equivariant mixed Hodge modules (on $Z$, with support on $X$). For
${\mathbb{D}}^{\mathbb{T}}$, one can use either the equivariant Grothendieck
duality on the Grothendieck group $K^{\mathbb{T}}_{0,X}(Z)$ induced from
Grothendieck duality on $Z$ (remembering the support in $X$), or the
equivariant Grothendieck duality on the Grothendieck group
$K^{\mathbb{T}}_{0}(X)$ induced from Grothendieck duality on $X$ (see [20] for
the foundations of equivariant Grothendieck duality). As in the nonequivariant
context, ${\mathbb{D}}^{\mathbb{T}}$ is extended to
$K^{\mathbb{T}}_{0}(X)[y^{\pm 1}]$ by $y\mapsto 1/y$. With the above
notations, this duality property of $\hbox{\rm DR}_{y}^{\mathbb{T}}$ implies
our second basic formula (cf. (5.13)):

(1.13) $\hbox{\rm
DR}^{\mathbb{T}}_{y}(\big{[}(j_{Q})_{*}{\mathbb{Q}}^{H}_{O_{Q}}\big{]})=(1+y)^{\dim(O_{Q})}\cdot(k_{Q})_{*}[{\mathcal{O}}_{V_{Q}}]_{\mathbb{T}},$

Let now $D^{\prime}$ be a ${\mathbb{T}}$-invariant Cartier divisor on
$X=X_{P}$, e.g., $D^{\prime}=\ell D=\ell D_{P}$ for $\ell\in{\mathbb{Z}}$. Let
$[{\mathcal{M}}]\in K_{0}({\rm MHM}^{\mathbb{T}}_{X}(Z))$ be fixed. Then

$\hbox{\rm
DR}^{\mathbb{T}}_{y}([{\mathcal{M}}])\otimes{\mathcal{O}}_{X}(D^{\prime}):=\hbox{\rm
DR}^{\mathbb{T}}_{y}([{\mathcal{M}}])\cdot[{\mathcal{O}}_{X}(D^{\prime})]\in
K_{0}^{\mathbb{T}}(X)[y^{\pm 1}],$

where we extend the $K^{0}_{\mathbb{T}}(X)$-module structure of
$K_{0}^{\mathbb{T}}(X)$ induced from the tensor product linearly over
${\mathbb{Z}}[y^{\pm 1}]$. For $a_{X}:X\to pt$ the constant map as above, we
define the ${\mathbb{T}}$-equivariant Hodge polynomial of
$(X,D^{\prime},[{\mathcal{M}}])$ as:

(1.14)
$\begin{split}\chi^{\mathbb{T}}_{y}(X,D^{\prime};[{\mathcal{M}}])&:=(a_{X})_{*}\left(\hbox{\rm
DR}^{\mathbb{T}}_{y}([{\mathcal{M}}])\otimes{\mathcal{O}}_{X}(D^{\prime})\right)\in
K_{0}^{\mathbb{T}}(pt)[y^{\pm 1}]=K^{0}_{\mathbb{T}}(pt)[y^{\pm 1}]\\\
&=:\chi^{\mathbb{T}}(X;\hbox{\rm
DR}^{\mathbb{T}}_{y}([{\mathcal{M}}])\otimes{\mathcal{O}}_{X}(D^{\prime}))\in{\mathbb{Z}}[M][y^{\pm
1}],\end{split}$

which only depends on $[{\mathcal{M}}]$. In the second equality of the above
definition, we use the isomorphism
$K^{0}_{\mathbb{T}}(pt)\simeq{\mathbb{Z}}[M]$ induced by the eigenspace
decomposition by characters $\chi^{m}\in{\mathbb{Z}}[M]$ (for $m\in M$ an
element in the character lattice of ${\mathbb{T}}$), see (4.2).

With these notations, we can now formulate the first main result of this paper
(see Theorems 6.4, 6.8, 5.2):

######  Theorem 1.2.

Let $X=X_{P}$ be the projective toric variety with ample Cartier divisor
$D=D_{P}$ associated to a full-dimensional lattice polytope $P\subset
M_{{\mathbb{R}}}\cong{\mathbb{R}}^{n}$, and let $\ell\in{\mathbb{Z}}_{>0}$.
Let $Z$ be an ambient ${\mathbb{T}}$-manifold containing $X$ as a
${\mathbb{T}}$-invariant subvariety. For any face $Q$ of $P$, let $x_{Q}\in
O_{Q}$ be a point in the orbit $O_{Q}$ of $X$, with inclusion map
$i_{x_{Q}}:\\{i_{x_{Q}}\\}\hookrightarrow Z$, and set
$f_{Q}(y)=\chi_{y}(i^{*}_{x_{Q}}[{\mathcal{M}}])$. Then for any Grothendieck
class $[{\mathcal{M}}]\in K_{0}({\rm MHM}^{\mathbb{T}}_{X}(Z))$ of a mixed
Hodge module (complex) on $Z$ with support in $X$, we have:

  * (a)

(1.15) $\begin{split}\chi^{\mathbb{T}}_{y}(X,\ell
D;[{\mathcal{M}}])&=\sum_{Q\preceq
P}f_{Q}(y)\cdot(1+y)^{\dim(Q)}\cdot\sum_{m\in{\rm Relint}({{\ell}}Q)\cap
M}\chi^{-m},\end{split}$

with ${\rm Relint}$ denoting the relative interior of a face.

  * (b)

(1.16) $\begin{split}\chi^{\mathbb{T}}_{y}(X,-\ell
D;[{\mathcal{M}}])&=\sum_{Q\preceq
P}f_{Q}(y)\cdot(-1-y)^{\dim(Q)}\cdot\sum_{m\in{\ell}Q\cap
M}\chi^{m}.\end{split}$

  * (c)

(1.17) $\begin{split}\chi^{\mathbb{T}}_{y}(X,0\cdot
D;[{\mathcal{M}}])&=\sum_{Q\preceq
P}\left(f_{Q}(y)\cdot(-1-y)^{\dim(Q)}\right)\cdot\chi^{0}.\end{split}$

Here, for $\ell\in{\mathbb{Z}}_{>0}$, $\ell D_{P}=D_{\ell P}$ is the ample
Cartier divisor corresponding to the dilated polytope $\ell P$, with the same
toric variety $X_{\ell P}=X_{P}$.

All the formulae of the above theorem rely on the decomposition formula (1.9),
together with well-known character decompositions of
$H^{*}(X;{\mathcal{O}}(\ell D))$ and $H^{*}(X;{\mathcal{O}}(\ell
D)\otimes\omega_{X})$, for any integer $\ell$, as in the classical work of
Danilov [12]. In addition, we also use the two basic identities (1.11) and
(1.13).

Furthermore, using the duality property (2.11) for $\hbox{\rm
DR}^{\mathbb{T}}_{y}$ as in Lemma 6.14, together with equivariant Serre
duality, and combining this with $(b)$, we obtain for
$\ell\in{\mathbb{Z}}_{>0}$ the following duality formula for the equivariant
Hodge polynomial:

(1.18) $\chi^{\mathbb{T}}_{y}(X,\ell
D;{\mathbb{D}}^{\mathbb{T}}_{Z}[{\mathcal{M}}])=\chi_{1/y}(X,-\ell
D;[{\mathcal{M}}])|_{m\mapsto-m}.$

Here, the involution $m\mapsto-m$ of ${\mathbb{Z}}[M]$ corresponds to the
involution $(-)^{\vee}$ on $K^{0}_{\mathbb{T}}(pt)$ induced from duality of
equivariant vector bundles on a point space, i.e., representation spaces.
Together with the purity property (1.10), the duality formula (1.18) applied
to ${\mathcal{M}}={IC^{\prime}}^{H}_{X}$ yields

(1.19) $\chi^{\mathbb{T}}_{y}(X,-\ell
D;[{IC^{\prime}}^{H}_{X}])=(-y)^{n}\cdot\chi^{\mathbb{T}}_{1/y}(X,\ell
D;[{IC^{\prime}}^{H}_{X}])|_{m\mapsto-m}.$

Let us finally explain the role of the chosen homogeneous polynomial function
$\varphi:M_{\mathbb{R}}\cong{\mathbb{R}}^{n}\to{\mathbb{C}}$ for proving the
properties of the generalized weighted Ehrhart polynomials
$E^{\varphi}_{P,f}(\ell,y)$ and, resp.,
$\widetilde{E}^{\varphi}_{P,f}(\ell,y)$, stated in Subsection 1.2. The
homogeneous polynomial $\varphi$ defines group homomorphisms

$\widetilde{E}^{\varphi}:K^{0}_{\mathbb{T}}(pt)\simeq{\mathbb{Z}}[M]\longrightarrow{\mathbb{C}},\
\ \
E^{\varphi}:K^{0}_{\mathbb{T}}(pt)\simeq{\mathbb{Z}}[M]\longrightarrow{\mathbb{C}}[y]$

via $\widetilde{E}^{\varphi}(\chi^{m}):=\varphi(-m)$, resp.,
${E}^{\varphi}(\chi^{m}):=\varphi(-(1+y)\cdot m)$. Since $\varphi$ is
homogeneous, the duality involution $m\mapsto-m$ on the left-hand side
corresponds to the multiplication by $(-1)^{\deg(\varphi)}$ on the right-hand
side. Linearly extending $\widetilde{E}^{\varphi},E^{\varphi}$ over Laurent
polynomials in $y$, we then have that

$\widetilde{E}^{\varphi}_{P,f}(\ell,y)=\widetilde{E}^{\varphi}\big{(}\chi^{\mathbb{T}}_{y}(X,\ell
D;[{\mathcal{M}}])\big{)}$

and

${E}^{\varphi}_{P,f}(\ell,y)={E}^{\varphi}\big{(}\chi^{\mathbb{T}}_{y}(X,\ell
D;[{\mathcal{M}}])\big{)}$

with $f_{Q}(y)=\chi_{y}(i^{*}_{x_{Q}}[{\mathcal{M}}])$. Then the formulae for
$\chi_{y}^{\mathbb{T}}(X,\ell D;[{\mathcal{M}}])$ stated in this subsection
translate via $E^{\varphi}$ and $\widetilde{E}^{\varphi}$ into the
corresponding properties for the generalized weighted Ehrhart polynomials
$E^{\varphi}_{P,f}(\ell,y)$ and, resp.,
$\widetilde{E}^{\varphi}_{P,f}(\ell,y)$, as stated in Subsection 1.2 (see also
Section 6).

To explain geometrically the polynomial behavior in $\ell$ of
$\widetilde{E}^{\varphi}_{P,f}(\ell,y)$, we have to go from $K$-theory to
equivariant homology via the equivariant Todd class transformation, using in
addition equivariant localization at the torus fixed points (as explained in
the Appendix).

###  1.4. Structure of the paper

The paper is structured as follows.

In Section 2, we give a general introduction to the equivariant $K$-theoretic
Hodge-Chern class transformation $\hbox{\rm DR}^{\mathbb{T}}_{y}$ of (classes
of) equivariant mixed Hodge modules. We explain here the basic calculus
(functoriality, module property, and duality), comparing it also with its
motivic counterpart from [2, 9].

Preparing the ground for the toric context, in Section 3 we compute these
equivariant Hodge-Chern classes in terms of a stratification, reducing them to
their motivic counterparts from [9], see Proposition 3.5.

In Section 4, for a ${\mathbb{T}}$-invariant Cartier divisor $D$ on
${\mathbb{T}}$-invariant compact algebraic subvariety $X$ of a smooth ambient
${\mathbb{T}}$-variety $Z$, we define for the class $[{\mathcal{M}}]\in
K_{0}({\rm MHM}^{\mathbb{T}}_{X}(Z))$ of an equivariant mixed Hodge module the
corresponding equivariant Hodge polynomial $\chi_{y}(X,D;[{\mathcal{M}}])$ of
the triple $(X,D;[{\mathcal{M}}])$.

In Section 5, we specialize to the toric context, and make the characteristic
class formulae from the previous sections much more explicit. We also prove
here the two basic formulae (1.11) and (1.13), as well as formula (1.17).

In Section 6, we prove parts $(a)$ and $(b)$ of Theorem 1.2, as well as a
technical result (Lemma 6.14) used to obtain the duality formula (1.18) in the
toric context.

Finally, in the Appendix we explain in a geometric way the polynomial behavior
in $\ell$ of the generalized weighted Ehrhart polynomials.

######  Acknowledgements.

L. Maxim acknowledges support from the project “Singularities and
Applications” - CF 132/31.07.2023 funded by the European Union -
NextGenerationEU - through Romania’s National Recovery and Resilience Plan. J.
Schürmann was funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) Project-ID 427320536 – SFB 1442, as well as under
Germany’s Excellence Strategy EXC 2044 390685587, Mathematics Münster:
Dynamics – Geometry – Structure. L. Maxim and J. Schürmann also thank the
Isaac Newton Institute for Mathematical Sciences for the support and
hospitality during the program “Equivariant methods in geometry” when work on
this paper was undertaken.

##  2\. Equivariant Hodge-Chern classes via mixed Hodge modules

Let $X$ be a complex algebraic variety with an algebraic
${\mathbb{T}}=({\mathbb{C}}^{*})^{n}$-action (e.g., the toric variety $X_{P}$
associated to a full-dimensional lattice polytope $P\subset
M_{\mathbb{R}}\cong{\mathbb{R}}^{n}$), and assume that $i:X\hookrightarrow Z$
is a closed ${\mathbb{T}}$-equivariant embedding of $X$ into an ambient
complex algebraic manifold $Z$ endowed with an algebraic ${\mathbb{T}}$-action
(if $X$ is a normal quasi-projective ${\mathbb{T}}$-variety, such an embedding
into a ${\mathbb{T}}$-equivariant smooth quasi-projective variety exists by
[30, Thm.1], see also [10, Thm.5.1.25] for any linear algebraic group action
on such a normal quasi-projective variety).

Consider the abelian category ${\rm MHM}^{{\mathbb{T}}}_{X}(Z)\subset{\rm
MHM}^{{\mathbb{T}}}(Z)$ of ${\mathbb{T}}$-equivariant mixed Hodge modules on
$Z$ with support in $X$, see [1, 31]. There is an exact forgetful functor
${\rm For}:{\rm MHM}^{{\mathbb{T}}}(Z)\to{\rm MHM}(Z)$, (which for the torus
${\mathbb{T}}$ is even fully faithful since ${\mathbb{T}}$ is connected, see
[1, Thm.5.2]). Let $K_{0}({\rm MHM}^{\mathbb{T}}_{X}(Z))\to K_{0}({\rm
MHM}^{\mathbb{T}}(Z))$ be the corresponding Grothendieck groups, and note
that, via cross-product, $K_{0}({\rm MHM}^{\mathbb{T}}(Z))$ is a module over
$K_{0}({\rm MHM}^{\mathbb{T}}(pt))$. In [1], Achar introduced a triangulated
category $D_{\rm MHM}^{b,{\mathbb{T}}}(Z)$, together with a bounded non-
degenerate t-structure whose heart is ${\rm MHM}^{{\mathbb{T}}}(Z)$ (note that
this is only defined for a smooth ${\mathbb{T}}$-variety $Z$, and it is not
the derived category of ${\rm MHM}^{{\mathbb{T}}}(Z)$). So one has a canonical
isomorphism of Grothendieck groups $K_{0}({\rm MHM}^{{\mathbb{T}}}(Z))\simeq
K_{0}(D_{\rm MHM}^{b,{\mathbb{T}}}(Z))$, and similarly for objects with
support contained in $X$. The category $D_{\rm MHM}^{b,{\mathbb{T}}}(Z)$ has
the same $6$-functor formalism as developed in [25] for $D^{b}{\rm MHM}(Z)$.
All of these functors commute with a corresponding forgetful functor ${\rm
For}:D_{\rm MHM}^{b,{\mathbb{T}}}(Z)\to D^{b}{\rm MHM}(Z)$.

We next discuss a ${\mathbb{T}}$-equivariant version of the Hodge-Chern class
of $X$, see also [14]. In the non-equivariant case, this was initially
introduced in [6], but see also [28] for a definition and study via mixed
Hodge modules.

Let $X$ be as before a closed ${\mathbb{T}}$-invariant subvariety of a complex
algebraic ${\mathbb{T}}$-manifold $Z$, and let ${\mathcal{M}}\in{\rm
MHM}^{\mathbb{T}}_{X}(Z)$ be a ${\mathbb{T}}$-equivariant mixed Hodge module
on $Z$ with support in $X$, e.g., ${\mathcal{M}}=IC_{X}^{H}$ the
${\mathbb{T}}$-equivariant intersection cohomology module (defined by the
equivariant intermediate extension of the constant Hodge module from the
regular part of $X$ to the ambient ${\mathbb{T}}$-variety $Z$). For
$p\in{\mathbb{Z}}$ consider de Rham complex $\hbox{\rm DR}_{Z}({\mathcal{M}})$
of the underlying algebraic left ${\mathcal{D}}_{Z}$-module ${\mathcal{M}}$
with its integrable connection $\nabla$:

$\hbox{\rm
DR}_{Z}({\mathcal{M}})=[{\mathcal{M}}\overset{\nabla}{\longrightarrow}{\mathcal{M}}\otimes_{{\mathcal{O}}_{Z}}\Omega^{1}_{Z}\overset{\nabla}{\longrightarrow}\cdots\overset{\nabla}{\longrightarrow}{\mathcal{M}}\otimes_{{\mathcal{O}}_{Z}}\Omega^{\dim(Z)}_{Z}]$

with ${\mathcal{M}}$ in degree $-\dim(Z)$, filtered by

$F_{p}\hbox{\rm
DR}_{Z}({\mathcal{M}})=[F_{p}{\mathcal{M}}\overset{\nabla}{\longrightarrow}F_{p+1}{\mathcal{M}}\overset{\nabla}{\longrightarrow}\cdots\overset{\nabla}{\longrightarrow}F_{p+n}{\mathcal{M}}\otimes_{{\mathcal{O}}_{Z}}\Omega^{\dim(Z)}_{Z}].$

The complex $Gr^{F}_{p}\hbox{\rm DR}_{Z}({\mathcal{M}})$ associated to the
above filtered de Rham complex $\hbox{\rm DR}_{Z}({\mathcal{M}})$ is a bounded
complex of ${\mathbb{T}}$-equivariant coherent sheaves on $Z$ with support in
$X$. (We use the same symbol for both the mixed Hodge module and the
underlying filtered (left) ${\mathcal{D}}_{Z}$-module.) Moreover,
$Gr^{F}_{p}\hbox{\rm DR}_{Z}({\mathcal{M}})\simeq 0$ for all but finitely many
integers $p$ and ${\mathcal{M}}$ fixed (as can be checked already
nonequivariantly).

######  Remark 2.1.

It follows from [24, Lem.3.2.6] that the complex $Gr^{F}_{p}\hbox{\rm
DR}_{Z}({\mathcal{M}})$ is a bounded complex of (equivariant) coherent
${\mathcal{O}}_{X}$-modules (not just supported on $X$), i.e., these
(equivariant) coherent ${\mathcal{O}}_{X}$-modules are annihilated by the
ideal sheaf ${\mathcal{I}}_{X}$ of $X$ in $Z$. ∎

The transformation $Gr^{F}_{p}\hbox{\rm DR}_{Z}$ induces a corresponding map
on Grothendieck groups since morphisms of (equivariant) mixed Hodge modules
are strict with respect to the Hodge filtration. For an equivariant complex
${\mathcal{M}}\in D_{{\rm MHM},X}^{b,{\mathbb{T}}}(Z)$ on $Z$ with support in
$X$, we perform the above construction for each cohomology object of
${\mathcal{M}}$. So one gets induced Grothendieck group transformations

$Gr^{F}_{p}\hbox{\rm DR}_{Z}:K_{0}(D_{{\rm MHM},X}^{b,{\mathbb{T}}}(Z))\simeq
K_{0}({\rm MHM}^{\mathbb{T}}_{X}(Z))\longrightarrow
K^{\mathbb{T}}_{0,X}(Z)\simeq K^{\mathbb{T}}_{0}(X),$

where $K^{\mathbb{T}}_{0,X}(Z)\simeq K^{\mathbb{T}}_{0}(X)$ is the
Grothendieck group of ${\mathbb{T}}$-equivariant coherent sheaves on ($Z$ with
support in) $X$. The fact that the transformation $Gr^{F}_{p}\hbox{\rm
DR}_{Z}$ takes values in $K^{\mathbb{T}}_{0}(X)$ follows directly from Remark
2.1. Alternatively, using the ambient variety $Z$, this can also be deduced
from the identification $K^{\mathbb{T}}_{0,X}(Z)\simeq K^{\mathbb{T}}_{0}(X)$
induced by the pushforward $i_{*}:K^{\mathbb{T}}_{0}(X)\to
K^{\mathbb{T}}_{0,X}(Z)$ for the closed embedding $i:X\hookrightarrow Z$.
Indeed, the right inverse of $i_{*}$ is $i^{*}:K^{\mathbb{T}}_{0}(Z)\to
K^{\mathbb{T}}_{0}(X)$, which is induced from the derived pullback (i.e.,
alternating sum of equivariant tor-sheaves), since $i$ is of finite tor-
dimension (as $Z$ is smooth). So $i_{*}$ is injective. Moreover, if
${\mathcal{F}}$ is an equivariant coherent sheaf on $Z$ supported on $X$, then
${\mathcal{F}}$ is annihilated by a finite power of the ideal sheaf
${\mathcal{I}}_{X}$ of $X$, and

$[{\mathcal{F}}]=\sum_{i\geq
0}[{\mathcal{I}}_{X}^{i+1}\cdot{\mathcal{F}}/{{\mathcal{I}}_{X}^{i}}\cdot{\mathcal{F}}]\in
i_{*}K^{\mathbb{T}}_{0}(X)\subset K^{\mathbb{T}}_{0,X}(Z),$

proving the surjectivity of $i_{*}$.

We can now make the following definition.

######  Definition 2.2 (Equivariant Hodge-Chern transformation).

The ${\mathbb{T}}$-equivariant Hodge-Chern class transformation of a complex
algebraic ${\mathbb{T}}$-variety $X$ equivariantly embedded in an ambient
complex algebraic ${\mathbb{T}}$-manifold $Z$ is defined as:

$\hbox{\rm DR}^{\mathbb{T}}_{y}:K_{0}({\rm
MHM}^{\mathbb{T}}_{X}(Z))\longrightarrow K^{\mathbb{T}}_{0}(X)[y^{\pm 1}]$
$\begin{split}\hbox{\rm
DR}^{\mathbb{T}}_{y}([{\mathcal{M}}])&:=\sum_{p}\,\big{[}Gr^{F}_{-p}\hbox{\rm
DR}_{Z}([{\mathcal{M}}])\big{]}_{\mathbb{T}}\cdot(-y)^{p}.\end{split}$

######  Remark 2.3.

By definition, $\hbox{\rm DR}^{\mathbb{T}}_{y}([{\mathcal{M}}])$ depends only
on the Grothendieck class $[{\mathcal{M}}]$, and not on the choice of a
representative ${\mathcal{M}}\in D^{b,{\mathbb{T}}}_{{\rm MHM},X}(Z)$ for this
class. Forgetting the ${\mathbb{T}}$-action, one recovers by definition the
Hodge-Chern class transformation $\hbox{\rm DR}_{y}$ from [6, 28]. ∎

######  Proposition 2.4 (Functoriality of $\hbox{\rm DR}^{\mathbb{T}}_{y}$).

Let $f\colon(Z,X)\to(Z^{\prime},X^{\prime})$ be a ${\mathbb{T}}$-equivariant
morphism of smooth complex algebraic ${\mathbb{T}}$-varieties $f:Z\to
Z^{\prime}$ restricting to a proper morphism $f\colon X\to X^{\prime}$ of
closed ${\mathbb{T}}$-invariant subvarieties. Then

$\hbox{\rm DR}^{\mathbb{T}}_{y}\circ f_{*}=f_{*}\circ\hbox{\rm
DR}^{\mathbb{T}}_{y}:K_{0}({\rm MHM}^{\mathbb{T}}_{X}(Z))\to
K^{\mathbb{T}}_{0,X^{\prime}}(Z^{\prime})[y^{\pm 1}].$

###### Proof.

This can be done exactly as explained in [14, Section 3]. ∎

######  Remark 2.5.

Under the identifications $K^{\mathbb{T}}_{0,X}(Z)\simeq
K^{\mathbb{T}}_{0}(X)$ and $K^{\mathbb{T}}_{0,X^{\prime}}(Z^{\prime})\simeq
K^{\mathbb{T}}_{0}(X^{\prime})$, the pushforward
$f_{*}:K^{\mathbb{T}}_{0,X}(Z)\to K^{\mathbb{T}}_{0,X^{\prime}}(Z^{\prime})$
corresponds to the usual pushforward $f_{*}:K^{\mathbb{T}}_{0}(X)\to
K^{\mathbb{T}}_{0}(X^{\prime})$. ∎

######  Remark 2.6.

The transformation $\hbox{\rm DR}^{\mathbb{T}}_{y}$ is (in the following
sense) independent of the choice of an ${\mathbb{T}}$-equivariant embedding of
$X$ in a smooth ambient ${\mathbb{T}}$-manifold $Z$. In the context of
Proposition 2.4, with $f\colon X\to X^{\prime}$ an isomorphism, the
pushforward $f_{*}\colon K_{0}({\rm MHM}^{\mathbb{T}}_{X}(Z))\to K_{0}({\rm
MHM}^{\mathbb{T}}_{X^{\prime}}(Z^{\prime}))$ is an isomorphism, e.g., using
Proposition 3.1 below. In particular, the transformation $\hbox{\rm
DR}^{\mathbb{T}}_{y}$ commutes with this isomorphism and the isomorphism
$f_{*}:K^{\mathbb{T}}_{0}(X)\to K^{\mathbb{T}}_{0}(X^{\prime})$, as in the
above remark. If we now consider two equivariant embeddings
$i:X\hookrightarrow Z$ and $i^{\prime}:X\hookrightarrow Z^{\prime}$, the above
observation can be applied to the diagonal embedding $X\hookrightarrow Z\times
Z^{\prime}$ (with the diagonal ${\mathbb{T}}$-action) and the projections on
the two factors to show that $\hbox{\rm DR}^{\mathbb{T}}_{y}$ is (up to
isomorphism) independent of the choice of an equivariant embedding of $X$ in a
smooth ambient ${\mathbb{T}}$-manifold. ∎

######  Example 2.7 (Degree of $\hbox{\rm DR}_{y}^{\mathbb{T}}$).

Let us illustrate the definition of the equivariant Hodge-Chern transformation
when $X=Z=pt$ is a point space. In this case, since ${\mathbb{T}}$ is
connected, we get by [31, Lem.1.2] that

${\rm MHM}^{\mathbb{T}}(pt)={\rm MHM}(pt)={\rm MHS}^{p},$

where ${\rm MHS}^{p}$ is the category of graded-polarizable mixed
${\mathbb{Q}}$-Hodge structures, with switching the increasing
${\mathcal{D}}$-module filtration to a decreasing Hodge filtration so that on
a point space this identification gives $Gr^{F}_{-p}\hbox{\rm
DR}_{Z}=Gr^{p}_{F}$.

By definition, for $H^{\bullet}\in D_{\rm MHM}^{b,{\mathbb{T}}}(pt)$ we have
that

$[H^{\bullet}]\in K_{0}(D_{\rm MHM}^{b,{\mathbb{T}}}(pt))=K_{0}({\rm
MHM}^{\mathbb{T}}(pt))=K_{0}({\rm MHS}^{p}),$

so $DR^{\mathbb{T}}_{y}([H^{\bullet}])$ does not see the
${\mathbb{T}}$-action, i.e.,

(2.1)
$DR^{\mathbb{T}}_{y}([H^{\bullet}])=\sum_{j,p}\,(-1)^{j}\dim_{{\mathbb{C}}}\text{\rm
Gr}_{F}^{p}H_{{\mathbb{C}}}^{j}\,(-y)^{p}=:\chi_{y}([H^{\bullet}])$

is the corresponding Hodge $\chi_{y}$-polynomial, defining a ring homomorphism

$\chi_{y}:K_{0}({\rm MHS}^{p})\longrightarrow K_{0}(pt)[y^{\pm
1}]={\mathbb{Z}}[y^{\pm 1}]\hookrightarrow K^{\mathbb{T}}_{0}(pt)[y^{\pm 1}].$

E.g., for $n\in{\mathbb{Z}}$, $\chi_{y}({\mathbb{Q}}(n))=(-y)^{-n}$, where
${\mathbb{Q}}(n)$ denotes the Tate Hodge structure of weight $-2n$ on the
vector space ${\mathbb{Q}}$. Similarly,

$\chi_{y}(X):=\chi_{y}([H^{\bullet}_{c}(X;{\mathbb{Q}}_{X}^{H})])$

is the (compactly supported) Hodge polynomial of $X$. E.g.,
$\chi_{y}(({\mathbb{C}}^{*})^{n})=(-y-1)^{n}$.

By functoriality, if $Z$ (and hence $X$) is compact, and $a_{Z}:Z\to pt$ is
the constant map to a point, we have for ${\mathcal{M}}\in
D^{b,{\mathbb{T}}}_{{\rm MHM},X}(Z)$ that $k_{*}{\mathcal{M}}\in D_{\rm
MHM}^{b,{\mathbb{T}}}(pt)$, so

(2.2)
$\chi_{y}(X;[{\mathcal{M}}]):=\chi_{y}([(a_{Z})_{*}{\mathcal{M}}])=\chi_{y}([H^{\bullet}(Z;{\mathcal{M}})])$

is the degree of the equivariant Hodge-Chern class of $[{\mathcal{M}}]$ (and
depends only of $[{\mathcal{M}}]$). ∎

######  Definition 2.8.

Let $K_{0}({\rm MHS}^{p})^{\rm Tate}$ be the subring of $K_{0}({\rm
MHS}^{p})=K_{0}({\rm MHM}^{\mathbb{T}}(pt))$ generated by
$u:=[{\mathbb{Q}}(-1)]$ and $u^{-1}:=[{\mathbb{Q}}(1)]$, with a surjective
ring homomorphism ${\mathbb{Z}}[u^{\pm 1}]\to K_{0}({\rm MHS}^{p})^{\rm
Tate}$. This is an isomorphism since its composition with $\chi_{y}$ is the
isomorphism ${\mathbb{Z}}[u^{\pm 1}]\to{\mathbb{Z}}[y^{\pm 1}],\ u\mapsto-y$.

######  Remark 2.9.

Via the description of $\hbox{\rm DR}^{\mathbb{T}}_{y}$ over a point space, it
follows that the transformation $\hbox{\rm DR}^{\mathbb{T}}_{y}$ commutes with
the cross-product $\boxtimes$ with a point space for the identification
$Z\times pt\simeq Z$, since the equivariant Hodge filtration on the product
${\mathcal{M}}\boxtimes H$ (with ${\mathcal{M}}\in{\rm
MHM}^{\mathbb{T}}_{X}(Z)$ and $H\in{\rm MHS}$) is the product filtration on
the underlying filtered ${\mathcal{D}}$-module, e.g., see the discussion in
[31, Sections 1.4 and 1.5]. Let us also note that the effect of the Tate twist
$(n)$ on $\hbox{\rm DR}^{\mathbb{T}}_{y}$ is (like in the case of $\hbox{\rm
DR}_{y}$) just multiplication by $(-y)^{-n}$, i.e., for ${\mathcal{M}}\in
D^{b,{\mathbb{T}}}_{\rm{MHM},X}(Z)$ with
${\mathcal{M}}(n):={\mathcal{M}}\boxtimes{\mathbb{Q}}(n)$, one has

(2.3) $\hbox{\rm
DR}^{\mathbb{T}}_{y}([{\mathcal{M}}(n)])=(-y)^{-n}\cdot\hbox{\rm
DR}^{{\mathbb{T}}}_{y}([{\mathcal{M}}]).$

∎

######  Remark 2.10.

Note that for $i:X\hookrightarrow Z$ a singular closed embedding into an
ambient smooth ${\mathbb{T}}$-variety, Achar’s $6$-functor calculus for
equivariant mixed Hodge modules cannot be used to define the constant mixed
Hodge complex on $X$ as a ${\mathbb{T}}$-equivariant object, e.g., via the
pullback of ${\mathbb{Q}}^{H}_{Z}$. Nevertheless, its Grothendieck class
$[{\mathbb{Q}}^{H}_{X}]\in K_{0}({\rm MHM}^{\mathbb{T}}_{X}(Z))$ can be
introduced via stratifications as follows. Let ${\mathcal{S}}_{X}$ be a
${\mathbb{T}}$-invariant algebraic stratification of $X\subset Z$ with smooth
strata, and denote by $j_{S,Z}:S\hookrightarrow Z$ the inclusion maps of
strata $S\in{\mathcal{S}}_{X}$. Then

(2.4)
$[{\mathbb{Q}}_{X}^{H}]:=\sum_{S\in{\mathcal{S}}_{X}}[(j_{S,Z})_{!}(j_{S,Z})^{*}{\mathbb{Q}}^{H}_{Z}]=\sum_{S\in{\mathcal{S}}_{X}}[(j_{S,Z})_{!}{\mathbb{Q}}^{H}_{S}]\in
K_{0}({\rm MHM}^{\mathbb{T}}_{X}(Z)),$

where ${\mathbb{Q}}^{H}_{S}\in D_{\rm MHM}^{b,{\mathbb{T}}}(S)$ is the
constant mixed Hodge complex on the stratum $S$. Then, by additivity,
$[{\mathbb{Q}}_{X}^{H}]$ is independent of the choice of the stratification.
Moreover, $\chi_{y}(X;[{\mathbb{Q}}^{H}_{X}])=\chi_{y}(X)$ is the Hodge
polynomial of $X$, as defined above. Similarly, we can introduce

$i^{*}:K_{0}({\rm MHM}^{\mathbb{T}}(Z))\to K_{0}({\rm
MHM}^{\mathbb{T}}_{X}(Z)),$

with $i^{*}\circ i_{*}=id$ on $K_{0}({\rm MHM}^{\mathbb{T}}_{X}(Z))$ and
$i_{*}:K_{0}({\rm MHM}^{\mathbb{T}}_{X}(Z))\to K_{0}({\rm
MHM}^{\mathbb{T}}(Z))$ defined by forgetting the support. So $i_{*}$ is
injective.

Alternatively, if $j:U=Z\setminus X\hookrightarrow Z$ is the inclusion map,
one may define the equivariant constant Hodge module complex of $X$ in the
ambient ${\mathbb{T}}$-manifold $Z$ (up to non-canonical isomorphism) as the
cone of the adjunction map

(2.5)
${\mathbb{Q}}^{H}_{X}:=Cone(j_{!}j^{*}{\mathbb{Q}}^{H}_{Z}\longrightarrow{\mathbb{Q}}^{H}_{Z})\in
D^{b,{\mathbb{T}}}_{{\rm{\rm MHM}},X}(Z).$

By construction, the equivariant Grothendieck class of ${\mathbb{Q}}^{H}_{X}$
agrees with (2.4). ∎

Let us now relate the transformation $\hbox{\rm DR}^{\mathbb{T}}_{y}$ to its
motivic counterpart, as used e.g., in [9]. Let $X$ be a
${\mathbb{T}}$-equivariant complex algebraic variety. The relative equivariant
motivic Grothendieck group $K^{\mathbb{T}}_{0}(var/X)$ of varieties over $X$
is the free abelian on isomorphism classes of ${\mathbb{T}}$-varieties over
$X$, modulo the usual scissor relations

$[Y\to X]=[U\to X]+[Y\setminus U\to X],$

for $U\subset X$ an open ${\mathbb{T}}$-invariant subvariety. If $X=pt$ is a
point space, then $K_{0}^{\mathbb{T}}(var/pt)$ is a ring with product given by
the cross-product of morphisms for $pt\times X\simeq X$, and the group
$K_{0}^{\mathbb{T}}(var/X)$ is a module over $K_{0}^{\mathbb{T}}(var/pt)$ with
respect to the cross product.

For any equivariant morphism $f:Y\to X$ of ${\mathbb{T}}$-varieties, there is
a well-defined push-forward $f_{!}:K_{0}^{\mathbb{T}}(var/Y)\to
K_{0}^{\mathbb{T}}(var/X)$ defined by composition, which is compatible with
the module structure over $K_{0}^{\mathbb{T}}(var/pt)$.

Assume now that $i:X\hookrightarrow Z$ is a ${\mathbb{T}}$-equivariant closed
embedding into an ambient smooth ${\mathbb{T}}$-manifold $Z$. Then, using the
same argument as in [14], there is a natural group homomorphism

$\chi^{\mathbb{T}}_{Hdg}:K^{{\mathbb{T}}}_{0}(var/X)\to K_{0}({\rm
MHM}^{\mathbb{T}}_{X}(Z))$

uniquely fixed by

$[g:Y\to X]\mapsto[(i\circ g)_{!}{\mathbb{Q}}^{H}_{Y}],$

where $Y$ is a smooth ${\mathbb{T}}$-variety. For singular $Y$, one decomposes
it into smooth ${\mathbb{T}}$-invariant subvarieties, and uses the usual
scissor relations to define $\chi^{\mathbb{T}}_{Hdg}([f:Y\to X])$, e.g., in
the notations of Remark 2.10, we have

$\chi^{\mathbb{T}}_{Hdg}([i:X\hookrightarrow Z])=[{\mathbb{Q}}^{H}_{X}]\in
K_{0}({\rm MHM}^{\mathbb{T}}_{X}(Z)).$

The transformation $\chi^{\mathbb{T}}_{Hdg}$ commutes with pushdown $f_{!}$
(with proper support) for morphisms of pairs
$f:(Z,X)\to(Z^{\prime},X^{\prime})$, with $Z,Z^{\prime}$ smooth. Hence, the
composition

$mC^{\mathbb{T}}_{y}:=DR^{\mathbb{T}}_{y}\circ\chi^{\mathbb{T}}_{Hdg}:K_{0}^{\mathbb{T}}(var/X)\to
K_{0}({\rm MHM}^{\mathbb{T}}_{X}(Z))\to K_{0}^{\mathbb{T}}(X)[y^{\pm 1}]$

is independent of the ambient smooth ${\mathbb{T}}$-variety $Z$, and it is
covariant for ${\mathbb{T}}$-equivariant proper morphisms. Moreover, if $X$ is
smooth and ${\mathcal{M}}={\mathbb{Q}}_{X}^{H}\in D_{\rm
MHM}^{b,{\mathbb{T}}}(X)$ is the constant Hodge module complex on $X$, then
the underlying ${\mathbb{T}}$-equivariant left ${\mathcal{D}}$-module is
${\mathcal{O}}_{X}$ with the trivial Hodge filtration, so that

$Gr^{F}_{-p}\hbox{\rm DR}_{X}({\mathbb{Q}}_{X}^{H})\simeq\Omega^{p}_{X}[-p]$

for $0\leq p\leq\dim(X)$, and it is $0$ otherwise. So,

$mC^{\mathbb{T}}_{y}([id_{X}])=\sum_{p=0}^{\dim(X)}[\Omega^{p}_{X}]_{\mathbb{T}}\cdot
y^{p}=:\Lambda_{y}(\Omega^{1}_{X}),$

the total $\Lambda$-class of the cotangent bundle of $X$. It then follows
that, for $X$ quasi-projective, $mC^{\mathbb{T}}_{y}$ induces the equivariant
motivic Chern class transformation defined in [2, Thm.4.2] on the
${\mathbb{T}}$-equivariant relative Grotendieck group
$K_{0}^{\mathbb{T}}(qproj/X)$ of ${\mathbb{T}}$-equivariant quasi-projective
varieties over $X$, as studied in the quasi-projective toric context in [9].

In general, if $i:X\hookrightarrow Z$ is a ${\mathbb{T}}$-equivariant closed
embedding into an ambient smooth ${\mathbb{T}}$-variety, we set

$mC^{\mathbb{T}}_{y}(X):=mC^{\mathbb{T}}_{y}([id_{X}])=\hbox{\rm
DR}^{\mathbb{T}}_{y}([{\mathbb{Q}}_{X}^{H}])$

for $[{\mathbb{Q}}_{X}^{H}]\in K^{\mathbb{T}}_{0}({\rm MHM}_{X}(Z))$ the class
defined as in (2.4). Moreover, if $X$ is pure-dimensional we set

$IC^{\mathbb{T}}_{y}(X):=\hbox{\rm
DR}^{\mathbb{T}}_{y}([{IC^{\prime}}_{X}^{H}]),$

with ${IC^{\prime}}_{X}^{H}:=IC_{X}^{H}[-\dim(X)]$ the (shifted) equivariant
intersection cohomology module of $X$, viewed as a (shifted)
${\mathbb{T}}$-equivariant Hodge module on $Z$. If $X$ is compact, the degree
of $IC^{\mathbb{T}}_{y}(X)$ is the $I\chi_{y}$-polynomial

$I\chi_{y}(X):=\chi_{y}(X;{IC^{\prime}}_{X}^{H}).$

If $X$ is moreover a rational homology manifold (e.g., a complex algebraic
$V$-manifold like a simplicial toric variety), then
$[{IC^{\prime}}_{X}^{H}]\simeq[{\mathbb{Q}}_{X}^{H}]$ (which also motivates
our shift convention).

######  Remark 2.11.

As shown in [26, Thm.4.2], one has the following identification in $D^{b}_{\rm
coh}(X)$:

(2.6) $\uuline{\Omega}^{p}_{X}[-p]\simeq Gr_{-p}^{F}\hbox{\rm
DR}_{Z}({\mathbb{Q}}^{H}_{X}),$

where

$\uuline{\Omega}^{p}_{X}:=Gr^{p}_{F}(\underline{\Omega}^{\bullet}_{X})[p]\in
D^{b}_{\rm coh}(X)$

are the graded parts of the filtered Du Bois complex
$(\underline{\Omega}^{\bullet}_{X},F)$ on $X$. Viewing the class
$[{\mathbb{Q}}^{H}_{X}]\in K_{0}({\rm MHM}^{\mathbb{T}}_{X}(Z))$, and
considering $Gr_{-p}^{F}\hbox{\rm DR}_{Z}$ as a transformation on this
Grothendieck group, one can take (2.6) as the definition of the equivariant
classes $[\uuline{\Omega}^{p}_{X}]_{\mathbb{T}}$ of the graded parts of the Du
Bois complex on $X$. This yields the formula

(2.7)
$mC^{\mathbb{T}}_{y}(X)=\sum_{p=0}^{\dim(X)}[\uuline{\Omega}_{X}^{p}]_{\mathbb{T}}\cdot
y^{p}\in K^{\mathbb{T}}_{0}(X)[y]$

in the language of equivariant mixed Hodge modules. Moreover, for such a
${\mathbb{T}}$-invariant closed subvariety $X$ in an ambient smooth
${\mathbb{T}}$-variety $Z$, we can use the equivariant de Rham complex and
Remark 2.1 together with (2.5) to define

$\uuline{\Omega}^{p}_{X}[-p]\simeq Gr_{-p}^{F}\hbox{\rm
DR}_{Z}({\mathbb{Q}}^{H}_{X})\in D^{b,{\mathbb{T}}}_{\rm coh}(X),$

as graded pieces of an equivariant Du Bois complex on $X$. If $X$ is,
moreover, a rational homology manifold, we have
${IC^{\prime}}^{H}_{X}\simeq{\mathbb{Q}}^{H}_{X}\in
D^{b,{\mathbb{T}}}_{{\rm{\rm MHM}},X}(Z)$, defining an equivariant Du Bois
complex on $X$ in terms of the equivariant intersection cohomology module, as
used e.g., in [19]. In the case of a rational homology manifold or if $X$ is
any quasi-projective toric variety, $\uuline{\Omega}^{p}_{X}$ is just the
${\mathbb{T}}$-equivariant coherent sheaf (sitting in degree zero, since this
is the case nonequivariantly) given by the ${\mathbb{T}}$-equivariant Zariski
sheaf $\widehat{\Omega}_{X}^{p}$ of $p$-forms on $X$. This holds even
equivariantly, since the adjunction map
$\uuline{\Omega}^{p}_{X}\to\widehat{\Omega}_{X}^{p}=j_{*}j^{*}\uuline{\Omega}^{p}_{X}=j_{*}\Omega_{X_{\rm
reg}}^{p}$ for the open inclusion $j$ of the regular part $X_{\rm reg}$ of $X$
is nonequivariantly an isomorphism of reflexive sheaves.

In [9], another approach based on simplicial resolutions was used to derive
(2.7), which in the quasi-projective toric situation yields (cf. [9,
Prop.3.5])

(2.8)
$mC^{\mathbb{T}}_{y}(X)=\sum_{p=0}^{\dim(X)}[\widehat{\Omega}_{X}^{p}]_{\mathbb{T}}\cdot
y^{p}\in K_{0}(X)[y].$

∎

Denote by ${\mathbb{L}}:=[{\mathbb{C}}\to pt]\in K_{0}^{\mathbb{T}}(var/pt)$
the Lefschetz motive (with ${\mathbb{T}}$ acting trivially on ${\mathbb{C}}$).
Then there is a natural group homomorphism on the localized Grothendieck group

$\chi^{\mathbb{T}}_{Hdg}:K^{{\mathbb{T}}}_{0}(var/X)[{\mathbb{L}}^{-1}]\to
K_{0}({\rm MHM}^{\mathbb{T}}_{X}(Z)),$

mapping ${\mathbb{L}}$ to $u=[{\mathbb{Q}}(-1)]\in K_{0}({\rm
MHS}^{p})=K_{0}({\rm MHM}^{\mathbb{T}}(pt)$. Under the natural forgetful
functors, this reduces to the corresponding localized nonequivariant
transformation (as used in [22]).

######  Remark 2.12 (Effect of duality).

As shown in [28, Cor.5.19, Rem.5.20], the non-equivariant transformation
$\hbox{\rm DR}_{y}$ of [6, 28] commutes with duality, i.e.,

(2.9) $\hbox{\rm DR}_{y}\circ{\mathbb{D}}_{X}={\mathbb{D}}\circ\hbox{\rm
DR}_{y},$

where ${\mathbb{D}}_{X}$ is induced from the duality of mixed Hodge modules,
and ${\mathbb{D}}$ on $K_{0}(X)$ is induced by Grothendieck duality (e.g., see
[18, Part 1, Sect.7], [20, Part 1]) extended to $K_{0}(X)[y^{\pm 1}]$ by
$y\mapsto 1/y$. A similar statement holds for the motivic transformation
$mC_{y}$, with duality on the localization $K_{0}(var/X)[{\mathbb{L}}^{-1}]$
defined as in [28, Section 4B] and [4]. All these duality transformations are
functorial for proper maps, so that for $X$ compact with $a_{X}:X\to pt$
proper, one gets

(2.10)
$\chi_{y}(X;{\mathbb{D}}_{X}[{\mathcal{M}}])=\chi_{1/y}(X;[{\mathcal{M}}]).$

Indeed, over a point space, Grothendieck duality and homological duality are
trivial, while the mixed Hodge module duality is the usual duality of mixed
Hodge structures, e.g., ${\mathbb{D}}_{pt}{\mathbb{Q}}(n)={\mathbb{Q}}(-n)$.∎

As explained in [14, Sect.3], the duality formula (2.9) also holds
equivariantly for complexes of mixed Hodge modules on $Z$ with support in $X$
(with ${\mathbb{D}}_{Z}$ instead of ${\mathbb{D}}_{X}$), at least on (Tate
twists of) elements in the image of $\chi^{\mathbb{T}}_{Hdg}$ (and this is the
only case we’ll be interested in later on), namely on such elements one has

(2.11) $\hbox{\rm
DR}^{\mathbb{T}}_{y}\circ{\mathbb{D}}^{\mathbb{T}}_{Z}={\mathbb{D}}^{\mathbb{T}}\circ\hbox{\rm
DR}^{\mathbb{T}}_{y},$

Here, ${\mathbb{D}}^{\mathbb{T}}_{Z}$ is induced from the exact Verdier
duality functor of equivariant mixed Hodge modules (on $Z$, with support on
$X$), see [1]. For ${\mathbb{D}}^{\mathbb{T}}$, one can use either the
equivariant Grothendieck duality on the Grothendieck group
$K^{\mathbb{T}}_{0,X}(Z)$ induced from Grothendieck duality on $Z$
(remembering the support in $X$), or the equivariant Grothendieck duality on
the Grothendieck group $K^{\mathbb{T}}_{0}(X)$ induced from Grothendieck
duality on $X$. These are identified by the isomorphism
$i_{*}:K^{\mathbb{T}}_{0}(X)\to K^{\mathbb{T}}_{0,X}(Z)$, since equivariant
Grothendieck duality commutes with proper pushforwards (see [20] for more
details). As in the nonequivariant context, ${\mathbb{D}}^{\mathbb{T}}$ is
extended to $K^{\mathbb{T}}_{0}(X)[y^{\pm 1}]$ by $y\mapsto 1/y$.

######  Remark 2.13.

To the authors’ knowledge, a corresponding duality transformation on the
localized equivariant group $K^{{\mathbb{T}}}_{0}(var/X)[{\mathbb{L}}^{-1}]$
is not currently available. ∎

##  3\. Computations via stratifications

Let us now indicate how to compute the equivariant Hodge-Chern classes in
terms of a ${\mathbb{T}}$-invariant stratification.

Let as before $X$ be a closed ${\mathbb{T}}$-invariant algebraic subvariety of
an algebraic ${\mathbb{T}}$-manifold $Z$, with inclusion map
$i:X\hookrightarrow Z$. Let ${\mathcal{M}}\in D^{b,{\mathbb{T}}}_{{\rm
MHM},X}(Z)$ be an equivariant complex of mixed Hodge modules on $Z$ with
support in $X$, with underlying bounded constructible complex
$K:={\it{rat}}({\rm For}({\mathcal{M}}))\in D^{b}_{c}(Z)$ supported on $X$.
Let ${\mathcal{S}}_{X}=\\{S\\}$ be a ${\mathbb{T}}$-invariant complex
algebraic stratification of $X$ so that for any stratum $S\in{\mathcal{S}}$,
$S$ is smooth, ${\bar{S}}\setminus S$ is a union of strata, and the sheaves
$L_{S,\ell}:={\mathcal{H}}^{\ell}K|_{S}$ are local systems on $S$ for any
$\ell\in{\mathbb{Z}}$. Let $U=Z\setminus X$ and
${\mathcal{S}}_{Z}={\mathcal{S}}_{X}\cup\\{U\\}$. Then ${\mathcal{S}}_{Z}$ is
an algebraic stratification of $Z$ satisfying the above properties for each
$S\in{\mathcal{S}}_{X}$, together with
$L_{U,\ell}:={\mathcal{H}}^{\ell}K|_{U}\simeq 0$. For each
$S\in{\mathcal{S}}_{X}$, consider the inclusions

$j_{S,Z}:S\overset{i_{S,{\bar{S}}}}{\hookrightarrow}{\bar{S}}\overset{i_{{\bar{S}},X}}{\hookrightarrow}X\overset{i}{\hookrightarrow}Z$

together with $i_{{\bar{S}},Z}=i\circ i_{{\bar{S}},X},$ so that

$j_{S,Z}=i_{{\bar{S}},Z}\circ i_{S,{\bar{S}}}.$

Under the above notations and assumptions, and using the $6$-functor formalism
from [1] we get as in [23, Prop.5.1.2] the following:

######  Proposition 3.1.

(3.1) $[{\mathcal{M}}]=\sum_{S\neq
U,\ell}\,(-1)^{\ell}\cdot\big{[}(j_{S,Z})_{!}L_{S,\ell}^{H}\big{]}\in
K_{0}({\rm MHM}^{\mathbb{T}}_{X}(Z)),$

where $L_{S,\ell}^{H}=H^{\ell+\dim(S)}(j_{S,Z})^{*}{\mathcal{M}}[-\dim(S)]$ is
the shifted equivariant smooth mixed Hodge module on $S$ with $L_{S,\ell}$ the
underlying local system of ${\rm For}(L_{S,\ell}^{H})$, and the summation is
over strata $S\in{\mathcal{S}}_{X}$.

In the notations of Proposition 3.1, we get the following.

######  Corollary 3.2.

Assume, moreover, that for any $S\in{\mathcal{S}}_{X}$, the stabilizer
${\mathbb{T}}_{s}\subset{\mathbb{T}}$ at a point $s\in S$ is connected. Then
each $L_{S,\ell}^{H}\simeq{\mathbb{Q}}^{H}_{S}\boxtimes L_{S,s,\ell}^{H}$ is a
constant variation of mixed Hodge structures on $S$ with stalk the mixed Hodge
structure $L_{S,s,\ell}^{H}$, and we have

(3.2)
$\begin{split}[{\mathcal{M}}]&=\sum_{S,\ell}\,(-1)^{\ell}\cdot\big{[}(j_{S,Z})_{!}{\mathbb{Q}}_{S}^{H}\big{]}\cdot[L^{H}_{S,s,\ell}]\\\
&=\sum_{S}\,\big{[}(j_{S,Z})_{!}{\mathbb{Q}}_{S}^{H}\big{]}\cdot
i_{s}^{*}[{\mathcal{M}}],\end{split}$

with $i_{s}:\\{s\\}\hookrightarrow Z$ the inclusion, for $s\in S$ chosen, and
the summation is over the strata $S\in{\mathcal{S}}_{X}$.

In particular,

(3.3) $\begin{split}\hbox{\rm
DR}^{\mathbb{T}}_{y}([{\mathcal{M}}])&=\sum_{S\in{\mathcal{S}}_{X}}\,\hbox{\rm
DR}^{\mathbb{T}}_{y}\big{[}(j_{S,Z})_{!}{\mathbb{Q}}_{S}^{H}\big{]}\cdot\chi_{y}(i_{s}^{*}[{\mathcal{M}}])\\\
&=\sum_{S\in{\mathcal{S}}_{X}}\,mC_{y}^{\mathbb{T}}([j_{S,Z}:S\hookrightarrow
Z])\cdot\chi_{y}(i_{s}^{*}[{\mathcal{M}}])\\\
&=\sum_{S\in{\mathcal{S}}_{X}}\,(i_{{\bar{S}},Z})_{*}mC^{\mathbb{T}}_{y}([i_{S,{\bar{S}}}:S\hookrightarrow{\bar{S}}])\cdot\chi_{y}(i_{s}^{*}[{\mathcal{M}}]).\end{split}$

###### Proof.

The fact that each $L^{H}_{S,\ell}$ is a constant variantion of mixed Hodge
structures on $S$ follows by [31, Lem.1.2] using the assumption that the
stabilizer ${\mathbb{T}}_{s}$ of $s\in S$ is connected. This proves (3.2). To
show (3.3), we apply the equivariant Hodge-Chern class transformation
$\hbox{\rm DR}_{y}^{\mathbb{T}}$ to (3.2), using the fact that $K_{0}({\rm
MHM}^{\mathbb{T}}_{X}(Z))$ is a $K_{0}({\rm MHM}^{\mathbb{T}}(pt))\simeq
K_{0}({\rm MHS}^{p})$-module, together with the functoriality of the
equivariant motivic Chern class transformation $mC_{y}^{\mathbb{T}}$ for the
closed embedding $i_{{\bar{S}},Z}$. ∎

######  Remark 3.3.

Formula (3.2) shows that under the connected ${\mathbb{T}}$-stabilizers
assumption, the class $[{\mathcal{M}}]$ belongs to the (free) $K_{0}({\rm
MHM}^{\mathbb{T}}(pt))=K_{0}({\rm MHS}^{p})$-submodule (respectively,
$K_{0}({\rm MHS}^{p})^{\rm Tate}$-submodule, if the stalks
$i_{s}^{*}[{\mathcal{M}}]$ are of Tate type) of $K_{0}({\rm
MHM}^{\mathbb{T}}_{X}(Z))$ generated by classes
$[(j_{S,Z})_{!}{\mathbb{Q}}_{S}^{H}\big{]}=\chi^{\mathbb{T}}_{Hdg}([j_{S,Z}:S\hookrightarrow
Z])$, $S\in{\mathcal{S}}_{X}$. In what follows, we regard
$\chi_{y}(i_{s}^{*}[{\mathcal{M}}])\in{\mathbb{Z}}[y^{\pm 1}]$ as a weight
(depending only on $[{\mathcal{M}}]$) attached to the stratum
$S\in{\mathcal{S}}_{X}$ containing $s$. In the case of a projective toric
variety $X=X_{P}$ associated to a full dimensional lattice polytope $P$ (as in
the Introduction), the subgroup

$\chi^{\mathbb{T}}_{Hdg}\left(\tau^{\mathbb{T}}(F^{{\mathbb{T}}}(X)[y^{\pm
1}])\right)\subset K^{\mathbb{T}}_{0}({\rm MHM}_{X}(Z))$

is the $K_{0}({\rm MHS}^{p})^{\rm Tate}$-submodule generated by classes
$[{\mathcal{M}}]$ whose stalks are of Tate type. ∎

######  Remark 3.4.

Under the assumption of connected ${\mathbb{T}}$-stabilizers, formula (3.3)
reduces the calculation of equivariant Hodge-Chern classes to the motivic
calculus of [9], up to assigning the above mentioned Laurent polynomial
weights $\chi_{y}(i_{s}^{*}[{\mathcal{M}}])$ to each stratum $S$, with $s\in
S$ chosen. For instance, the connected stabilizers assumption is automatically
satisfied in the toric context, where such stabilizers are tori.∎

By functoriality, the terms $mC^{\mathbb{T}}_{y}([S\hookrightarrow{\bar{S}}])$
appearing in formula (3.3) are computed via resolutions of singularities as
follows (see [9, Thm.3.3], adapting ideas from [32, Thm.5.1, Rem.5.2] to the
equivariant context):

######  Proposition 3.5.

Let $i_{S,V}:S\hookrightarrow V$ be a smooth partial
${\mathbb{T}}$-equivariant compactification of a stratum $S$ so that
$D:=V\setminus S$ is a ${\mathbb{T}}$-invariant simple normal crossing
divisor, and $i_{S,{\bar{S}}}=\pi_{V}\circ i_{S,V}$ for a proper morphism
$\pi_{V}:V\to{\bar{S}}$. Then we have

(3.4)
$mC^{\mathbb{T}}_{y}([S\hookrightarrow{\bar{S}}])=(\pi_{V})_{*}\big{[}\mathcal{O}_{V}(-D)\otimes\Lambda_{y}\Omega_{V}^{1}(\log(D)\big{]}_{\mathbb{T}}\in
K^{\mathbb{T}}_{0}({\bar{S}})[y].$

These formulae become more explicit and much simplified in the toric context,
as it will be discussed in Section 5 below.

##  4\. Twisted equivariant Hodge-Chern classes and their degrees

Assume now that a torus ${\mathbb{T}}$ with character lattice $M$ acts
linearly on a finite dimensional complex vector space $W$, with eigenspace

(4.1) $W_{\chi^{m}}:=\\{w\in W\mid t\cdot w=\chi^{m}(t)w\ \text{ for all}\
t\in{\mathbb{T}}\\},$

for $\chi^{m}$ a character of ${\mathbb{T}}$. Then, as in [11, Prop.1.1.2],
one has an eigenspace decomposition

$W=\bigoplus_{m\in M}W_{\chi^{m}}.$

This induces an isomorphism

(4.2)
$K^{0}_{\mathbb{T}}(pt)\overset{\simeq}{\longrightarrow}\mbox{$\mathbb{Z}$}[M],\
\ [W]_{\mathbb{T}}\mapsto\sum_{m\in
M}\dim_{\mathbb{C}}W_{\chi^{m}}\cdot\chi^{m},$

identifying the duality involution $(-)^{\vee}$ on the left hand side with the
involution $m\mapsto-m$ on characters. If $X$ is a compact algebraic
${\mathbb{T}}$-variety, with a ${\mathbb{T}}$-equivariant coherent sheaf
${\mathcal{F}}$, then the cohomology spaces $H^{i}(X;{\mathcal{F}})$ are
finite dimensional ${\mathbb{T}}$-representations, vanishing for $i$ large
enough. The $K$-theoretic Euler characteristic of ${\mathcal{F}}$ is then
defined as:

(4.3)
$\begin{split}\chi^{\mathbb{T}}(X,{\mathcal{F}})&:=(a_{X})_{*}[{\mathcal{F}}]_{\mathbb{T}}=[H^{\bullet}(X;{\mathcal{F}})]_{\mathbb{T}}:=\sum_{i}(-1)^{i}[H^{i}(X;{\mathcal{F}})]_{\mathbb{T}}\in
K^{0}_{\mathbb{T}}(pt)\\\ &=\sum_{m\in
M}\sum_{i=0}^{\dim(X)}(-1)^{i}\dim_{\mathbb{C}}H^{i}(X;{\mathcal{F}})_{\chi^{m}}\cdot\chi^{m}\in{\mathbb{Z}}[M],\end{split}$

with $a_{X}:X\to pt$ the constant map to a point. Note that
$\chi^{\mathbb{T}}(X,{\mathcal{F}})$ depends only on the class
$[{\mathcal{F}}]\in K_{0}^{\mathbb{T}}(X)$.

Let now $X$ be a closed ${\mathbb{T}}$-invariant compact algebraic subvariety
of an algebraic ${\mathbb{T}}$-manifold $Z$, and let $D$ be a
${\mathbb{T}}$-invariant Cartier divisor on $X$. Let $[{\mathcal{M}}]\in
K_{0}({\rm MHM}^{\mathbb{T}}_{X}(Z))$ be fixed. Then

$\hbox{\rm
DR}^{\mathbb{T}}_{y}([{\mathcal{M}}])\otimes{\mathcal{O}}_{X}(D):=\hbox{\rm
DR}^{\mathbb{T}}_{y}([{\mathcal{M}}])\cdot[{\mathcal{O}}_{X}(D)]\in
K_{0}^{\mathbb{T}}(X)[y^{\pm 1}],$

where we extend the $K^{0}_{\mathbb{T}}(X)$-module structure of
$K_{0}^{\mathbb{T}}(X)$ induced from the tensor product linearly over
${\mathbb{Z}}[y^{\pm 1}]$. For $a_{X}:X\to pt$ the constant map as above, we
can then define the ${\mathbb{T}}$-equivariant Hodge polynomial of
$(X,D,[{\mathcal{M}}])$ as:

(4.4)
$\begin{split}\chi^{\mathbb{T}}_{y}(X,D;[{\mathcal{M}}])&:=(a_{X})_{*}\left(\hbox{\rm
DR}^{\mathbb{T}}_{y}([{\mathcal{M}}])\otimes{\mathcal{O}}_{X}(D)\right)\in
K_{0}^{\mathbb{T}}(pt)[y^{\pm 1}]=K^{0}_{\mathbb{T}}(pt)[y^{\pm 1}]\\\
&=:\chi^{\mathbb{T}}(X;\hbox{\rm
DR}^{\mathbb{T}}_{y}([{\mathcal{M}}])\otimes{\mathcal{O}}_{X}(D))\in{\mathbb{Z}}[M][y^{\pm
1}],\end{split}$

which only depends on $[{\mathcal{M}}]$. In other words,
$\chi^{\mathbb{T}}_{y}(X,D;[{\mathcal{M}}])$ is an incarnation (in terms of
the characters of ${\mathbb{T}}$) of the degree of the Hodge-Chern class of
$[{\mathcal{M}}]$ twisted by ${\mathcal{O}}_{X}(D)$.

######  Remark 4.1.

If $D=0$,

(4.5)
$\begin{split}\chi^{\mathbb{T}}_{y}(X,0;[{\mathcal{M}}])&=(a_{X})_{*}\hbox{\rm
DR}^{\mathbb{T}}_{y}([{\mathcal{M}}])=\chi_{y}(X;[{\mathcal{M}}])\in{\mathbb{Z}}[y^{\pm
1}]\\\ &=\chi_{y}(X;[{\mathcal{M}}])\cdot\chi^{0}\in{\mathbb{Z}}[M][y^{\pm
1}],\end{split}$

is the usual Hodge polynomial of $[{\mathcal{M}}]$ as in (2.2). ∎

######  Example 4.2.

If $X$ is smooth and ${\mathcal{M}}={\mathbb{Q}}_{X}^{H}$ is the constant
Hodge module complex (viewed as a Hodge module complex on $Z$), then

$\chi^{\mathbb{T}}_{y}(X,D;[{\mathbb{Q}}_{X}^{H}])=\sum_{p=0}^{\dim(X)}\chi^{\mathbb{T}}\left(X,\Omega^{p}_{X}\otimes{\mathcal{O}}_{X}(D)\right)\cdot
y^{p},$

which is exactly $\chi^{\mathbb{T}}_{y}(X,{\mathcal{O}}_{X}(D))$, the
equivariant Hirzebruch polynomial of $D$, e.g., see [9, Sect.3.2].∎

######  Example 4.3.

Assume $X$ is a toric variety or a complex algebraic $V$-manifold. Then

$\chi^{\mathbb{T}}_{y}(X,D;[{\mathbb{Q}}_{X}^{H}])=\sum_{p=0}^{\dim(X)}\chi^{\mathbb{T}}\left(X,\widehat{\Omega}^{p}_{X}\otimes{\mathcal{O}}_{X}(D)\right)\cdot
y^{p},$

where $\widehat{\Omega}_{X}^{p}$ denotes the corresponding sheaf of Zariski
differential $p$-forms, is the Hodge polynomial of $D$ considered in [9,
Sect.3.2] in the toric context. ∎

##  5\. Characteristic class formulae in toric geometry

In this section, we specialize the formulae of previous sections to the case
of toric varieties (e.g., associated to a full-dimensional lattice polytope),
which are known to be Whitney stratified by the orbits of the torus action.

Let us first introduce some notation. Let $X=X_{\Sigma}$ be a quasi-projective
toric variety associated to the fan $\Sigma\subset N_{\mathbb{R}}\cong
N\otimes{\mathbb{R}}={\mathbb{R}}^{n}$, with torus
${\mathbb{T}}=N\otimes_{\mathbb{Z}}{\mathbb{C}}^{*}=({\mathbb{C}}^{*})^{n}$
having $M$, the dual of the lattice $N\cong{\mathbb{Z}}^{n}$, as the character
lattice. The action of the torus ${\mathbb{T}}$ on itself extends to an
algebraic action of ${\mathbb{T}}$ on $X$ with finitely many orbits
$O_{\sigma}$, one for each cone $\sigma\in\Sigma$. In fact, to a
$k$-dimensional cone $\sigma\in\Sigma$ there corresponds an
$(n-k)$-dimensional torus-orbit $O_{\sigma}$$\ \cong({\mathbb{C}}^{*})^{n-k}$,
and the orbit closure $V_{\sigma}$ of $O_{\sigma}$ is itself a toric variety
and a ${\mathbb{T}}$-invariant subvariety of $X$. Let
$N(\sigma)=N/N_{\sigma},$ with $N_{\sigma}$ denoting the sublattice of $N$
spanned by the points in $\sigma\cap N$. Let
${\mathbb{T}}_{N(\sigma)}=N(\sigma)\otimes_{\mathbb{Z}}{\mathbb{C}}^{*}$ be
the torus associated to $N(\sigma)$. For each cone $\nu\in\Sigma$ containing
$\sigma$ (in short, $\sigma\preceq\nu$), let $\overline{\nu}$ be the image
cone in $N(\sigma)_{\mathbb{R}}$ under the quotient map $N_{\mathbb{R}}\to
N(\sigma)_{\mathbb{R}}$. Then

(5.1) $Star(\sigma)=\\{\overline{\nu}\subseteq
N(\sigma)_{\mathbb{R}}\mid\sigma\preceq\nu\\}$

is a fan in $N(\sigma)_{\mathbb{R}}$, with associated toric variety isomorphic
to $V_{\sigma}$ (see, e.g., [11, Prop.3.2.7]), and ${\mathbb{T}}$ acts on
$V_{\sigma}$ via the morphism

(5.2) ${\mathbb{T}}\to{\mathbb{T}}_{N(\sigma)}$

induced by the quotient map $N\to N(\sigma)$.

Before specializing formula (3.3) to this toric context, we need the following
key additional input derived in [9]:

######  Proposition 5.1.

[9, Prop.3.8] Let $X=X_{\Sigma}$ be a quasi-projective toric variety defined
by the fan $\Sigma$. For any cone $\sigma\in\Sigma$, with orbit $O_{\sigma}$
and inclusion $i_{\sigma}:O_{\sigma}\hookrightarrow V_{\sigma}$ in the orbit
closure, we have:

(5.3)
$mC^{\mathbb{T}}_{y}([i_{\sigma}])=(1+y)^{\dim(O_{\sigma})}\cdot[\omega_{V_{\sigma}}]_{\mathbb{T}},$

where $\omega_{V_{\sigma}}=\widehat{\Omega}^{\dim(O_{\sigma})}_{V_{\sigma}}$
is the canonical sheaf on the toric variety $V_{\sigma}$, viewed as a
${\mathbb{T}}$-equivariant sheaf via the quotient map
${\mathbb{T}}\to{\mathbb{T}}_{N(\sigma)}$ given by (5.2).

###### Proof.

This follows from Proposition 3.5 by using a toric resolution $V$ of
$V_{\sigma}$, together with the fact that the locally free sheaf
$\Omega_{V}^{1}(\log(D))$ is ${\mathbb{T}}_{N(\sigma)}$-equivariantly, hence
also ${\mathbb{T}}$-equivariantly, a trivial sheaf of rank $\dim(O_{\sigma})$,
and there is a ${\mathbb{T}}_{N(\sigma)}$-equivariant, hence also
${\mathbb{T}}$-equivariant isomorphism
$R(\pi_{V})_{*}{\mathcal{O}}_{V}(-D)\simeq\omega_{V_{\sigma}}$. ∎

In the following we assume that $X=X_{\Sigma}$ is a quasi-projective toric
variety. Then [30] guarantees the existence of some smooth
${\mathbb{T}}$-equivariant ambient variety $Z$ with a closed
${\mathbb{T}}$-equivariant embedding $i:X\hookrightarrow Z$. Note also that
the torus stabilizer at any point in a torus orbit of $X$ is itself a torus,
hence connected.

Together with formula (3.3), Proposition 5.1 then gives the following:

######  Theorem 5.2.

Let $X=X_{\Sigma}$ be a quasi-projective toric variety defined by the fan
$\Sigma$, with torus ${\mathbb{T}}$. For each cone $\sigma\in\Sigma$ with
orbit $O_{\sigma}$, let $k_{\sigma}:V_{\sigma}\hookrightarrow Z$ be the
inclusion of the orbit closure in the smooth ambient ${\mathbb{T}}$-manifold
$Z$ mentioned above, and fix a point $x_{\sigma}\in O_{\sigma}$ with inclusion
$i_{x_{\sigma}}:\\{x_{\sigma}\\}\hookrightarrow Z$. Let ${\mathcal{M}}\in
D^{b,{\mathbb{T}}}_{X}(Z)$ be a ${\mathbb{T}}$-equivariant mixed Hodge module
complex on $Z$ with support in $X$. Then:

(5.4) $\hbox{\rm
DR}^{\mathbb{T}}_{y}([{\mathcal{M}}])=\sum_{\sigma\in\Sigma}\,\chi_{y}(i_{x_{\sigma}}^{*}[{\mathcal{M}}])\cdot(1+y)^{\dim(O_{\sigma})}\cdot(k_{\sigma})_{*}[\omega_{V_{\sigma}}]_{\mathbb{T}}.$

In particular, by taking degrees,

(5.5)
$\chi_{y}(X;[{\mathcal{M}}])=\sum_{\sigma\in\Sigma}\,\chi_{y}(i_{x_{\sigma}}^{*}[{\mathcal{M}}])\cdot(-1-y)^{\dim(O_{\sigma})}.$

Theorem 5.2 yields the following:

######  Corollary 5.3.

Let $X=X_{\Sigma}$ be the quasi-projective toric variety defined by a fan
$\Sigma$, with a chosen closed ${\mathbb{T}}$-equivariant embedding into a
smooth ambient ${\mathbb{T}}$-variety $Z$. For each cone $\sigma\in\Sigma$
with orbit $O_{\sigma}$, fix a point $x_{\sigma}\in O_{\sigma}$. Then we have
the following formula:

(5.6) $\hbox{\rm
DR}^{\mathbb{T}}_{y}([{IC^{\prime}}_{X}^{H}])=\sum_{\sigma\in\Sigma}\,\chi_{y}({IC^{\prime}}_{X}^{H}|_{x_{\sigma}})\cdot(1+y)^{\dim(O_{\sigma})}\cdot(k_{\sigma})_{*}[\omega_{V_{\sigma}}]_{\mathbb{T}}.$

Here, ${IC}_{X}^{H}\in{\rm MHM}^{\mathbb{T}}_{X}(Z)$ is regarded as a
${\mathbb{T}}$-equivariant Hodge module on the ambient ${\mathbb{T}}$-manifold
$Z$, and each class $[\omega_{V_{\sigma}}]_{\mathbb{T}}$ is pushed into
$K_{0,X}^{\mathbb{T}}(Z)\simeq K_{0}^{\mathbb{T}}(X)$.

######  Remark 5.4.

In particular, if $X_{\Sigma}$ is a simplicial toric variety then
$\chi_{y}({IC^{\prime}}_{X}^{H}|_{x_{\sigma}})=\chi_{y}({\mathbb{Q}}_{X}^{H}|_{x_{\sigma}})=1$
for all $\sigma\in\Sigma$, and one recovers in this case a result from [9].
Also note that formula (5.6) for $X$ an affine toric variety also fits with
recent results of the paper [19]. ∎

For future reference, we reformulate here (5.3) in the language of equivariant
mixed Hodge modules. Let $k_{\sigma}:V_{\sigma}\hookrightarrow Z$ be the orbit
closure embedding into the smooth ambient ${\mathbb{T}}$-manifold $Z$
constructed in [30]. Let $j_{\sigma}:=k_{\sigma}\circ
i_{\sigma}:O_{\sigma}\hookrightarrow Z$.

######  Corollary 5.5.

In the above notations,

(5.7) $\hbox{\rm
DR}^{\mathbb{T}}_{y}(\big{[}(j_{\sigma})_{!}{\mathbb{Q}}^{H}_{O_{\sigma}}\big{]})=(1+y)^{\dim(O_{\sigma})}\cdot(k_{\sigma})_{*}[\omega_{V_{\sigma}}]_{\mathbb{T}}.$

###### Proof.

This follows immediately by applying the proper pushforward $(k_{\sigma})_{*}$
to both sides of (5.3), using the functoriality of $mC^{\mathbb{T}}_{y}$ and
the definition of $\chi^{\mathbb{T}}_{Hdg}$. Indeed,

$(1+y)^{\dim(O_{\sigma})}\cdot(k_{\sigma})_{*}[\omega_{V_{\sigma}}]_{\mathbb{T}}=(k_{\sigma})_{*}mC^{\mathbb{T}}_{y}([i_{\sigma}])=mC^{\mathbb{T}}_{y}([j_{\sigma}])=\hbox{\rm
DR}_{y}^{\mathbb{T}}(\big{[}(j_{\sigma})_{!}{\mathbb{Q}}^{H}_{O_{\sigma}}]).$

∎

Before computing $\hbox{\rm
DR}^{\mathbb{T}}_{y}(\big{[}(j_{\sigma})_{*}{\mathbb{Q}}^{H}_{O_{\sigma}}\big{]})$,
we need the following lemma.

######  Lemma 5.6.

For $\sigma\in\Sigma$ with orbit $O_{\sigma}$ and orbit closure $V_{\sigma}$,
we have for the equivariant dualizing complex that

(5.8)
${\mathbb{D}}^{\mathbb{T}}({\mathcal{O}}_{V_{\sigma}})\simeq\omega_{V_{\sigma}}[\dim(O_{\sigma})].$

In particular, by equivariant biduality, we get

(5.9)
${\mathbb{D}}^{\mathbb{T}}[\omega_{V_{\sigma}}]_{\mathbb{T}}=(-1)^{\dim(O_{\sigma})}\cdot[{\mathcal{O}}_{V_{\sigma}}]_{\mathbb{T}}.$

###### Proof.

Let $\widetilde{V}_{\sigma}\overset{f_{\sigma}}{\to}V_{\sigma}$ be a toric
resolution of singularities of $V_{\sigma}$. Then $\widetilde{V}_{\sigma}$ is
a toric variety obtained by refining the fan of $V_{\sigma}$. Let
$O_{\sigma}\hookrightarrow\widetilde{V}_{\sigma}$ be the natural open
inclusion, with complement the simple normal crossing divisor $D_{\sigma}$
whose irreducible components correspond to the rays in the fan of
$\widetilde{V}_{\sigma}$. Note that $D_{\sigma}$ is a
${\mathbb{T}}_{N(\sigma)}$-invariant (hence a ${\mathbb{T}}$-invariant)
divisor.

Furthermore, note that the canonical sheaf $\omega_{\widetilde{V}_{\sigma}}$
on the toric variety $\widetilde{V}_{\sigma}$ is precisely given by
${\mathcal{O}}_{\widetilde{V}_{\sigma}}(-D_{\sigma})\simeq\omega_{\widetilde{V}_{\sigma}}$
as ${\mathbb{T}}_{N(\sigma)}$-equivariant, hence also
${\mathbb{T}}$-equivariant sheaves, e.g., see [11, Thm.8.2.3]. And since the
toric morphism $f_{\sigma}:\widetilde{V}_{\sigma}\to V_{\sigma}$ is induced by
a refinement of the fan of $V_{\sigma}$, it follows from [11, Thm.8.2.15] that
there is a ${\mathbb{T}}_{N(\sigma)}$-equivariant, hence also
${\mathbb{T}}$-equivariant, isomorphism:

(5.10)
${f_{\sigma}}_{*}\omega_{\widetilde{V}_{\sigma}}\simeq\omega_{V_{\sigma}}.$

Altogether, we obtain the following sequence of isomorphisms:

(5.11)
$\begin{split}{\mathbb{D}}^{\mathbb{T}}({\mathcal{O}}_{V_{\sigma}})&\simeq{\mathbb{D}}^{\mathbb{T}}\left((Rf_{\sigma})_{*}\big{(}{\mathcal{O}}_{\widetilde{V}_{\sigma}}\big{)}\right)\\\
&\simeq(Rf_{\sigma})_{*}\big{(}{\mathbb{D}}^{\mathbb{T}}({\mathcal{O}}_{\widetilde{V}_{\sigma}})\big{)}\\\
&\simeq(Rf_{\sigma})_{*}\big{(}\omega_{\widetilde{V}_{\sigma}}[\dim(O_{\sigma})]\big{)}\\\
&\simeq\omega_{V_{\sigma}}[\dim(O_{\sigma})],\end{split}$

where the first isomorphism follows since ${V}_{\sigma}$ has only rational
singularities (as in the proof of [9, Prop.3.5]); the second isomorphism
follows since equivariant Grothendieck duality commutes with proper
pushforward (cf. [20, Thm.25.2]); the third isomorphism follows by

(5.12)
${\mathbb{D}}^{\mathbb{T}}({\mathcal{O}}_{\widetilde{V}_{\sigma}})\simeq\omega_{\widetilde{V}_{\sigma}}[\dim(O_{\sigma})],$

for the smooth variety $\widetilde{V}_{\sigma}$, cf. [20, Thm.28.11]; finally,
the last isomorphism is a consequence of (5.10) and the vanishing of the
higher derived image sheaves of $\omega_{\widetilde{V}_{\sigma}}$, i.e.,
$R^{i}(f_{\sigma})_{*}\omega_{\widetilde{V}_{\sigma}}=0$, for all $i>0$, see
[11, Thm.9.3.12].

By using equivariant biduality, cf. [20, Prop.31.9, Lem.31.11], we get from
(5.8) that

${\mathcal{O}}_{V_{\sigma}}\simeq({\mathbb{D}}^{\mathbb{T}}\omega_{V_{\sigma}})[-\dim(O_{\sigma})],$

which yields (5.9). ∎

In the notations of the above Corollary 5.5, we then have the following

######  Proposition 5.7.

For $j_{\sigma}:O_{\sigma}\hookrightarrow Z$ the orbit inclusion into the
ambient ${\mathbb{T}}$-manifold $Z$,

(5.13) $\hbox{\rm
DR}^{\mathbb{T}}_{y}(\big{[}(j_{\sigma})_{*}{\mathbb{Q}}^{H}_{O_{\sigma}}\big{]})=(1+y)^{\dim(O_{\sigma})}\cdot(k_{\sigma})_{*}[{\mathcal{O}}_{V_{\sigma}}]_{\mathbb{T}}.$

###### Proof.

Since $O_{\sigma}$ is smooth, ${\mathbb{Q}}^{H}_{O_{\sigma}}$ is a (shifted)
${\mathbb{T}}$-equivariant pure Hodge module of weight $\dim(O_{\sigma})$, so

${\mathbb{D}}^{\mathbb{T}}_{O_{\sigma}}{\mathbb{Q}}^{H}_{O_{\sigma}}\simeq{\mathbb{Q}}^{H}_{O_{\sigma}}[2\dim(O_{\sigma}](\dim(O_{\sigma})).$

Using (2.3) and the fact that the even shift does not affect the Grothendieck
class, we get

(5.14) $\hbox{\rm
DR}^{\mathbb{T}}_{y}(\big{[}(j_{\sigma})_{*}{\mathbb{Q}}^{H}_{O_{\sigma}}\big{]})=(-y)^{\dim(O_{\sigma})}\cdot\hbox{\rm
DR}^{\mathbb{T}}_{y}({\mathbb{D}}^{\mathbb{T}}_{Z}\big{[}(j_{\sigma})_{!}{\mathbb{Q}}^{H}_{O_{\sigma}}\big{]}).$

Since $(j_{\sigma})_{!}{\mathbb{Q}}^{H}_{O_{\sigma}}\in{\rm
Image}(\chi^{\mathbb{T}}_{Hdg})$, Remark 2.12 gives

(5.15) $\begin{split}\hbox{\rm
DR}^{\mathbb{T}}_{y}({\mathbb{D}}^{\mathbb{T}}_{Z}\big{[}(j_{\sigma})_{!}{\mathbb{Q}}^{H}_{O_{\sigma}}\big{]})&=\left({\mathbb{D}}^{\mathbb{T}}\left(\hbox{\rm
DR}^{\mathbb{T}}_{y}(\big{[}(j_{\sigma})_{!}{\mathbb{Q}}^{H}_{O_{\sigma}}\big{]})\right)\right)_{|_{y\mapsto
1/y}}\\\
&\overset{\eqref{drtc}}{=}\left({\mathbb{D}}^{\mathbb{T}}\left((1+y)^{\dim(O_{\sigma})}\cdot(k_{\sigma})_{*}[\omega_{V_{\sigma}}]_{\mathbb{T}}\right)\right)_{|_{y\mapsto
1/y}}\\\
&=(1+\frac{1}{y})^{\dim(O_{\sigma})}\cdot(k_{\sigma})_{*}{\mathbb{D}}^{\mathbb{T}}[\omega_{V_{\sigma}}]_{\mathbb{T}}\\\
&\overset{\eqref{Sed}}{=}y^{-\dim(O_{\sigma})}\cdot(1+y)^{\dim(O_{\sigma})}\cdot(-1)^{\dim(O_{\sigma})}\cdot(k_{\sigma})_{*}[{\mathcal{O}}_{V_{\sigma}}]_{\mathbb{T}}\\\
&=(-y)^{-\dim(O_{\sigma})}\cdot(1+y)^{\dim(O_{\sigma})}\cdot(k_{\sigma})_{*}[{\mathcal{O}}_{V_{\sigma}}]_{\mathbb{T}}.\end{split}$

Here, in the first two equalities we use a linear extension of equivariant
Grothedieck duality over ${\mathbb{Z}}[y^{\pm 1}]$. Formula (5.13) is now a
consequence of (5.14) and (5.15). ∎

##  6\. Applications to weighted Ehrhart theory

Let now $M\cong{\mathbb{Z}}^{n}$ be a lattice and $P\subset
M_{{\mathbb{R}}}\cong{\mathbb{R}}^{n}$ be a full-dimensional lattice polytope
with associated projective toric variety $X=X_{P}$, inner normal fan
$\Sigma=\Sigma_{P}$ and ample Cartier divisor $D=D_{P}$. One has (e.g., see
[11, Prop.4.3.3])

(6.1) $\Gamma(X_{P};{\mathcal{O}}_{X_{P}}(D_{P}))=\bigoplus_{m\in P\cap
M}{\mathbb{C}}\cdot\chi^{m}\subset{\mathbb{C}}[M],$

where $\chi^{m}$ denotes the character defined by $m\in M$, and
${\mathbb{C}}[M]$ is the coordinate ring of the torus ${\mathbb{T}}$. Also,
since $D_{P}$ is ample,

(6.2) $H^{i}(X_{P};{\mathcal{O}}_{X_{P}}(D_{P}))=0,\ {\rm for\ all}\ i>0.$

Moreover, one has (e.g., see [11, Prop.9.2.7])

(6.3) $H^{i}(X_{P};{\mathcal{O}}_{X_{P}}(-D_{P}))=0,\ {\rm for\ all}\ i\neq
n,$

and

(6.4) $H^{n}(X_{P};{\mathcal{O}}_{X_{P}}(-D_{P}))=\bigoplus_{m\in{\rm
Int}(P)\cap M}{\mathbb{C}}\cdot\chi^{-m},$

with ${\rm Int}({P})$ the interior of $P$. Also, by equivariant toric Serre
duality [11, Thm.9.2.10] (fitting with our description of the equivariant
dualizing complex in (5.8)),

(6.5) $\bigoplus_{m\in{\rm Int}(P)\cap
M}{\mathbb{C}}\cdot\chi^{m}=\left(H^{n}(X_{P};{\mathcal{O}}_{X_{P}}(-D_{P}))\right)^{\vee}\simeq
H^{0}(X_{P};{\mathcal{O}}_{X_{P}}(D_{P})\otimes\omega_{X_{P}}),$

with $\omega_{X_{P}}=\widehat{\Omega}^{\dim(P)}_{X_{P}}$ the equivariant
dualizing and canonical sheaf of $X_{P}$, together with
$H^{i}(X_{P};{\mathcal{O}}_{X_{P}}(D_{P})\otimes\omega_{X_{P}})=0$ for all
$i\neq 0$.

######  Remark 6.1.

For any ${\mathbb{T}}$-invariant Cartier divisor $D$ on an $n$-dimensional
complete toric variety $X$ with torus ${\mathbb{T}}$, one has the
nondegenerate equivariant pairing

$H^{i}(X;{\mathcal{O}}_{X}(-D))\times
H^{n-i}(X;{\mathcal{O}}_{X}(D)\otimes\omega_{X})\longrightarrow
H^{n}(X;\omega_{X})\longrightarrow{\mathbb{C}}$

induced by the ${\mathbb{T}}$-equivariant sheaf isomorphism

${\mathcal{O}}_{X}(-D)\otimes{\mathcal{O}}_{X}(D)\otimes\omega_{X}\longrightarrow\omega_{X}$

and the ${\mathbb{T}}$-equivariant trace map
$\omega_{X}[n]\to\omega_{pt}={\mathbb{C}}$. This induces the equivariant Serre
duality

$\left(H^{i}(X;{\mathcal{O}}_{X}(-D))\right)^{\vee}\simeq
H^{n-i}(X;{\mathcal{O}}_{X}(D)\otimes\omega_{X}).$

∎

######  Remark 6.2.

For a cone $\sigma_{Q}\in\Sigma_{P}$ associated to a face $Q$ of $P$, the
corresponding orbit closure $V_{\sigma_{Q}}$ can be identified with the toric
variety $X_{Q}$, with corresponding lattice polytope defined as follows (cf.
[11, Prop.3.2.9]): translate $P$ by a vertex $m_{0}$ of $Q$ so that the origin
is a vertex of $Q_{0}:=Q-m_{0}$; while this translation by $m_{0}$ does not
change $\Sigma_{P}$ or $X_{P}$, $Q_{0}$ is now a full-dimensional polytope in
$Span(Q_{0})$ relative to the lattice $Span(Q_{0})\cap M$, and $X_{Q}$ is the
associated toric variety. However, if one wishes to apply the above formulas
to the toric variety $X_{Q}$ associated to a face $Q$ of $P$, one needs to
work with the corresponding divisor

(6.6) $D_{Q_{0}}={D_{P-m_{0}}}|_{X_{Q}}=D_{P}|_{X_{Q}}+div(\chi^{m_{0}}).$

∎

Formulae (6.1)-(6.4) correspond to the weight decomposition of the respective
cohomology groups, e.g.,

(6.7) $H^{i}(X_{P};{\mathcal{O}}_{X_{P}}(D_{P}))=\bigoplus_{m\in
M}H^{i}(X_{P};{\mathcal{O}}_{X_{P}}(D_{P}))_{m},$

as in [11, Sect.9.1]. To relate this weight decomposition with the eigenspace
decomposition of (4.1), we recall that the torus ${\mathbb{T}}$ acts on
${\mathbb{C}}[M]$ as follows: if $t\in{\mathbb{T}}$ and $f\in{\mathbb{C}}[M]$,
then $t\cdot f\in{\mathbb{C}}[M]$ is given by $p\mapsto f(t^{-1}\cdot p)$, for
$p\in{\mathbb{T}}$ (see [11, pag.18]). In particular,
$t\cdot\chi^{m}=\chi^{m}(t^{-1})\chi^{m}$, so that

$H^{i}(X;{\mathcal{O}}_{X}(D))_{m}=H^{i}(X;{\mathcal{O}}_{X}(D))_{\chi^{-m}}.$

Let us note that if for a positive integer $\ell\in{\mathbb{Z}}_{>0}$ we
consider the dilated polytope

$\ell P:=\\{\ell\cdot u\ |\ u\in P\\},$

then the toric varieties corresponding to $P$ and, resp., $\ell P$, are the
same, but $D_{\ell P}=\ell D_{P}$. Together with (6.1)-(6.5), this gives for
such $\ell\in{\mathbb{Z}}_{>0}$:

(6.8) $\chi^{\mathbb{T}}(X_{P},{\mathcal{O}}_{X_{P}}(\ell
D_{P}))=\sum_{m\in\ell P\cap M}\chi^{-m},$ (6.9)
$\chi^{\mathbb{T}}(X_{P},{\mathcal{O}}_{X_{P}}(-\ell
D_{P}))=(-1)^{n}\cdot\sum_{m\in{\rm Int}(\ell P)\cap M}\chi^{m}$

and

(6.10) $\chi^{\mathbb{T}}(X_{P},{\mathcal{O}}_{X_{P}}(\ell
D_{P})\otimes\omega_{X_{P}})=\sum_{m\in{\rm Int}(\ell P)\cap M}\chi^{-m}.$

More generally, we have by [9, Rem.4.9], applied to the polytope $\ell P$, the
following identity:

(6.11)
$\chi^{\mathbb{T}}(X_{P},mC^{\mathbb{T}}_{y}(X_{P})\otimes{\mathcal{O}}_{X_{P}}(\ell
D_{P}))=\sum_{Q\preceq P}(1+y)^{\dim(Q)}\cdot\sum_{m\in{\rm Relint}(\ell
Q)\cap M}\chi^{-m}\in{\mathbb{Z}}[M]\otimes_{\mathbb{Z}}{\mathbb{Z}}[y].$

where ${\rm Relint}$ denotes the relative interior, and the first summation is
over the faces $Q$ of the lattice polytope $P$.

For further reference, let us also include here the following.

######  Lemma 6.3.

In the above notations, with $\ell\in{\mathbb{Z}}_{>0}$,

(6.12) $\chi^{\mathbb{T}}(X_{P},{\mathcal{O}}_{X_{P}}(-\ell
D_{P})\otimes\omega_{X_{P}})=(-1)^{n}\cdot\sum_{m\in\ell P\cap M}\chi^{m}$

###### Proof.

By toric equivariant Serre duality [11, Thm.9.2.10] and Remark 6.1, together
with (6.1) and (6.2),

(6.13) $\bigoplus_{m\in P\cap
M}{\mathbb{C}}\cdot\chi^{-m}=\left(H^{0}(X_{P};{\mathcal{O}}_{X_{P}}(D_{P}))\right)^{\vee}\simeq
H^{n}(X_{P};{\mathcal{O}}_{X_{P}}(-D_{P})\otimes\omega_{X_{P}}),$

and $H^{i}(X_{P};{\mathcal{O}}_{X_{P}}(-D_{P})\otimes\omega_{X_{P}})=0$ for
all $i\neq n$. The assertion follows by reverting to the weight decomposition
of these cohomology groups, as indicated above. ∎

Making use of Theorem 5.2, we can now generalize formula (6.11) as follows.

######  Theorem 6.4.

Let $X=X_{P}$ be the projective toric variety with ample Cartier divisor
$D=D_{P}$ associated to a full-dimensional lattice polytope $P\subset
M_{{\mathbb{R}}}\cong{\mathbb{R}}^{n}$, and let $\ell\in{\mathbb{Z}}_{>0}$.
Let $Z$ be an ambient ${\mathbb{T}}$-manifold containing $X$ as a
${\mathbb{T}}$-invariant subvariety. Then for any Grothendieck class
$[{\mathcal{M}}]\in K_{0}({\rm MHM}^{\mathbb{T}}_{X}(Z))$ of a mixed Hodge
module (complex) on $Z$ with support in $X$, we have

(6.14) $\begin{split}\chi^{\mathbb{T}}_{y}(X,\ell
D;[{\mathcal{M}}])&=\sum_{Q\preceq
P}\chi_{y}(i_{x_{Q}}^{*}[{\mathcal{M}}])\cdot(1+y)^{\dim(Q)}\cdot\sum_{m\in{\rm
Relint}({{\ell}}Q)\cap M}\chi^{-m},\end{split}$

where the first summation on the right is over the faces $Q$ of $P$, and
$x_{Q}\in O_{\sigma_{Q}}$ is a point in the orbit of $X$ associated to (the
cone $\sigma_{Q}$ in the inner normal fan of $P$ corresponding to) the face
$Q$, with inclusion map $i_{x_{Q}}:\\{i_{x_{Q}}\\}\hookrightarrow Z$.

###### Proof.

For a face $Q$ of $P$, denote by $k_{Q}:V_{\sigma_{Q}}:=X_{Q}\hookrightarrow
X\hookrightarrow Z$ the inclusion in the ambient ${\mathbb{T}}$-manifold $Z$
of the orbit closure associated to the (cone $\sigma_{Q}$ of the) face $Q$.
Note that we have $\dim_{\mathbb{R}}(Q)=\dim_{\mathbb{R}}(O_{\sigma_{Q}}).$
Let ${\mathbb{T}}_{Q}$ be the quotient torus of ${\mathbb{T}}$ corresponding
to $X_{Q}$, as in (5.2).

In what follows, we will denote all constant maps to a point by $a$,
regardless of their domains (which should be understood from the context).
Then by (4.4) and (5.4) the following equality holds:

$\begin{split}\chi^{\mathbb{T}}_{y}(X,\ell
D;{\mathcal{M}})&=a_{*}\left(\hbox{\rm
DR}^{\mathbb{T}}_{y}([{\mathcal{M}}])\otimes{\mathcal{O}}_{X}(\ell
D)\right)\\\ &=\sum_{Q\preceq
P}\chi_{y}(i_{x_{Q}}^{*}[{\mathcal{M}}])\cdot(1+y)^{\dim(Q)}\cdot
a_{*}\left((k_{Q})_{*}[\omega_{X_{Q}}]_{\mathbb{T}}\otimes{\mathcal{O}}_{X}(\ell
D)\right).\end{split}$

So it remains to prove that, for any face $Q$ of $P$, we have that:

(6.15)
$\chi^{\mathbb{T}}(X,(k_{Q})_{*}[\omega_{X_{Q}}]_{\mathbb{T}}\otimes{\mathcal{O}}_{X}(\ell
D))=a_{*}\left((k_{Q})_{*}[\omega_{X_{Q}}]_{\mathbb{T}}\otimes{\mathcal{O}}_{X}(\ell
D)\right)=\sum_{m\in{\rm Relint}(\ell Q)\cap M}\chi^{-m}.$

This follows from the projection formula and (6.10), by making use of formula
(6.6). Indeed,

$\begin{split}a_{*}\left((k_{Q})_{*}[\omega_{X_{Q}}]_{\mathbb{T}}\otimes{\mathcal{O}}_{X}(\ell
D)\right)&=a_{*}(k_{Q})_{*}\left([\omega_{X_{Q}}]_{\mathbb{T}}\otimes
k_{Q}^{*}{\mathcal{O}}_{X}(\ell D)\right)\\\
&\overset{(\ref{tran})}{=}a_{*}\left([\omega_{X_{Q}}]_{\mathbb{T}}\otimes{\mathcal{O}}_{X_{Q}}(\ell
D_{Q_{0}}-div(\chi^{\ell m_{0}}))\right)\\\ &=\chi^{-\ell m_{0}}\cdot
a_{*}\left([\omega_{X_{Q}}]_{\mathbb{T}}\otimes{\mathcal{O}}_{X_{Q}}(\ell
D_{Q_{0}})\right)\\\ &\overset{({\ast})}{=}\chi^{-\ell m_{0}}\cdot
a_{*}\left([\omega_{X_{Q}}]_{{\mathbb{T}}_{Q}}\otimes{\mathcal{O}}_{X_{Q}}(\ell
D_{Q_{0}})\right)\\\ &\overset{(\ref{am6})}{=}\chi^{-\ell
m_{0}}\cdot\left(\sum_{m\in{\rm Relint}\ \ell(Q-m_{0})\cap
M}\chi^{-m}\right)\\\ &=\sum_{m\in{\rm Relint}(\ell Q)\cap
M}\chi^{-m},\end{split}$

where $(\ast)$ uses [9, Prop.3.1]. Here $m_{0}$ is a vertex of $Q$ as in
(6.6), so that $Q_{0}:=Q-m_{0}$ is a full-dimensional lattice polytope in
$Span(Q_{0})$ relative to the lattice $Span(Q_{0})\cap M$, with $X_{Q}$ the
associated toric variety. ∎

######  Remark 6.5.

For future use, we include here the following formula, which one gets as in
the proof of (6.15), but using ${\mathcal{O}}_{X_{Q}}$ instead of
$\omega_{X_{Q}}$, together with (6.8):

(6.16)
$\chi^{\mathbb{T}}(X,(k_{Q})_{*}[{\mathcal{O}}_{X_{Q}}]_{\mathbb{T}}\otimes{\mathcal{O}}_{X}(\ell
D))=a_{*}\left((k_{Q})_{*}[{\mathcal{O}}_{X_{Q}}]_{\mathbb{T}}\otimes{\mathcal{O}}_{X}(\ell
D)\right)=\sum_{m\in\ell Q\cap M}\chi^{-m}.$

∎

######  Definition 6.6 (Generalized weighted Ehrhart polynomials).

Let $\varphi:M_{\mathbb{R}}\cong{\mathbb{R}}^{n}\to{\mathbb{C}}$ be a
homogeneous polynomial function, and $\ell\in{\mathbb{Z}}_{>0}$. With the
above notations, the expression

(6.17) $E^{\varphi}_{P,{\mathcal{M}}}(\ell,y):=\sum_{Q\preceq
P}\chi_{y}(i_{x_{Q}}^{*}[{\mathcal{M}}])\cdot(1+y)^{\dim(Q)+\deg(\varphi)}\cdot\sum_{m\in{\rm
Relint}({{\ell}}Q)\cap M}\varphi(m)$

is obtained from $\chi^{\mathbb{T}}_{y}(X,\ell D;{\mathcal{M}})$ by
substituting $\chi^{-m}$ by
$\varphi\left((1+y)m\right)=(1+y)^{\deg(\varphi)}\cdot\varphi(m)$, for each
$m$. As we will show later on in the Appendix (but see also [8, Prop.4.1]),
$E^{\varphi}_{P,{\mathcal{M}}}(\ell,y)$ is obtained by evaluating a polynomial
$E^{\varphi}_{P,{\mathcal{M}}}(z,y)$ at $z=\ell$. We call this polynomial the
generalized weighted Ehrhart polynomial of $P$ and $\varphi$ with coefficients
induced from $[{\mathcal{M}}]\in K_{0}({\rm MHM}^{\mathbb{T}}_{X}(Z))$.
Similarly, one can define

(6.18) $\widetilde{E}^{\varphi}_{P,{\mathcal{M}}}(\ell,y):=\sum_{Q\preceq
P}\chi_{y}(i_{x_{Q}}^{*}[{\mathcal{M}}])\cdot(1+y)^{\dim(Q)}\cdot\sum_{m\in{\rm
Relint}({{\ell}}Q)\cap M}\varphi(m)$

which obtained from $\chi^{\mathbb{T}}_{y}(X,\ell D;{\mathcal{M}})$ by
substituting $\chi^{-m}$ by $\varphi\left(m\right)$, for each $m$. Since
$\varphi$ is homogeneous, this substitution fits with the duality involution
$\chi^{m}\mapsto\chi^{-m}$ for $m\in M$, via
$\varphi(-m)=(-1)^{\deg(\varphi)}\cdot\varphi(m)$.

More generally, for a Laurent polynomial weight function

$f:\\{\text{faces of }P\\}\to{\mathbb{Z}}[y^{\pm 1}],\ \ \ Q\mapsto f_{Q}(y)$

defined on the set of faces of $P$, we define as in the Introduction an
associated generalized weighted Ehrhart polynomial by

(6.19) $\begin{split}E^{\varphi}_{P,f}(\ell,y)&:=\sum_{Q\preceq
P}f_{Q}(y)\cdot(1+y)^{\dim(Q)+\deg(\varphi)}\cdot\sum_{m\in{\rm
Relint}({{\ell}}Q)\cap M}\varphi(m)\\\
&=:(1+y)^{\deg(\varphi)}\cdot\widetilde{E}^{\varphi}_{P,f}(\ell,y).\end{split}$

######  Remark 6.7.

Note that any Grothendieck class $[{\mathcal{M}}]\in K_{0}({\rm
MHM}^{\mathbb{T}}_{X}(Z))$ of a equivariant mixed Hodge module (complex) on
$Z$ with support in $X=X_{P}$, induces a weight function on the faces of $P$
defined by $f_{Q}(y)=\chi_{y}(i_{x_{Q}}^{*}[{\mathcal{M}}])$, with $x_{Q}\in
O_{\sigma_{Q}}$ a point in the orbit of $X_{P}$ associated to (the cone
$\sigma_{Q}$ in the inner normal fan of $P$ corresponding to) the face $Q$ and
$i_{x_{Q}}:\\{x_{Q}\\}\to Z$ the inclusion map. Moreover, any weight function
$f$ can be obtained in this way from the class of such a mixed Hodge module
complex ${\mathcal{M}}$, e.g., by using direct sums and shifts of classes of
$(j_{Q})_{!}{\mathbb{Q}}^{H}_{O_{\sigma_{Q}}}(n_{Q})$, with
$n_{Q}\in{\mathbb{Z}}$ and $j_{Q}:O_{\sigma_{Q}}\hookrightarrow Z$ the
inclusion, regarded as ${\mathbb{T}}$-equivariant objects on $Z$ (or, using
the subgroup $\chi^{\mathbb{T}}_{Hdg}\left(\tau(F^{{\mathbb{T}}}(X)[y^{\pm
1}])\right)$ as in the Introduction). In particular, the expressions
$E^{\varphi}_{P,f}(\ell,y)$ and $\widetilde{E}^{\varphi}_{P,f}(\ell,y)$ are
then also obtained by evaluating a polynomial $E^{\varphi}_{P,f}(z,y)$, resp.,
$\widetilde{E}^{\varphi}_{P,f}(z,y)$ at $z=\ell$. ∎

Before studying the generalized weighted Ehrhart polynomial
$E^{\varphi}_{P,f}(\ell,y)$, resp., $\widetilde{E}^{\varphi}_{P,f}(\ell,y)$,
in more detail, let us also mention the following result, similar to Theorem
6.4.

######  Theorem 6.8.

In the notations of Theorem 6.4, we have for $\ell\in{\mathbb{Z}}_{>0}$:

(6.20) $\begin{split}\chi^{\mathbb{T}}_{y}(X,-\ell
D;[{\mathcal{M}}])&=\sum_{Q\preceq
P}\chi_{y}(i_{x_{Q}}^{*}[{\mathcal{M}}])\cdot(-1-y)^{\dim(Q)}\cdot\sum_{m\in{\ell}Q\cap
M}\chi^{m}.\end{split}$

###### Proof.

The proof is very similar to that of Theorem 6.4, so we only sketch it below.
In view of (5.4), this amounts to showing that for any face $Q$ of $P$, and
using notations from loc.cit., we have the following formula:

(6.21)
$\begin{split}\chi^{\mathbb{T}}(X,(k_{Q})_{*}[\omega_{X_{Q}}]_{\mathbb{T}}\otimes{\mathcal{O}}_{X}(-\ell
D))&=a_{*}\left((k_{Q})_{*}[\omega_{X_{Q}}]_{\mathbb{T}}\otimes{\mathcal{O}}_{X}(-\ell
D)\right)\\\ &=(-1)^{\dim(Q)}\cdot\sum_{m\in\ell Q\cap M}\chi^{m}.\end{split}$

Indeed,

$\begin{split}a_{*}\left((k_{Q})_{*}[\omega_{X_{Q}}]_{\mathbb{T}}\otimes{\mathcal{O}}_{X}(-\ell
D)\right)&=a_{*}(k_{Q})_{*}\left([\omega_{X_{Q}}]_{\mathbb{T}}\otimes
k_{Q}^{*}{\mathcal{O}}_{X}(-\ell D)\right)\\\
&\overset{(\ref{tran})}{=}a_{*}\left([\omega_{X_{Q}}]_{\mathbb{T}}\otimes{\mathcal{O}}_{X_{Q}}(-\ell
D_{Q_{0}}+div(\chi^{\ell m_{0}}))\right)\\\ &=\chi^{\ell m_{0}}\cdot
a_{*}\left([\omega_{X_{Q}}]_{\mathbb{T}}\otimes{\mathcal{O}}_{X_{Q}}(-\ell
D_{Q_{0}})\right)\\\ &\overset{({\ast})}{=}\chi^{\ell m_{0}}\cdot
a_{*}\left([\omega_{X_{Q}}]_{{\mathbb{T}}_{Q}}\otimes{\mathcal{O}}_{X_{Q}}(-\ell
D_{Q_{0}})\right)\\\ &\overset{(\ref{am56})}{=}\chi^{\ell
m_{0}}\cdot\left((-1)^{\dim(Q)}\cdot\sum_{m\in\ell(Q-m_{0})\cap
M}\chi^{m}\right)\\\ &=(-1)^{\dim(Q)}\cdot\sum_{m\in\ell Q\cap
M}\chi^{m},\end{split}$

where $(\ast)$ uses again [9, Prop.3.1]. ∎

We can now justify the reciprocity formula (1.4) without relying on Brion-
Vergne’s combinatorial work [8], as well as the constant term formula (1.2):

######  Corollary 6.9.

Let $P\subset M_{\mathbb{R}}\cong{\mathbb{R}}^{n}$ be a full-dimensional
lattice polytope, with
$\varphi:M_{\mathbb{R}}\cong{\mathbb{R}}^{n}\to{\mathbb{C}}$ a homogeneous
polynomial function. For any Laurent polynomial weight vector
$\\{f_{Q}\\}_{Q\preceq P}$ defined on the set of faces of $P$, we have

(6.22) $E^{\varphi}_{P,f}(-\ell,y)=\sum_{Q\preceq
P}f_{Q}(y)\cdot(-1-y)^{\dim(Q)+\deg(\varphi)}\cdot\sum_{m\in{{\ell}}Q\cap
M}\varphi(m)$

and

(6.23) $\widetilde{E}^{\varphi}_{P,f}(-\ell,y)=\sum_{Q\preceq
P}f_{Q}(y)\cdot(-1-y)^{\dim(Q)}\cdot(-1)^{\deg(\varphi)}\cdot\sum_{m\in{{\ell}}Q\cap
M}\varphi(m).$

In particular,

(6.24) $E^{\varphi}_{P,f}(0,y)=\sum_{Q\preceq
P}f_{Q}(y)\cdot(-1-y)^{\dim(Q)+\deg(\varphi)}\cdot\varphi(0)$

and

(6.25) $\widetilde{E}^{\varphi}_{P,f}(0,y)=\sum_{Q\preceq
P}f_{Q}(y)\cdot(-1-y)^{\dim(Q)}\cdot(-1)^{\deg(\varphi)}\cdot\varphi(0).$

###### Proof.

As in Remark 6.7, choose a $[{\mathcal{M}}]\in K_{0}({\rm
MHM}^{\mathbb{T}}_{X}(Z))$ with
$f_{Q}(y)=\chi_{y}(i_{x_{Q}}^{*}[{\mathcal{M}}])$, such that
$E^{\varphi}_{P,f}(\ell,y)=E^{\varphi}_{P,{\mathcal{M}}}(\ell,y)$. From to the
definition of $E^{\varphi}_{P,{\mathcal{M}}}(\ell,y)$ in (6.17),
$E^{\varphi}_{P,{\mathcal{M}}}(-\ell,y)$ is obtained from
$\chi^{\mathbb{T}}_{y}(X,-\ell D;{\mathcal{M}})$ by substituting each
$\chi^{m}$ by
$\varphi\left((1+y)\cdot(-m)\right)=(-1-y)^{\deg(\varphi)}\cdot\varphi(m)$.
Formula (6.22) follows then by applying this rule to the expression for
$\chi^{\mathbb{T}}_{y}(X,-\ell D;{\mathcal{M}})$ given by (6.20). The constant
term formula (6.24) is then an easy consequence, by setting $\ell=0$ in
(6.22), or by combining (4.5) and (5.5). Similarly, for
$\widetilde{E}^{\varphi}_{P,f}(-\ell,y)$ and
$\widetilde{E}^{\varphi}_{P,f}(0,y)$. ∎

######  Example 6.10.

Let us next give some examples for Theorem 6.4, formulated in terms of our
generalized weighted Ehrhart polynomials
$E^{\varphi}_{P,{\mathcal{M}}}(\ell,y)$ via Definition 6.6. Let $X=X_{P}$ be
the projective toric variety with inner normal fan $\Sigma=\Sigma_{P}$ and
ample Cartier divisor $D=D_{P}$ associated to a full-dimensional lattice
polytope $P\subset M_{{\mathbb{R}}}\cong{\mathbb{R}}^{n}$, and let
$\ell\in{\mathbb{Z}}_{>0}$. For a face $Q$ of $P$ denote by
$j_{Q}:O_{\sigma_{Q}}\hookrightarrow Z$ the inclusion of the orbit associated
to the (cone of the) face $Q$.

  1. (a)

For ${\mathcal{M}}=(j_{Q})_{!}{\mathbb{Q}}^{H}_{O_{\sigma_{Q}}}$, we get

(6.26)
$E^{\varphi}_{P,{\mathcal{M}}}(\ell,y)=(1+y)^{\dim(Q)+\deg(\varphi)}\cdot\sum_{m\in{\rm
Relint}({{\ell}}Q)\cap M}\varphi(m),$

with
$E^{\varphi}_{P,{\mathcal{M}}}(0,y)=(-y-1)^{\dim(Q)+\deg(\varphi)}\cdot\varphi(0)=(-y-1)^{\deg(\varphi)}\cdot\chi_{y}(O_{\sigma_{Q}})\cdot\varphi(0)$.

In particular, by evaluating $E^{\varphi}_{P,{\mathcal{M}}}(\ell,y)$ at $y=0$,
we get the polynomial

$E^{\varphi}_{P,{\mathcal{M}}}(\ell,0)=\sum_{m\in{\rm Relint}({{\ell}}Q)\cap
M}\varphi(m)$

summing up the values of $\varphi$ at lattice points in the relative interior
of the dilated face $\ell Q$ (i.e., the Euler-Maclaurin sum for the relative
interior of $\ell Q$), with
$E^{\varphi}_{P,{\mathcal{M}}}(0,0)=(-1)^{\dim(Q)+\deg(\varphi)}\cdot\varphi(0)$.

  2. (b)

Let $X^{\prime}:=X_{P^{\prime}}$ be a torus-invariant closed algebraic subset
of $X=X_{P}$ corresponding to a polytopal subcomplex $P^{\prime}\subseteq P$
(i.e., a closed union of faces of $P$), with
$[{\mathcal{M}}]=[{\mathbb{Q}}_{X^{\prime}}^{H}]$ pushed forward to the
ambient ${\mathbb{T}}$-manifod $Z$ containing $X_{P}$ as a
${\mathbb{T}}$-invariant closed subvariety. Then

(6.27) $E^{\varphi}_{P,{\mathcal{M}}}(\ell,y)=\sum_{Q\preceq
P^{\prime}}(1+y)^{\dim(Q)+\deg(\varphi)}\cdot\sum_{m\in{\rm
Relint}({{\ell}}Q)\cap M}\varphi(m)\in{\mathbb{Z}}[y],$

where the summation is over the faces $Q$ of $P^{\prime}$. Hence,

$E^{\varphi}_{P,{\mathcal{M}}}(\ell,0)=\sum_{m\in\ell P^{\prime}\cap
M}\varphi(m)$

is the corresponding Euler-Maclaurin sum for $\ell P^{\prime}$, and

$E^{\varphi}_{P,{\mathcal{M}}}(0,y)=\sum_{Q\preceq
P^{\prime}}(-y-1)^{\dim(Q)+\deg(\varphi)}\cdot\varphi(0)=(-1)^{\deg(\varphi)}\cdot\chi_{y}(X^{\prime})\cdot\varphi(0),$

which gives that

$E^{\varphi}_{P,{\mathcal{M}}}(0,0)=(-1)^{\deg(\varphi)}\cdot\chi(P^{\prime})\cdot\varphi(0),$

with $\chi(P^{\prime})$ the topological Euler characteristic of $P^{\prime}$.

  3. (c)

For ${\mathcal{M}}=(j_{Q})_{*}{\mathbb{Q}}^{H}_{O_{\sigma_{Q}}}$, we get by
(5.13) and (6.16) that

(6.28)
$E^{\varphi}_{P,{\mathcal{M}}}(\ell,y)=(1+y)^{\dim(Q)+\deg(\varphi)}\cdot\sum_{m\in\ell
Q\cap M}\varphi(m),$

with
$E^{\varphi}_{P,{\mathcal{M}}}(0,y)=(1+y)^{\deg(\varphi)}\cdot\chi_{y}(V_{\sigma_{Q}})\cdot\varphi(0)$,

$E^{\varphi}_{P,{\mathcal{M}}}(\ell,0)=\sum_{m\in\ell Q\cap M}\varphi(m)$

the Euler-Maclaurin sum for $\ell Q$, and
$E^{\varphi}_{P,{\mathcal{M}}}(0,0)=\chi(Q)\cdot\varphi(0)$. ∎

Let $X=X_{P}$ be as before the projective toric variety with inner normal fan
$\Sigma=\Sigma_{P}$ and ample Cartier divisor $D=D_{P}$ associated to a full-
dimensional lattice polytope $P\subset M_{{\mathbb{R}}}\cong{\mathbb{R}}^{n}$.
Let $X\hookrightarrow Z$ be the equivariant embedding of $X$ into an ambient
${\mathbb{T}}$-manifold, as in [30]. Then another important example is
provided by ${\mathcal{M}}={IC^{\prime}}^{H}_{X}$. This is a shifted
${\mathbb{T}}$-equivariant pure Hodge module on $Z$ with support in $X$. As
explained in the Introduction, in the case when $P$ contains the origin in its
interior (so that its polar polytope $P^{\circ}$ is defined), the stalk
contributions $\chi_{y}({IC^{\prime}}_{X}^{H}|_{x_{Q}})$ have the following
combinatorial description (see, e.g., [5, 16, 13, 27], etc):

(6.29)
$\chi_{y}({IC^{\prime}}_{X}^{H}|_{x_{Q}})=\widetilde{g}_{Q}(-y):=g_{Q^{\circ}}(-y),$

with $Q$ a face of $P$ and $Q^{\circ}$ the corresponding face of the polar
polytope $P^{\circ}$, and $g_{Q^{\circ}}(t)\in{\mathbb{Z}}[t]$ the Stanley
$g$-polynomial of $Q^{\circ}$, cf. [29, Sect.3]. In the notations of
Definition 6.6, we then get for
$E^{\varphi}_{P,IC^{\prime}}(\ell,y):=E^{\varphi}_{P,{IC^{\prime}}^{H}_{X}}(\ell,y)$
the following.

######  Corollary 6.11.

Let $X=X_{P}$ is the projective toric variety with ample Cartier divisor
$D=D_{P}$ associated to a full-dimensional lattice polytope $P\subset
M_{{\mathbb{R}}}\cong{\mathbb{R}}^{n}$, with
$\varphi:M_{\mathbb{R}}\cong{\mathbb{R}}^{n}\to{\mathbb{C}}$ a homogeneous
polynomial function. Assume that $P$ contains the origin in its interior.
Then, for $\ell\in{\mathbb{Z}}_{>0}$,

(6.30) $\begin{split}E^{\varphi}_{P,IC^{\prime}}(\ell,y)&=\sum_{Q\preceq
P}\widetilde{g}_{Q}(-y)\cdot(1+y)^{\dim(Q)+\deg(\varphi)}\cdot\sum_{m\in{\rm
Relint}({{\ell}}Q)\cap M}\varphi(m),\end{split}$

and

(6.31) $E^{\varphi}_{P,IC^{\prime}}(-\ell,y)=\sum_{Q\preceq
P}\widetilde{g}_{Q}(-y)\cdot(-1-y)^{\dim(Q)+\deg(\varphi)}\cdot\sum_{m\in{{\ell}}Q\cap
M}\varphi(m).$

where the summation is over the faces $Q$ of $P$. In particular,

(6.32) $E^{\varphi}_{P,IC^{\prime}}(0,y)=\sum_{Q\preceq
P}\widetilde{g}_{Q}(-y)\cdot(-1-y)^{\dim(Q)+\deg(\varphi)}\cdot\varphi(0).$

Similar formulas hold for $\widetilde{E}^{\varphi}_{P,IC^{\prime}}(\ell,y)$ by
deleting the factor $(1+y)^{\deg(\varphi)}$.

######  Remark 6.12.

  * (a)

Formula (6.30) equals $G_{\varphi}(\ell,y)$ from [3, formula (14)].

  * (b)

If the full-dimensional polytope $P$ is simple, then $X=X_{P}$ is a simplicial
toric variety, hence $[{IC^{\prime}}_{X}^{H}]=[{\mathbb{Q}}^{H}_{X}]\in
K_{0}^{\mathbb{T}}({\rm MHM}_{X}(Z))$. So, in this case,
$\chi_{y}({IC^{\prime}}_{X}^{H}|_{x_{Q}})=1$ (hence $\widetilde{g}_{Q}(-y)=1$,
if $P$ contains the origin in its interior).

∎

We next discuss the effect of duality on generalized weighted Ehrhart
polynomials, proving a generalized reciprocity formula. This takes the form of
a purity statement for pure equivariant Hodge modules (like $IC_{X}^{H}$).

######  Theorem 6.13 (Reciprocity and Purity).

In the above notations, for any Grothendieck class $[{\mathcal{M}}]\in{\rm
Image}(\chi^{\mathbb{T}}_{Hdg})\subset K_{0}({\rm MHM}^{\mathbb{T}}_{X}(Z))$,
we have

(6.33)
$\begin{split}E^{\varphi}_{P,{\mathcal{M}}}(-\ell,y)=(-y)^{\deg(\varphi)}\cdot
E^{\varphi}_{P,{\mathbb{D}}^{\mathbb{T}}_{Z}{\mathcal{M}}}(\ell,\frac{1}{y}),\end{split}$

and

(6.34)
$\widetilde{E}^{\varphi}_{P,{\mathcal{M}}}(-\ell,y)=(-1)^{\deg(\varphi)}\cdot\widetilde{E}^{\varphi}_{P,{\mathbb{D}}^{\mathbb{T}}_{Z}{\mathcal{M}}}(\ell,\frac{1}{y}).$

In particular, if ${\mathcal{M}}$ is such a self-dual pure equivariant Hodge
module of weight $n$ on $X=X_{P}$, then the following purity property holds:

(6.35) $E^{\varphi}_{P,{\mathcal{M}}}(-\ell,y)=(-y)^{n+\deg(\varphi)}\cdot
E^{\varphi}_{P,{\mathcal{M}}}(\ell,\frac{1}{y}),$

and

(6.36)
$\widetilde{E}^{\varphi}_{P,{\mathcal{M}}}(-\ell,y)=(-y)^{n}\cdot(-1)^{\deg(\varphi)}\cdot\widetilde{E}^{\varphi}_{P,{\mathcal{M}}}(\ell,\frac{1}{y}).$

Before proving Theorem 6.13, we need the following.

######  Lemma 6.14.

Under the assumptions of Theorem 6.13 and for $\ell\in{\mathbb{Z}}_{>0}$, we
have:

(6.37) $\chi^{\mathbb{T}}_{y}(X,\ell
D;{\mathbb{D}}^{\mathbb{T}}_{Z}[{\mathcal{M}}])=\sum_{Q\preceq
P}f_{Q}(\frac{1}{y})\cdot(-y)^{-\dim(Q)}\cdot(1+y)^{\dim(Q)}\cdot\sum_{m\in{{\ell}}Q\cap
M}\chi^{-m},$

with $f_{Q}(y)=\chi_{y}(i^{*}_{x_{Q}}[{\mathcal{M}}])$.

###### Proof.

First, by (5.4) and (5.7), we have

(6.38) $\begin{split}\hbox{\rm
DR}^{\mathbb{T}}_{y}([{\mathcal{M}}])&=\sum_{Q\preceq
P}\,\chi_{y}(i^{*}_{x_{Q}}[{\mathcal{M}}])\cdot(1+y)^{\dim(Q)}\cdot(k_{Q})_{*}[\omega_{V_{\sigma_{Q}}}]\\\
&=\sum_{Q\preceq P}\,f_{Q}(y)\cdot\hbox{\rm
DR}^{\mathbb{T}}_{y}\left([(j_{Q})_{!}{\mathbb{Q}}^{H}_{O_{\sigma_{Q}}}]\right),\end{split}$

with $j_{Q}:O_{\sigma_{Q}}\hookrightarrow Z$ and
$k_{Q}:V_{\sigma_{Q}}=X_{Q}\hookrightarrow Z$ the inclusion maps. Moreover, as
in (5.14), we have:

(6.39) $\hbox{\rm
DR}^{\mathbb{T}}_{y}\left({\mathbb{D}}^{\mathbb{T}}_{Z}\left([(j_{Q})_{!}{\mathbb{Q}}^{H}_{O_{\sigma_{Q}}}]\right)\right)=(-y)^{-\dim(Q)}\cdot\hbox{\rm
DR}^{\mathbb{T}}_{y}\left([(j_{Q})_{*}{\mathbb{Q}}^{H}_{O_{\sigma_{Q}}}]\right).$

Since $\hbox{\rm DR}^{\mathbb{T}}_{y}$ commutes with dualities, up to
exchanging $y$ by $1/y$, we then get from (6.38) and (6.39) that

(6.40) $\begin{split}\hbox{\rm
DR}^{\mathbb{T}}_{y}({\mathbb{D}}^{\mathbb{T}}_{Z}[{\mathcal{M}}])&=\sum_{Q\preceq
P}\,f_{Q}(\frac{1}{y})\cdot(-y)^{-\dim(Q)}\cdot\hbox{\rm
DR}^{\mathbb{T}}_{y}\left([(j_{Q})_{*}{\mathbb{Q}}^{H}_{O_{\sigma_{Q}}}]\right)\\\
&\overset{\eqref{drtz}}{=}\sum_{Q\preceq
P}\,f_{Q}(\frac{1}{y})\cdot(-y)^{-\dim(Q)}\cdot(1+y)^{\dim(Q)}\cdot(k_{Q})_{*}[{\mathcal{O}}_{X_{Q}}]_{\mathbb{T}}.\end{split}$

Therefore, with $a$ denoting as before the constant map to a point,

(6.41) $\begin{split}\chi^{\mathbb{T}}_{y}(X,\ell
D;{\mathbb{D}}^{\mathbb{T}}_{Z}[{\mathcal{M}}])&=a_{*}\left(\hbox{\rm
DR}^{\mathbb{T}}_{y}({\mathbb{D}}^{\mathbb{T}}_{Z}[{\mathcal{M}}])\otimes{\mathcal{O}}_{X}(\ell
D)\right)\\\ &=\sum_{Q\preceq
P}\,f_{Q}(\frac{1}{y})\cdot(-y)^{-\dim(Q)}\cdot(1+y)^{\dim(Q)}\cdot
a_{*}\left((k_{Q})_{*}[{\mathcal{O}}_{X_{Q}}]_{\mathbb{T}}\otimes{\mathcal{O}}_{X}(\ell
D)\right)\\\ &\overset{\eqref{lem2b}}{=}\sum_{Q\preceq
P}\,f_{Q}(\frac{1}{y})\cdot(-y)^{-\dim(Q)}\cdot(1+y)^{\dim(Q)}\cdot\sum_{m\in\ell
Q\cap M}\chi^{-m}.\end{split}$

∎

###### Proof of Theorem 6.13.

First, we get from (6.37) that

$E^{\varphi}_{P,{\mathbb{D}}^{\mathbb{T}}_{Z}{\mathcal{M}}}(\ell,{y})=\sum_{Q\preceq
P}\,f_{Q}(\frac{1}{y})\cdot(-y)^{-\dim(Q)}\cdot(1+y)^{\dim(Q)}\cdot(1+y)^{\deg(\varphi)}\cdot\sum_{m\in\ell
Q\cap M}\varphi(m),$

with $f_{Q}(y)=\chi_{y}(i^{*}_{x_{Q}}[{\mathcal{M}}])$ as before. Substituting
$1/y$ in place of $y$, we then get

(6.42)
$\begin{split}E^{\varphi}_{P,{\mathbb{D}}^{\mathbb{T}}_{Z}{\mathcal{M}}}(\ell,\frac{1}{y})&=\sum_{Q\preceq
P}\,f_{Q}({y})\cdot(-y)^{\dim(Q)}\cdot(1+\frac{1}{y})^{\dim(Q)+\deg(\varphi)}\cdot\sum_{m\in\ell
Q\cap M}\varphi(m)\\\ &=\sum_{Q\preceq P}\,f_{Q}({y})\cdot
y^{-\deg(\varphi)}\cdot(-1)^{\dim(Q)}\cdot(1+{y})^{\dim(Q)+\deg(\varphi)}\cdot\sum_{m\in\ell
Q\cap M}\varphi(m)\\\ &=\sum_{Q\preceq P}\,f_{Q}({y})\cdot
y^{-\deg(\varphi)}\cdot(-1)^{\deg(\varphi)}\cdot(-1-{y})^{\dim(Q)+\deg(\varphi)}\cdot\sum_{m\in\ell
Q\cap M}\varphi(m)\\\ &=(-y)^{-\deg(\varphi)}\cdot\sum_{Q\preceq
P}f_{Q}(y)\cdot(-1-y)^{\dim(Q)+\deg(\varphi)}\cdot\sum_{m\in{{\ell}}Q\cap
M}\varphi(m)\\\ &\overset{\eqref{recip}}{=}(-y)^{-\deg(\varphi)}\cdot
E^{\varphi}_{P,f}(-\ell,y).\end{split}$

Formula (6.35) follows immediately by combining (6.33) with (2.3).

The formulas for the polynomials $\widetilde{E}$ are obtained similarly. ∎

######  Example 6.15.

The reciprocity formula (6.33) can be applied to all cases of Example 6.10. In
the (dual) cases $(a)$ and $(c)$, for
${\mathcal{M}}=(j_{Q})_{!}{\mathbb{Q}}^{H}_{O_{\sigma_{Q}}}$ with
${\mathbb{D}}^{\mathbb{T}}_{Z}[{\mathcal{M}}]=[(j_{Q})_{*}{\mathbb{Q}}^{H}_{O_{\sigma_{Q}}}(\dim(Q))]\in
K^{\mathbb{T}}_{0}({\rm MHM}_{X}(Z))$, formula (6.33) specializes to the
classical Ehrhart reciprocity for $Q$ from [8, Prop.4.1]. Indeed,

$E^{\varphi}_{P,{\mathcal{M}}}(\ell,y)=(1+y)^{\dim(Q)+\deg(\varphi)}\cdot\sum_{m\in{\rm
Relint}({{\ell}}Q)\cap M}\varphi(m)$

and

$E^{\varphi}_{P,{\mathbb{D}}^{\mathbb{T}}_{X}{\mathcal{M}}}(\ell,\frac{1}{y})=(1+y)^{\dim(Q)+\deg(\varphi)}\cdot\left((-1)^{\dim(Q)+\deg(\varphi)}\sum_{m\in{{\ell}}Q\cap
M}\varphi(m)\right).$

∎

######  Example 6.16 ($IC$-reciprocity).

Assume $P$ contains the origin in its interior. Then for
${\mathcal{M}}={IC^{\prime}}_{X}^{H}$, corresponding to the weight function
$f_{Q}(y)=\widetilde{g}_{Q}(-y)$, our general purity formula (6.35) recovers
via Corollary 6.11 the reciprocity formula proved [3, Theorems 1.3 and 2.6] by
combinatorial methods.∎

##  7\. Appendix

In this section we give a geometric proof of the following result (without
relying on the combinatorial work of Brion-Vergne [8]).

######  Theorem 7.1.

Let $P\subset M_{R}\cong{\mathbb{R}}^{n}$ be a full-dimensional lattice
polytope. Let $\varphi:M_{\mathbb{R}}\cong{\mathbb{R}}^{n}\to{\mathbb{C}}$ be
a homogeneous polynomial function, and $\ell\in{\mathbb{Z}}_{>0}$. Then for
any Laurent polynomial weight function

$f:\\{\text{faces of }P\\}\to{\mathbb{Z}}[y^{\pm 1}],\ \ \ Q\mapsto f_{Q}(y)$

defined on the set of faces of $P$, the expression

(7.1) $E^{\varphi}_{P,f}(\ell,y):=\sum_{Q\preceq
P}f_{Q}(y)\cdot(1+y)^{\dim(Q)+\deg(\varphi)}\cdot\sum_{m\in{\rm
Relint}({{\ell}}Q)\cap
M}\varphi(m)=(1+y)^{\deg(\varphi)}\cdot\widetilde{E}^{\varphi}_{P,f}(\ell,y)$

is obtained by evaluating a polynomial $E^{\varphi}_{P,f}(z,y)$ at $z=\ell$.

Let $X=X_{P}$ be the toric variety with torus ${\mathbb{T}}$ and ample Cartier
divisor $D=D_{P}$ associated to the lattice polytope $P$. Let $Z$ be an
ambient smooth ${\mathbb{T}}$-variety. In view of Remark 6.7, it suffices to
prove the assertion in the above theorem for
$E^{\varphi}_{P,{\mathcal{M}}}(\ell,y)$, as defined in (6.17), i.e., assuming
that $f_{Q}(y)=\chi_{y}(i_{x_{Q}}^{*}[{\mathcal{M}}])$, for a certain class
$[{\mathcal{M}}]\in K_{0}({\rm MHM}^{\mathbb{T}}_{X}(Z))$ of a mixed Hodge
module (complex) on $Z$ with support in $X=X_{P}$, with $x_{Q}\in
O_{\sigma_{Q}}$ a point in the orbit of $X$ associated to (the cone
$\sigma_{Q}$ in the inner normal fan of $P$ corresponding to) the face $Q$,
and $i_{x_{Q}}:\\{x_{Q}\\}\to Z$ the inclusion map. Our proof in this case is
based on localization techniques developed in [9]. We first introduce some
terminology.

Recall that the (Borel-type) rational equivariant cohomology of $X$ is defined
as

(7.2)
$H^{*}_{\mathbb{T}}(X;\mbox{$\mathbb{Q}$}):=H^{*}(E{\mathbb{T}}\times_{\mathbb{T}}X;\mbox{$\mathbb{Q}$}),$

where ${\mathbb{T}}\hookrightarrow E{\mathbb{T}}\to B{\mathbb{T}}$ is the
universal principal ${\mathbb{T}}$-bundle, i.e.,
$B{\mathbb{T}}=(B\mbox{$\mathbb{C}$}^{*})^{n}=({\mathbb{C}}P^{\infty})^{n}$
and $E{\mathbb{T}}$ is contractible. In particular, if $X=pt$ is a point
space,

(7.3)
$H^{*}_{\mathbb{T}}(pt;{\mathbb{Q}})=H^{*}(B{\mathbb{T}};{\mathbb{Q}})\simeq{\mathbb{Q}}[t_{1},\ldots,t_{n}]=:(\Lambda_{\mathbb{T}})_{{\mathbb{Q}}}.$

Similarly, one can define equivariant homology groups
$H_{*}^{\mathbb{T}}(X;\mbox{$\mathbb{Q}$})$ as in [7, 15], using suitable
finite-dimensional approximations of $B{\mathbb{T}}$. Note that these
equivariant homology groups can be non-zero also in negative degrees. As in
the non-equivariant context, there is a cap product operation

$\cap:H^{*}_{\mathbb{T}}(X;\mbox{$\mathbb{Q}$})\times
H_{*}^{\mathbb{T}}(X;\mbox{$\mathbb{Q}$})\longrightarrow
H_{*}^{\mathbb{T}}(X;\mbox{$\mathbb{Q}$}),$

together with an equivariant Poincaré duality if $X$ is simplicial. In
particular, $H^{*}_{\mathbb{T}}(X;\mbox{$\mathbb{Q}$})$, resp.,
$H_{*}^{\mathbb{T}}(X;\mbox{$\mathbb{Q}$})$, is a
$(\Lambda_{\mathbb{T}})_{{\mathbb{Q}}}$-algebra, resp.,
$(\Lambda_{\mathbb{T}})_{{\mathbb{Q}}}$-module.

For $m\in M$, an element in the character lattice of the torus ${\mathbb{T}}$,
one can view the corresponding character
$\chi^{m}:{\mathbb{T}}\to{\mathbb{C}}^{*}$ as a ${\mathbb{T}}$-equivariant
line bundle ${\mathbb{C}}_{\chi^{m}}$ over a point space $pt$, where the
${\mathbb{T}}$-action on ${\mathbb{C}}$ is induced via $\chi^{m}$. This gives
rise to an abelian group isomorphism

$(M,+)\simeq(Pic_{\mathbb{T}}(pt),\otimes)\>.$

Recall that $m\mapsto-m$ corresponds to the duality involution $(-)^{\vee}$.
Taking the first equivariant Chern class $c^{1}_{\mathbb{T}}$ (or the dual
$-c^{1}_{\mathbb{T}}=c^{1}_{\mathbb{T}}\circ(-)^{\vee}$) of
${\mathbb{C}}_{\chi^{m}}$ gives an isomorphism

(7.4) $c=c^{1}_{\mathbb{T}},\>\text{resp.},\>s=-c^{1}_{\mathbb{T}}:\>M\simeq
H_{\mathbb{T}}^{2}(pt;\mbox{$\mathbb{Z}$})$

and

(7.5) $c,\>\text{resp.},\>s:Sym_{\mbox{$\mathbb{Q}$}}(M)\simeq
H_{\mathbb{T}}^{*}(pt;\mbox{$\mathbb{Q}$})=(\Lambda_{\mathbb{T}})_{\mbox{$\mathbb{Q}$}}\>,$

with
$Sym_{\mbox{$\mathbb{Q}$}}(M)=\bigoplus_{k=0}^{\infty}Sym^{k}_{\mbox{$\mathbb{Q}$}}(M)$
the (rational) symmetric algebra of $M$. So if $m_{i}$ ($i=1,\dots,n$) is a
basis of $M\simeq\mbox{$\mathbb{Z}$}^{n}$, then
$H_{\mathbb{T}}^{*}(pt;\mbox{$\mathbb{Q}$})=(\Lambda_{\mathbb{T}})_{\mbox{$\mathbb{Q}$}}\simeq\mbox{$\mathbb{Q}$}[t_{1},\dots,t_{n}],$
with $t_{i}=\pm c^{1}_{\mathbb{T}}({\mathbb{C}}_{\chi^{m_{i}}})$ for
$i=1,\dots,n$.

Let

$\mbox{$\mathbb{Q}$}\\{t_{1},\dots,t_{n}\\}\simeq(H^{*}_{\mathbb{T}}(pt;\mbox{$\mathbb{Q}$}))^{an}=:(\Lambda^{an}_{\mathbb{T}})_{\mbox{$\mathbb{Q}$}}\subset(\widehat{\Lambda}_{\mathbb{T}})_{\mbox{$\mathbb{Q}$}}\simeq{\mathbb{Q}}[[t_{1},\ldots,t_{n}]]$

be the subring of convergent power series (around zero) with rational
coefficients, i.e., after pairing with $z\in
N_{\mbox{$\mathbb{K}$}}=N\otimes_{\mbox{$\mathbb{Z}$}}\mbox{$\mathbb{K}$}$
(for $\mbox{$\mathbb{K}$}=\mbox{$\mathbb{R}$},\mbox{$\mathbb{C}$}$) one gets a
convergent power series function in $z$ around zero, whose corresponding
Taylor polynomials have rational coefficients. Here, the degree completion
$(\widehat{\Lambda}_{\mathbb{T}})_{\mbox{$\mathbb{Q}$}}$ of
$(\Lambda_{\mathbb{T}})_{\mbox{$\mathbb{Q}$}}$ corresponds to formal power
series with rational coefficients.

Recall that if ${\mathcal{F}}$ is a ${\mathbb{T}}$-equivariant coherent sheaf
on $X$, then the $K$-theoretic Euler characteristic of ${\mathcal{F}}$ is
defined as in (4.3) by:

(7.6) $\begin{split}\chi^{\mathbb{T}}(X,{\mathcal{F}})&=\sum_{m\in
M}\sum_{i=0}^{\dim(X)}(-1)^{i}\dim_{\mathbb{C}}H^{i}(X;{\mathcal{F}})_{\chi^{m}}\cdot\chi^{m}\in{\mathbb{Z}}[M],\end{split}$

By applying the equivariant Chern character ring homomorphism

${\rm ch}^{\mathbb{T}}:{\mathbb{Z}}[M]\simeq
K^{0}_{\mathbb{T}}(pt)\hookrightarrow(\Lambda^{an}_{\mathbb{T}})_{\mbox{$\mathbb{Q}$}}\subset(\widehat{\Lambda}_{\mathbb{T}})_{\mbox{$\mathbb{Q}$}}$

given by $m\mapsto e^{c(m)}$, we further get the cohomological Euler
characteristic of ${\mathcal{F}}$:

(7.7) $\chi^{\mathbb{T}}(X,{\mathcal{F}})=\sum_{m\in
M}\sum_{i=0}^{\dim(X)}(-1)^{i}\dim_{\mathbb{C}}H^{i}(X;{\mathcal{F}})_{\chi^{m}}\cdot
e^{c(m)}\in(\Lambda^{an}_{\mathbb{T}})_{\mbox{$\mathbb{Q}$}}\subset(\widehat{\Lambda}_{\mathbb{T}})_{\mbox{$\mathbb{Q}$}}\>.$

Consider now the fixed-point set $X^{\mathbb{T}}$ of the torus action; this
can be identified with the set of vertices of the lattice polytope $P$. By [7,
Lem.8.4, Lem.8.5], the inclusion $\alpha:X^{\mathbb{T}}\hookrightarrow X$
induces an injective homomorphism of
$H^{*}_{\mathbb{T}}(pt;{\mathbb{Q}})$-modules

$\alpha_{*}:\widehat{H}_{*}^{\mathbb{T}}(X^{\mathbb{T}};{\mathbb{Q}})\hookrightarrow\widehat{H}_{*}^{\mathbb{T}}(X;{\mathbb{Q}})$

which becomes an isomorphism

$\alpha_{*}:\widehat{H}_{*}^{\mathbb{T}}(X^{\mathbb{T}};{\mathbb{Q}})_{L}\simeq\widehat{H}_{*}^{\mathbb{T}}(X;{\mathbb{Q}})_{L}$

upon localization at the multiplicative subset
$L\subset(\Lambda_{\mathbb{T}})_{\mathbb{Q}}=H^{*}_{\mathbb{T}}(pt;{\mathbb{Q}})$
generated by the elements $\pm c(m)$, for $0\neq m\in M$ (cf. [7, Cor.8.9]).
Here, we use the completion

$\widehat{H}_{*}^{\mathbb{T}}(X;{\mathbb{Q}}):=\prod_{i\leq\dim(X)}{H}_{i}^{\mathbb{T}}(X;{\mathbb{Q}}).$

Since

$\widehat{H}_{*}^{\mathbb{T}}(X^{\mathbb{T}};{\mathbb{Q}})=\bigoplus_{x\in
X^{\mathbb{T}}}\widehat{H}_{*}^{\mathbb{T}}(x;{\mathbb{Q}}),$

one gets via the isomorphism $\alpha_{*}$ a projection map of
$H^{*}_{\mathbb{T}}(pt;{\mathbb{Q}})$-modules, called the homological
localization map at $x$,

$pr_{x}:\widehat{H}_{*}^{\mathbb{T}}(X;{\mathbb{Q}})_{L}\simeq\bigoplus_{x\in
X^{\mathbb{T}}}\widehat{H}_{*}^{\mathbb{T}}(x;{\mathbb{Q}})_{L}\longrightarrow\widehat{H}_{*}^{\mathbb{T}}(x;{\mathbb{Q}})_{L}=L^{-1}(\widehat{\Lambda}_{\mathbb{T}})_{\mathbb{Q}}.$

Moreover, as in [9, Sect.4.2], if $i_{x}:\\{x\\}\hookrightarrow X$ is the
inclusion map, then for any
$a\in\widehat{H}^{*}_{\mathbb{T}}(X;{\mathbb{Q}})_{L}$ and
$b\in\widehat{H}_{*}^{\mathbb{T}}(X;{\mathbb{Q}})_{L}$, we have:

(7.8) $pr_{x}(a\cap b)=i_{x}^{*}(a)\cap pr_{x}(b).$

We then have the following result from [9, Sect.4.2], adapted to the notations
of our setting.

######  Proposition 7.2.

Let $X=X_{P}$ be the projective toric variety associated to the full-
dimensional lattice polytope $P\subset M_{\mathbb{R}}\cong{\mathbb{R}}^{n}$,
with ample Cartier divisor $D=D_{P}$. Then for ${\mathcal{F}}$ a
${\mathbb{T}}$-coherent sheaf on $X$, the cohomological Euler characteristic
$\chi^{\mathbb{T}}(X,{\mathcal{O}}_{X}(D)\otimes{\mathcal{F}})$ can be
calculated via localization at the ${\mathbb{T}}$-fixed points as:

(7.9) $\chi^{\mathbb{T}}(X,{\mathcal{O}}_{X}(D)\otimes{\mathcal{F}})=\sum_{v\
{\rm vertex\ of}\ P}pr_{x_{v}}({\rm
td}_{*}^{\mathbb{T}}([{\mathcal{O}}_{X}(D)\otimes{\mathcal{F}}]),$

with each summand

(7.10) $pr_{x_{v}}({\rm
td}_{*}^{\mathbb{T}}([{\mathcal{O}}_{X}(D)\otimes{\mathcal{F}}])\in
L^{-1}(\Lambda^{an}_{\mathbb{T}})_{\mathbb{Q}},$

where $x_{v}$ is the ${\mathbb{T}}$-fixed point of $X$ corresponding to the
vertex $v$ of $P$, and ${\rm
td}_{*}^{\mathbb{T}}:K^{\mathbb{T}}_{0}(X)\longrightarrow\widehat{H}^{\mathbb{T}}_{2*}(X;{\mathbb{Q}})$
is the equivariant Riemann-Roch map of Edidin-Graham [15] (see also [7]).

######  Remark 7.3.

The above homological localization formula is in fact a consequence of a
similar localization formula in equivariant $K$-theory, using the fact that
the equivariant Chern character induces a ring homomorphism of localized rings

${\rm ch}^{\mathbb{T}}:S^{-1}{\mathbb{Z}}[M]\simeq
S^{-1}K^{0}_{\mathbb{T}}(pt)\longrightarrow
L^{-1}(\Lambda^{an}_{\mathbb{T}})_{\mathbb{Q}},$

with $S$ the multiplicative system generated by elements $1-\chi^{m}$ for
$0\neq m\in M$. Note that for a homogeneous polynomial function
$\varphi:M_{\mathbb{R}}\to{\mathbb{C}}$, this property does not hold for the
corresponding group homomorphisms $\widetilde{E}^{\varphi}$ and $E^{\varphi}$
as defined in the Introduction. For this reason, we use exponential sums and
their Taylor expansions (as in [8]), as we shall explain below. ∎

Recall the identification $s:Sym_{\mbox{$\mathbb{Q}$}}(M)\simeq
H_{\mathbb{T}}^{*}(pt;\mbox{$\mathbb{Q}$})=:(\Lambda_{\mathbb{T}})_{\mbox{$\mathbb{Q}$}}$.
Let us fix $z\in
N_{\mathbb{C}}:=N\otimes_{\mathbb{Z}}{\mathbb{C}}={\rm{Hom}}_{\mathbb{R}}(M_{\mathbb{R}},{\mathbb{C}})$.
By the universal property of $Sym$, $z$ induces an ${\mathbb{R}}$-algebra
homomorphism

$\langle-,z\rangle:Sym_{\mbox{$\mathbb{R}$}}(M)\to{\mathbb{C}}\>,$

by which we can view $\langle p,z\rangle$ for $z$ now variable and $p\in
Sym_{\mbox{$\mathbb{R}$}}(M)$, resp.,
$p\in(Sym_{\mbox{$\mathbb{R}$}}(M))^{an}$ fixed, as a ${\mathbb{C}}$-valued
polynomial on $N_{\mathbb{R}}$, resp., as a convergent power series function
around zero in $N_{\mathbb{R}}$.

We can now complete the proof of Theorem 7.1.

###### Proof of Theorem 7.1.

From the geometric viewpoint, it is more natural to prove the assertion for
$\widetilde{E}^{\varphi}_{P,f}(\ell,y)$.

By applying the Chern character ${\rm ch}^{\mathbb{T}}$ to formula (6.14), we
obtain its cohomological counterpart as follows:

(7.11) $\begin{split}\sum_{Q\preceq
P}f_{Q}(y)\cdot(1+y)^{\dim(Q)}\cdot\sum_{m\in{\rm Relint}({{\ell}}Q)\cap
M}e^{s(m)}&=\chi^{\mathbb{T}}_{y}(X,\ell D;[{\mathcal{M}}])\\\
&=\chi^{\mathbb{T}}(X;\hbox{\rm
DR}^{\mathbb{T}}_{y}([{\mathcal{M}}])\otimes{\mathcal{O}}_{X}(\ell D))\\\
&\overset{\eqref{eqec}}{=}\sum_{v\ {\rm vertex}}pr_{x_{v}}({\rm
td}_{*}^{\mathbb{T}}(\hbox{\rm
DR}^{\mathbb{T}}_{y}([{\mathcal{M}}])\otimes{\mathcal{O}}_{X}(\ell D)))\\\
&\overset{(*)}{=}\sum_{v\ {\rm vertex}}pr_{x_{v}}(e^{\ell[D]}\cap{\rm
td}_{*}^{\mathbb{T}}(\hbox{\rm DR}^{\mathbb{T}}_{y}([{\mathcal{M}}])))\\\
&\overset{\eqref{proj}}{=}\sum_{v\ {\rm vertex}}e^{\ell\cdot
i_{{x_{v}}}^{*}[D]}\cap pr_{x_{v}}({\rm td}_{*}^{\mathbb{T}}(\hbox{\rm
DR}^{\mathbb{T}}_{y}([{\mathcal{M}}]))),\end{split}$

where $i_{{x_{v}}}:\\{x_{v}\\}\hookrightarrow X$ is the inclusion of the
${\mathbb{T}}$-fixed point of $X$ corresponding to the vertex $v$ of $P$, with
$f_{Q}(y)=\chi_{y}(i_{x_{Q}}^{*}[{\mathcal{M}}])$, and $(*)$ makes use of the
module property for ${\rm td}_{*}^{\mathbb{T}}$.

Applying $\langle-,z\rangle$ to both sides of (7.11), we obtain

(7.12) $\begin{split}\sum_{Q\preceq
P}f_{Q}(y)\cdot(1+y)^{\dim(Q)}\cdot\sum_{m\in{\rm Relint}({{\ell}}Q)\cap
M}e^{\langle m,z\rangle}&=\sum_{v\ {\rm vertex}}e^{\langle\ell\cdot
i_{{x_{v}}}^{*}[D],z\rangle}\cdot\langle pr_{x_{v}}({\rm
td}_{*}^{\mathbb{T}}(\hbox{\rm
DR}^{\mathbb{T}}_{y}([{\mathcal{M}}]))),z\rangle.\end{split}$

Let us now fix a vertex $v$ of $P$. By (7.10), one also has that

$pr_{x_{v}}({\rm td}_{*}^{\mathbb{T}}(\hbox{\rm
DR}^{\mathbb{T}}_{y}([{\mathcal{M}}])))\in
L^{-1}(\Lambda^{an}_{\mathbb{T}})_{\mathbb{Q}},$

since $e^{\ell\cdot
i_{{x_{v}}}^{*}[D]}\in(\Lambda^{an}_{\mathbb{T}})_{\mathbb{Q}}$ is invertible.
Then $\langle pr_{x_{v}}({\rm td}_{*}^{\mathbb{T}}(\hbox{\rm
DR}^{\mathbb{T}}_{y}([{\mathcal{M}}]))),z\rangle$ is a finite linear
combination of terms of the form $\frac{h(z)}{\prod_{i=1}^{k}\langle
m_{i},z\rangle}$, with $0\neq m_{i}\in M$ and $h$ analytic in $z$ near $0$.

As in [8, Prop.4.1], we next replace $z$ by $tz$ in (7.12), with $0\neq
t\in{\mathbb{R}}$ small and $z$ generic (in the sense that $\langle
m_{i},z\rangle\neq 0$, $i=1,\ldots k$), and consider the expansion into
Laurent series in $t$ on both sides of the equality. We then obtain exactly as
in [8, Prop.4.1] the polynomial behavior in $\ell$ of
$\widetilde{E}^{\varphi}_{P,{\mathcal{M}}}(\ell,y)$ for the polynomial
function $\varphi(m)=\langle m,z\rangle^{r}$, for any given non-negative
integer $r$ and $z$ outside a finite number of hyperplanes. This then implies
the statement for any polynomial $\varphi(m)$. ∎

######  Remark 7.4.

If we want to estimate the degree in $\ell$ of the generalized weighted
Ehrhart polynomial in Theorem 7.1 to be at most $deg(\varphi)+d$ for some
$d\in{\mathbb{Z}}_{>0}$, we need to control uniformly the number $k\leq d$ of
linear factors appearing in the above localization. In fact, we can choose
$d=n=\dim(P)$, as we will now explain. By linearity and using the previous
notations, it suffices to work with
${\mathcal{M}}=(j_{Q})_{!}{\mathbb{Q}}^{H}_{O_{\sigma_{Q}}}$, for all faces
$Q$ of $P$. By using a resolution of singularities of the orbit closure and
functoriality, one can further reduce to the case of a smooth projective toric
variety $X^{\prime}$ of dimension at most $n^{\prime}\leq n$. In this smooth
context, a more precise formula for the homological localization $pr_{x}$ to a
torus fixed point $x$ is available via equivariant Poincaré duality as (e.g.,
see [9, Sect.4.2]):

$pr_{x}=\frac{i_{x}^{*}}{c_{n}^{\mathbb{T}}(N_{x})},$

with $i_{x}:\\{x\\}\hookrightarrow X^{\prime}$ the inclusion map and
$c_{n}^{\mathbb{T}}(N_{x})$ the top equivariant Chern class of the normal
bundle $N_{x}$ of $x$ in $X^{\prime}$. Taking a character decomposition of
$N_{x}=\bigoplus_{i=1}^{n^{\prime}}{\mathbb{C}}_{\chi^{m_{i}}}$, we get
$c_{n}^{\mathbb{T}}(N_{x})=\prod_{i=1}^{n^{\prime}}c(m_{i})$. ∎

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