Regard $X=X\times pt$, with the ${\mathbb{T}}$-action on the left
corresponding to the product action of ${\mathbb{T}}^{\prime}$ on $X$ and
${\mathbb{T}}^{\prime\prime}$ on $pt$ on the right.

These splittings then induce the following factorizations:

  * (a)

(equivariant K-theory)

(109) $\displaystyle K^{0}_{\mathbb{T}}(X)$ $\displaystyle\simeq
K^{0}_{{\mathbb{T}}^{\prime}}(X)\otimes_{\mathbb{Z}}K^{0}_{{\mathbb{T}}^{\prime\prime}}(pt)$
(110) $\displaystyle{\mathbb{Z}}[M]=K^{0}_{\mathbb{T}}(pt)$
$\displaystyle\simeq
K^{0}_{{\mathbb{T}}^{\prime}}(pt)\otimes_{\mathbb{Z}}K^{0}_{{\mathbb{T}}^{\prime\prime}}(pt)={\mathbb{Z}}[M^{\prime}]\otimes_{\mathbb{Z}}{\mathbb{Z}}[M^{\prime\prime}]$
(111) $\displaystyle K_{0}^{\mathbb{T}}(X)$ $\displaystyle\simeq
K_{0}^{{\mathbb{T}}^{\prime}}(X)\otimes_{\mathbb{Z}}K_{0}^{{\mathbb{T}}^{\prime\prime}}(pt),$

with (109) and (110) compatible under pullback, and (109) and (111) compatible
under the module structure induced from the tensor product.

  * (b)

(equivariant (co)homology)

(112) $\displaystyle H^{*}_{\mathbb{T}}(X,{\mathbb{Q}})$ $\displaystyle\simeq
H^{*}_{{\mathbb{T}}^{\prime}}(X,{\mathbb{Q}})\otimes_{\mathbb{Q}}(\Lambda_{{\mathbb{T}}^{\prime\prime}})_{\mathbb{Q}}$
(113) $\displaystyle
Sym_{\mathbb{Q}}(M)=(\Lambda_{{\mathbb{T}}})_{\mathbb{Q}}$
$\displaystyle\simeq(\Lambda_{{\mathbb{T}}^{\prime}})_{\mathbb{Q}}\otimes_{\mathbb{Q}}(\Lambda_{{\mathbb{T}}^{\prime\prime}})_{\mathbb{Q}}=Sym_{\mathbb{Q}}(M^{\prime})\otimes_{\mathbb{Q}}Sym_{\mathbb{Q}}(M^{\prime\prime})$
(114) $\displaystyle H_{*}^{\mathbb{T}}(X,{\mathbb{Q}})$ $\displaystyle\simeq
H_{*}^{{\mathbb{T}}^{\prime}}(X,{\mathbb{Q}})\otimes_{\mathbb{Q}}(\Lambda_{{\mathbb{T}}^{\prime\prime}})_{\mathbb{Q}},$

with (112) and (113) compatible under pullback, and (112) and (114) compatible
under the module structure induced from the cap product. Similar statements
hold for the corresponding completions, as well as for equivariant Chow groups
and their completions.

The above factorizations are compatible with the corresponding equivariant
Chern character and Todd class transformations.

Tensoring with the distinguished elements ${\mathbb{C}}_{\chi^{0}}\in
K^{0}_{{\mathbb{T}}^{\prime\prime}}(pt)$ and
$1\in(\Lambda_{{\mathbb{T}}^{\prime\prime}})_{\mathbb{Q}}$ induces inclusions
of the ${\mathbb{T}}^{\prime}$-equivariant groups on $X$ (or pt) into the
corresponding ${\mathbb{T}}$-equivariant groups on $X$ (or pt). Moreover,
these inclusions are independent of the choice of the compatible splittings of
${\mathbb{T}}$ and $M$, respectively.

###### Proof.

The factorizations of (a) and (b) of (i) follow from suitable Künneth
formulae.

Statement (ii) follows by construction from the multiplicativity under cross-
products of the equivariant Chern character and equivariant Todd class
transformation, see [23].

It remains to justify the independence statement of part (iii). Two choices of
splittings of ${\mathbb{T}}$ differ by a group homomorphism
$h:{\mathbb{T}}^{\prime}\to{\mathbb{T}}^{\prime\prime}$ acting on product
classes
$[-]_{{\mathbb{T}}^{\prime}}\otimes{\mathbb{C}}_{\chi^{m^{\prime\prime}}}$ of
type (a) with $m^{\prime\prime}\in M^{\prime\prime}$ via (compare
[35][Prop.2.14]):

$[-]_{{\mathbb{T}}^{\prime}}\otimes{\mathbb{C}}_{\chi^{m^{\prime\prime}}}\mapsto\left([-]_{{\mathbb{T}}^{\prime}}\cdot{\mathbb{C}}_{\chi^{m^{\prime\prime}\circ
h}}\right)\otimes{\mathbb{C}}_{\chi^{m^{\prime\prime}}}.$

So for $m^{\prime\prime}=0$, such an $h$ acts trivially. Similarly, by
choosing a basis $m_{1}^{\prime\prime},\ldots,m_{r}^{\prime\prime}$ of
$M^{\prime\prime}$, one gets a basis
$\prod_{i=1}^{r}c(m^{\prime\prime}_{i})^{k_{i}}$, $k_{i}\geq 0$, of the
${\mathbb{Q}}$-vector space
$(\Lambda_{{\mathbb{T}}^{\prime\prime}})_{\mathbb{Q}}$. Then $h$ acts on
product classes
$[-]_{{\mathbb{T}}^{\prime}}\otimes\prod_{i=1}^{r}c(m^{\prime\prime}_{i})^{k_{i}}$
of type (b) via

$[-]_{{\mathbb{T}}^{\prime}}\otimes\prod_{i=1}^{r}c(m^{\prime\prime}_{i})^{k_{i}}\mapsto\left([-]_{{\mathbb{T}}^{\prime}}\cdot\prod_{i=1}^{r}c(m^{\prime\prime}_{i}\circ
h)^{k_{i}}\right)\otimes\prod_{i=1}^{r}c(m^{\prime\prime}_{i})^{k_{i}}.$

So on product classes of type $[-]_{{\mathbb{T}}^{\prime}}\otimes 1$ (i.e.,
with all $k_{i}$’s being $0$) it acts trivially. ∎

######  Remark 3.12.

Results of Proposition 3.11 dealing only with rational equivariant
(co)homology hold with the same proof under the weaker assumption of a
splitting of rational vector spaces
$M_{\mathbb{Q}}=M^{\prime}_{\mathbb{Q}}\oplus M^{\prime\prime}_{\mathbb{Q}}$.
Here $M_{\mathbb{Q}}:=M\otimes_{\mathbb{Z}}{\mathbb{Q}}$, and similarly for
$M^{\prime}_{\mathbb{Q}}$, $M^{\prime\prime}_{\mathbb{Q}}$.

As a consequence, we have the following.

######  Corollary 3.13.

In the setup of Proposition 3.11, let ${\mathcal{E}},{\mathcal{F}}$ be a
${\mathbb{T}}^{\prime}$-equivariant vector bundle, resp., coherent sheaf on
$X$, with induced ${\mathbb{T}}$-action via the projection
${\mathbb{T}}\to{\mathbb{T}}^{\prime}$. Then

(115) ${\rm ch}^{\mathbb{T}}([{\mathcal{E}}]_{\mathbb{T}})={\rm
ch}^{{\mathbb{T}}^{\prime}}([{\mathcal{E}}]_{{\mathbb{T}}^{\prime}})\otimes 1\
\ \text{\rm and}\ \ \ {\rm
td}_{*}^{\mathbb{T}}([{\mathcal{F}}]_{\mathbb{T}})={\rm
td}_{*}^{{\mathbb{T}}^{\prime}}([{\mathcal{F}}]_{{\mathbb{T}}^{\prime}})\otimes
1.$

###### Proof.

Choose a splitting
${\mathbb{T}}={\mathbb{T}}^{\prime}\oplus{\mathbb{T}}^{\prime\prime}$. Then,
by part (ii) of Proposition 3.11, we get:

$\begin{split}{\rm td}_{*}^{\mathbb{T}}([{\mathcal{F}}]_{\mathbb{T}})&={\rm
td}_{*}^{{\mathbb{T}}^{\prime}}([{\mathcal{F}}]_{{\mathbb{T}}^{\prime}})\otimes{\rm
td}_{*}^{{\mathbb{T}}^{\prime\prime}}({\mathbb{C}}_{\chi^{0}})\\\ &={\rm
td}_{*}^{{\mathbb{T}}^{\prime}}([{\mathcal{F}}]_{{\mathbb{T}}^{\prime}})\otimes
1,\end{split}$

and similarly for the equivariant Chern character. ∎

In particular, in the notations of Proposition 3.8, a choice of splitting of
the surjection of tori ${\mathbb{T}}\to{\mathbb{T}}^{\prime}:=T_{N(\sigma)}$
yields an identification

(116) ${\rm td}_{*}^{\mathbb{T}}([\omega_{V_{\sigma}}]_{\mathbb{T}})={\rm
td}_{*}^{{\mathbb{T}}^{\prime}}([\omega_{V_{\sigma}}]_{{\mathbb{T}}^{\prime}}),$

which is independent of the splitting.

###  3.2. Generalized equivariant Hirzebruch-Riemann-Roch

Let $X_{\Sigma}$ be a projective toric variety defined by a fan $\Sigma$, with
$X:=X_{\Sigma^{\prime}}\subset X_{\Sigma}$ a ${\mathbb{T}}$-invariant closed
algebraic subset of $X_{\Sigma}$ defined by a star-closed subset
$\Sigma^{\prime}\subset\Sigma$. Let $D$ be a ${\mathbb{T}}$-invariant Cartier
divisor on $X_{\Sigma}$. The equivariant Hirzebruch polynomial of $D|_{X}$ is
defined by the formula:

(117)
$\chi^{\mathbb{T}}_{y}(X,{\mathcal{O}}_{X_{\Sigma}}(D)|_{X}):=\sum_{p=0}^{\dim(X)}\chi^{\mathbb{T}}(X,\widetilde{\Omega}^{p}_{X}\otimes{\mathcal{O}}_{X_{\Sigma}}(D)|_{X})\cdot
y^{p}\in(\Lambda^{an}_{\mathbb{T}})_{\mbox{$\mathbb{Q}$}}[y]\subset(\widehat{\Lambda}_{\mathbb{T}})_{\mbox{$\mathbb{Q}$}}[y]\>,$

with $\widetilde{\Omega}^{p}_{X}$ denoting as in Remark 3.6 the sheaf of
Ishida $p$-forms on $X$.

Then we have the following result:

######  Theorem 3.14.

(generalized equivariant Hirzebruch-Riemann-Roch)
In the above setup, the equivariant Hirzebruch polynomial of $D|_{X}$ is
computed by the formula:

(118)
$\begin{split}\chi^{\mathbb{T}}_{y}(X,{\mathcal{O}}_{X_{\Sigma}}(D)|_{X})&=\int_{X}{\rm
ch}^{\mathbb{T}}({\mathcal{O}}_{X_{\Sigma}}(D)|_{X})\cap
T^{\mathbb{T}}_{y*}(X).\end{split}$

###### Proof.

This follows as in [36][Theorem 2.4], by making use of Proposition 3.5, Remark
3.6 and the module property (81) of ${\rm td}_{*}^{\mathbb{T}}$. ∎

As a consequence, we obtain the following weighted version of formula (75):

######  Corollary 3.15.

Let $P$ be a full-dimensional lattice polytope with associated projective
toric variety $X_{P}$ and ample Cartier divisor $D=D_{P}$. Let
$X:=X_{P^{\prime}}$ be the ${\mathbb{T}}$-invariant closed algebraic subset of
$X_{P}$ corresponding to a polytopal subcomplex $P^{\prime}\subset P$ (i.e., a
closed union of faces of $P$). Then:

(119) $\chi^{\mathbb{T}}_{y}(X,{\mathcal{O}}_{X_{P}}(D)|_{X})=\sum_{E\preceq
P^{\prime}}(1+y)^{\dim(E)}\cdot\sum_{m\in{\rm Relint}(E)\cap M}e^{s(m)}.$

###### Proof.

For a face $E$ of $P^{\prime}$, denote by
$i_{E}:V_{\sigma_{E}}:=X_{E}\hookrightarrow X$ the inclusion of the orbit
closure associated to the (cone of the) face $E$. Note that we have
$\dim(E)=\dim(O_{\sigma_{E}}).$ Let ${\mathbb{T}}^{\prime}:=T_{N(\sigma_{E})}$
be the quotient torus of ${\mathbb{T}}$ corresponding to $X_{E}$.

Then, by Theorem 3.14, Proposition 3.7 and Proposition 3.8, the following
equality holds:

$\begin{split}\chi^{\mathbb{T}}_{y}(X,{\mathcal{O}}_{X_{P}}(D)|_{X})&=\int_{X}{\rm
ch}^{\mathbb{T}}({\mathcal{O}}_{X_{P}}(D)|_{X})\cap T^{\mathbb{T}}_{y*}(X)\\\
&=\sum_{E\preceq P^{\prime}}(1+y)^{\dim(E)}\int_{X}{\rm
ch}^{\mathbb{T}}({\mathcal{O}}_{X_{P}}(D)|_{X})\cap(i_{E})_{*}{\rm
td}^{\mathbb{T}}_{*}([\omega_{X_{E}}]_{\mathbb{T}}).\end{split}$

It remains to prove that for any face $E$ of $P^{\prime}$, we have that:

(120) $\int_{X}{\rm
ch}^{\mathbb{T}}({\mathcal{O}}_{X_{P}}(D)|_{X})\cap(i_{E})_{*}{\rm
td}^{\mathbb{T}}_{*}([\omega_{X_{E}}]_{\mathbb{T}})=\sum_{m\in{\rm
Relint}(E)\cap M}e^{s(m)}.$

This follows from the functorial properties of the cap product and formula
(43). Indeed,

$\begin{split}\int_{X}{\rm
ch}^{\mathbb{T}}({\mathcal{O}}_{X_{P}}(D)|_{X})\cap(i_{E})_{*}{\rm
td}^{\mathbb{T}}_{*}([\omega_{X_{E}}]_{\mathbb{T}})&=\int_{X_{E}}(i_{E})^{*}{\rm
ch}^{\mathbb{T}}({\mathcal{O}}_{X_{P}}(D)|_{X})\cap{\rm
td}^{\mathbb{T}}_{*}([\omega_{X_{E}}]_{\mathbb{T}})\\\ &=\int_{X_{E}}{\rm
ch}^{\mathbb{T}}((i_{E})^{*}({\mathcal{O}}_{X_{P}}(D)|_{X}))\cap{\rm
td}^{\mathbb{T}}_{*}([\omega_{X_{E}}]_{\mathbb{T}})\\\
&\overset{(\ref{tran})}{=}\int_{X_{E}}{\rm
ch}^{\mathbb{T}}({\mathcal{O}}_{X_{E}}(D_{E_{0}}-div(\chi^{m_{0}})))\cap{\rm
td}^{\mathbb{T}}_{*}([\omega_{X_{E}}])\\\ &=e^{-c(m_{0})}\int_{X_{E}}{\rm
ch}^{\mathbb{T}}({\mathcal{O}}_{X_{E}}(D_{E_{0}}))\cap{\rm
td}^{\mathbb{T}}_{*}([\omega_{X_{E}}]_{\mathbb{T}})\\\
&\overset{({*})}{=}e^{-c(m_{0})}\int_{X_{E}}{\rm
ch}^{{\mathbb{T}}^{\prime}}({\mathcal{O}}_{X_{E}}(D_{E_{0}}))\cap{\rm
td}^{{\mathbb{T}}^{\prime}}_{*}([\omega_{X_{E}}]_{{\mathbb{T}}^{\prime}})\\\
&\overset{(\ref{lpc2})}{=}e^{s(m_{0})}\cdot\left(\sum_{m\in{\rm
Relint}(E-m_{0})\cap M}e^{s(m)}\right)\\\ &=\sum_{m\in{\rm Relint}(E)\cap
M}e^{s(m)},\end{split}$

where (*) uses Proposition 3.11 and Corollary 3.13. Here $m_{0}$ is a vertex
of $E$ as in (43), so that $E_{0}:=E-m_{0}$ is a full-dimensional lattice
polytope in $Span(E_{0})$ relative to the lattice $Span(E_{0})\cap M$, with
$X_{E}$ the associated toric variety. ∎

######  Remark 3.16.

For future use, we include here the following formula, which one gets as in
the proof of (120), but using ${\mathcal{O}}_{X_{E}}$ instead of
$\omega_{X_{E}}$:

(121) $\int_{X}{\rm ch}^{\mathbb{T}}({\mathcal{O}}_{X_{P}}(D)|_{X})\cap{\rm
td}^{\mathbb{T}}_{*}((i_{E})_{*}[{\mathcal{O}}_{X_{E}}]_{\mathbb{T}})=\sum_{m\in
E\cap M}e^{s(m)}.$

This is also a special case of Corollary 3.15 for $y=0$ and $P^{\prime}=E$.

As another application, we employ Theorem 3.14 in the context of a globally
generated ${\mathbb{T}}$-invariant Cartier divisor $D$ on a projective toric
variety $X$ with associated torus ${\mathbb{T}}$. Let $P_{D}\subset
M_{\mathbb{R}}$ be the lattice polytope corresponding to $D$, and let $X_{D}$
be the toric variety of the lattice polytope $P_{D}$, defined via the
corresponding generalized fan as in [16][Prop.6.2.3]. By [16][Thm.6.2.8],
there is a proper toric morphism $f:X\to X_{D}$, induced by the corresponding
lattice projection $N\to N_{D}$ given by dividing out by the minimal cone of
the generalized fan of $P_{D}$. In particular, $f:X\to X_{D}$ is a toric
fibration. Let $M_{D}\hookrightarrow M$ be the associated inclusion of dual
lattices. Choosing a vertex $m_{0}$ of $P_{D}$, we get that
$P^{\prime}:=P_{D}-m_{0}\subset M_{D}$ is a full-dimensional lattice polytope.
Let $D^{\prime}$ be the ample divisor on $X_{D}$ associated to $P^{\prime}$,
with $\Sigma^{\prime}$ the inner normal fan of $P^{\prime}$ (defining
$X_{D}$), so that there is a one-to-one correspondence between cones
$\sigma^{\prime}\in\Sigma^{\prime}$ and faces $E^{\prime}$ of the lattice
polytope $P^{\prime}$, and by translation to the faces $E$ of $P_{D}$. Then,
by the proof of [16][Thm.6.2.8], one gets that
${\mathcal{O}}(D-div(\chi^{m_{0}}))\simeq f^{*}{\mathcal{O}}(D^{\prime})$, as
${\mathbb{T}}$-equivariant sheaves.

Following the notations of Proposition 3.9, let $U\subset X$ be a closed
${\mathbb{T}}$-invariant subset with associated multiplicities
$d_{\ell}(U/\sigma^{\prime})$, denoted here via the above correspondence as
$d_{\ell}(U/E)$. We can now prove the following.

######  Corollary 3.17.

With the above notations,

(122) $\chi^{\mathbb{T}}_{y}(U,{\mathcal{O}}_{X}(D)|_{U})=\sum_{E\preceq
P_{D}}\left(\sum_{\ell\geq 0}(-1)^{\ell}\cdot
d_{\ell}(U/E)\cdot(1+y)^{\ell+\dim(E)}\right)\cdot\sum_{m\in{\rm
Relint}(E)\cap M}e^{s(m)}.$

###### Proof.

By the generalized Hirzebruch-Riemann-Roch Theorem 3.14, the projection
formula, and functoriality of Hirzebruch classes, we have

$\begin{split}\chi^{\mathbb{T}}_{y}(U,{\mathcal{O}}_{X}(D)|_{U})&=\int_{U}{\rm
ch}^{\mathbb{T}}({\mathcal{O}}_{X}(D)|_{U})\cap T^{\mathbb{T}}_{y*}(U)\\\
&=\int_{X}{\rm ch}^{\mathbb{T}}({\mathcal{O}}_{X}(D))\cap
T^{\mathbb{T}}_{y*}([U\hookrightarrow X])\\\ &=e^{s(m_{0})}\int_{X}{\rm
ch}^{\mathbb{T}}(f^{*}{\mathcal{O}}_{X}(D^{\prime}))\cap
T^{\mathbb{T}}_{y*}([U\hookrightarrow X])\\\ &=e^{s(m_{0})}\int_{X_{D}}{\rm
ch}^{\mathbb{T}}({\mathcal{O}}_{X}(D^{\prime}))\cap
f_{*}T^{\mathbb{T}}_{y*}([U\hookrightarrow X])\\\
&\overset{(\ast)}{=}e^{s(m_{0})}\left(\sum_{E^{\prime}\preceq
P^{\prime}}\sum_{\ell\geq 0}(-1)^{\ell}\cdot
d_{\ell}(U/E^{\prime})\cdot(1+y)^{\ell+\dim(E^{\prime})}\cdot\sum_{m\in{\rm
Relint}(E^{\prime})\cap M}e^{s(m)}\right)\\\ &=\sum_{E\preceq
P_{D}}\sum_{\ell\geq 0}(-1)^{\ell}\cdot
d_{\ell}(U/E)\cdot(1+y)^{\ell+\dim(E)}\cdot\sum_{m\in{\rm Relint}(E)\cap
M}e^{s(m)},\end{split}$

where the equality $({\ast})$ follows by using (108) and (120). ∎

######  Example 3.18 (Rigidity of the equivariant $\chi_{y}$-genus).

If, in Corollary 3.17, we take $D=0$, then $P_{D}=\\{0\\}\subset M\subset
M_{\mathbb{R}}$, so $d_{\ell}(U/\\{0\\})$ is just the number of
$\ell$-dimensional torus orbits in $U$. Hence, formula (122) becomes in this
case

(123) $\chi^{\mathbb{T}}_{y}(U,{\mathcal{O}}_{U})=\left(\sum_{\ell\geq
0}(-1)^{\ell}\cdot d_{\ell}(U/\\{0\\})\cdot(1+y)^{\ell}\right)\cdot e^{s(0)}.$

Note that this formula does not take into account any equivariant information.
So, forgetting the ${\mathbb{T}}$-action,

$\left(\sum_{\ell\geq 0}(-1)^{\ell}\cdot
d_{\ell}(U/\\{0\\})\cdot(1+y)^{\ell}\right)=\chi_{y}(U,{\mathcal{O}}_{U}).$

In particular, for $U=X$ and $y=0$, this becomes
$\chi(X,{\mathcal{O}}_{X})=1$. ∎

######  Remark 3.19.

Note that if $D$ is an ample ${\mathbb{T}}$-invariant Cartier divisor, then
the above morphism $f$ is the identity of $X$, so formula (122) reduces to
(119) since the multiplicities $d_{\ell}(U/E)$ are given as follows:
$d_{\ell}(U/E)=0$ for $\ell>0$, and $d_{0}(U/E)$ is either $1$ or $0$,
depending wether the orbit associated to $E$ is included or not in $U$.

######  Remark 3.20.

By forgetting the ${\mathbb{T}}$-action in formula (122), we get the following
weighted lattice point counting for lattice polytopes associated to globally
generated ${\mathbb{T}}$-invariant Cartier divisors. More precisely, in the
above notations, we get the following generalization of Theorem 2.3 and of
[36][Cor.1.8].

(124) $\chi_{y}(U,{\mathcal{O}}_{X}(D)|_{U})=\sum_{E\preceq
P_{D}}\left(\sum_{\ell\geq 0}(-1)^{\ell}\cdot
d_{\ell}(U/E)\cdot(1+y)^{\ell+\dim(E)}\right)\cdot|{\rm Relint}(E)\cap M|.$

Here, it is enough to assume that $X$ is complete, since the non-equivariant
Hirzebruch class transformation is defined in this generality.

###  3.3. Equivariant Hirzebruch and Todd characteristic classes of simplicial
toric varieties

The previously mentioned formulae of Section 2.4 for the Todd class and
Hirzebruch classes of quasi-projective simplicial toric varieties hold with
the same proofs (up to some small modifications, as explained below) for the
equivariant versions of these characteristic classes, compare also with
[24][Rem.4.3]. A different proof of all these results, independent of the
equivariant version of the Lefschetz-Riemann-Roch from [24][Thm.3.1 and
Rem.4.3], will be given in Section 4.3 via localization at the torus fixed
points. Nevertheless, the approach of this section is useful to derive global
expressions for the equivariant characteristic classes of interest, which will
then be localized.

Let $X:=X_{\Sigma}$ be a simplicial quasi-projective toric variety associated
to a fan $\Sigma$ of full-dimensional convex support, e.g., $X=X_{P}$ is the
semi-projective simplicial toric variety corresponding to a simple full-
dimensional lattice polyhedron $P\subset M_{\mathbb{R}}$. This includes the
cases of full-dimensional simple lattice polytopes, as well as full-
dimensional rational pointed polyhedral cones. The assumption that $\Sigma$ is
of full-dimensional convex support implies that $X$ contains no torus factors,
so we can use the Cox construction as described in Section 2.1.2. Moreover,
this allows us to make use of Remark 2.8.

In addition, we also prove formulae for the equivariant Hirzebruch classes
$T_{y*}^{\mathbb{T}}([U\hookrightarrow X])$, with $U$ the open complement of a
${\mathbb{T}}$-invariant divisor $D_{K}:=\bigcup_{\rho\in K}D_{\rho}$, for
$K\subset\Sigma(1)$. Similarly, we indicate here the argument even for the
Hirzebruch classes of an orbit closure $V_{\tau}$, for $\tau\in\Sigma$,
pointing out the needed modifications in the proof.

The arguments in this section are based on the Cox construction. With the
notations from Subsection 2.1.2, let $\pi:W:={\mathbb{C}}^{r}\setminus
Z(\Sigma)\to X$ be the toric morphism, with $G$ the kernel of the
corresponding map of tori
$\gamma:\widetilde{{\mathbb{T}}}:=({\mathbb{C}}^{*})^{r}\to{\mathbb{T}}$ (here
$r=|\Sigma(1)|$). Since $X$ is a simplicial toric variety containing no torus
factor, it follows that $X$ is the geometric quotient $W/G$, with $G$ acting
with finite stabilizers. If $\tau$ is a cone of $\Sigma$, then the orbit
closure $V_{\tau}$ is the image under $\pi$ of a linear subspace
$W_{\tau}\subset W$. We have the following formula (in which we omit the
symbols for pushforwards under closed embeddings).

######  Lemma 3.21.

(125) ${\rm
td}_{*}^{\mathbb{T}}((\pi_{*}\Lambda_{y}T^{*}_{W_{\tau}})^{G})=(1+y)^{r-n}\cdot{\rm
td}_{*}^{\mathbb{T}}(mC_{y}^{\mathbb{T}}(V_{\tau}))\in\widehat{H}_{*}^{\mathbb{T}}(X;{\mathbb{Q}})[y]\simeq\widehat{H}^{*}_{\mathbb{T}}(X;{\mathbb{Q}})[y].$

###### Proof.

By (65), it suffices to check formula (125) after restriction to each
$U_{\sigma}$, with $U_{\sigma}$ the ${\mathbb{T}}$-invariant open affine
subset of $X$ containing the corresponding ${\mathbb{T}}$-fixed point
$x_{\sigma}$, for $\sigma\in\Sigma(n)$. We have
$\pi^{-1}(U_{\sigma})\simeq{\mathbb{C}}^{n}\times({\mathbb{C}}^{*})^{r-n}$,
with
$\widetilde{{\mathbb{T}}}\simeq{\mathbb{T}}\times({\mathbb{C}}^{*})^{r-n}=:{\mathbb{T}}\times{\mathbb{T}}^{\prime}$
acting on the respective factors, and the factor ${\mathbb{C}}^{n}$
corresponding to the rays of $\sigma$ and
${\mathbb{T}}^{\prime}=({\mathbb{C}}^{*})^{r-n}$ acting freely by
multiplication on itself. Similarly, $G\simeq
G_{\sigma}\times{\mathbb{T}}^{\prime}$, with $G_{\sigma}\subset{\mathbb{T}}$ a
finite subgroup. So, above $U_{\sigma}$, $\pi$ can be factorized as a
composition of the free quotient
$\widetilde{\pi}:{\mathbb{C}}^{n}\times({\mathbb{C}}^{*})^{r-n}\to{\mathbb{C}}^{n}$
by the ${\mathbb{T}}^{\prime}$-action, followed by a finite quotient map
$\pi=\pi_{\sigma}:{\mathbb{C}}^{n}\to{\mathbb{C}}^{n}/{G_{\sigma}}=U_{\sigma}$.
Similarly, $\pi^{-1}(U_{\sigma}\cap V_{\tau})\simeq
L_{\tau}\times({\mathbb{C}}^{*})^{r-n}$, with
$L_{\tau}\subset{\mathbb{C}}^{n}$ a linear subspace. Then

$\big{(}\widetilde{\pi}_{*}\Lambda_{y}T^{*}_{L_{\tau}\times({\mathbb{C}}^{*})^{r-n}}\big{)}^{{\mathbb{T}}^{\prime}}=\Lambda_{y}T^{*}_{L_{\tau}}\cdot(1+y)^{r-n},$

since $T^{*}_{({\mathbb{C}}^{*})^{r-n}}\simeq({\mathbb{C}}^{*})^{r-n}\times
T^{*}_{id}$, with ${\mathbb{T}}^{\prime}$ acting by the co-adjoint action on
the cotangent space $T^{*}_{id}$ of ${\mathbb{T}}^{\prime}$ at the identity
element $id\in{\mathbb{T}}^{\prime}$. This is the trivial
${\mathbb{T}}^{\prime}$-bundle of rank $r-n$, since ${\mathbb{T}}^{\prime}$ is
an abelian group, so that
$\Lambda_{y}T^{*}_{({\mathbb{C}}^{*})^{r-n}}=(1+y)^{r-n}$. Finally, in
$K_{0}^{\mathbb{T}}(U_{\sigma})[y]$, we have by Proposition 3.5 and Corollary
3.13 that

$((\pi_{\sigma})_{*}\Lambda_{y}T^{*}_{L_{\tau}})^{G_{\sigma}}=mC^{\mathbb{T}}_{y}(V_{\tau}\cap
U_{\sigma}).$

The desired formula (125) follows by applying the equivariant Todd
transformation. ∎

By the equivariant version of the Lefschetz-Riemann-Roch theorem (see
[24][Thm.3.1 and Rem.4.3]), applied to the left-hand side of formula (125),
the proof of the formulae of Section 2.4 for the Todd class and Hirzebruch
classes of projective simplicial toric varieties hold with the same proofs in
the equivariant setting. For later applications, we work with the
cohomological images ${\rm
td}_{*}^{\mathbb{T}}(X)\in\widehat{H}^{*}_{\mathbb{T}}(X;{\mathbb{Q}})$ and
${T}^{\mathbb{T}}_{y*}(X)\in\widehat{H}^{*}_{\mathbb{T}}(X;{\mathbb{Q}})[y]$,
resp.,
$\widehat{T}^{\mathbb{T}}_{y*}(X)\in\widehat{H}^{*}_{\mathbb{T}}(X;{\mathbb{Q}}[y]):=\prod_{i\geq
0}H^{i}_{\mathbb{T}}(X;{\mathbb{Q}}[y])$ of these classes under equivariant
Poincaré duality. In the notations of Section 2.1.2, one has the following:

######  Theorem 3.22.

Let $X:=X_{\Sigma}$ be a simplicial quasi-projective toric variety associated
to a fan $\Sigma$ of full-dimensional convex support. Then the equivariant
Hirzebruch classes ${T}^{\mathbb{T}}_{y*}(X)$ and
$\widehat{T}^{\mathbb{T}}_{y*}(X)$ are computed by:

(126) ${T}^{\mathbb{T}}_{y*}(X)=(1+y)^{n-r}\cdot\sum_{g\in
G_{\Sigma}}\prod_{\rho\in\Sigma(1)}\frac{F_{\rho}\cdot\big{(}1+y\cdot
a_{\rho}(g)\cdot e^{-F_{\rho}}\big{)}}{1-a_{\rho}(g)\cdot
e^{-F_{\rho}}}\in\widehat{H}^{*}_{\mathbb{T}}(X;{\mathbb{Q}})[y]\>,$

(127) $\widehat{T}^{\mathbb{T}}_{y*}(X)=\sum_{g\in
G_{\Sigma}}\prod_{\rho\in\Sigma(1)}\frac{F_{\rho}\cdot\big{(}1+y\cdot
a_{\rho}(g)\cdot e^{-F_{\rho}(1+y)}\big{)}}{1-a_{\rho}(g)\cdot
e^{-F_{\rho}(1+y)}}\in\widehat{H}^{*}_{\mathbb{T}}(X;{\mathbb{Q}}[y])\>,$

with $r=|\Sigma(1)|$, and $F_{\rho}=[D_{\rho}]_{\mathbb{T}}$ denoting the
equivariant fundamental class of the ${\mathbb{T}}$-invariant divisor
$D_{\rho}$ corresponding to the ray $\rho\in\Sigma(1)$.

######  Corollary 3.23.

If $X:=X_{\Sigma}$ is a smooth quasi-projective toric variety associated to a
fan $\Sigma$ of full-dimensional convex support, then $G_{\Sigma}$ is the
trivial group. Hence, the equivariant Hirzebruch classes of $X$ are given by:

(128)
${T}^{\mathbb{T}}_{y*}(X)=(1+y)^{n-r}\cdot\prod_{\rho\in\Sigma(1)}\frac{F_{\rho}\cdot\big{(}1+y\cdot
e^{-F_{\rho}}\big{)}}{1-e^{-F_{\rho}}}$ (129)
$\widehat{T}^{\mathbb{T}}_{y*}(X)=\prod_{\rho\in\Sigma(1)}\frac{F_{\rho}\cdot\big{(}1+y\cdot
e^{-F_{\rho}(1+y)}\big{)}}{1-e^{-F_{\rho}(1+y)}}\>.$

######  Remark 3.24.

In particular, by setting $y=0$ in Theorem 3.22, we recover the equivariant
Todd class formula of [13] given by the following expression in
$\widehat{H}^{*}_{\mathbb{T}}(X;{\mathbb{Q}})$:

(130) $\begin{split}{\rm td}_{*}^{\mathbb{T}}(X)&=\sum_{g\in
G_{\Sigma}}\prod_{\rho\in\Sigma(1)}\frac{F_{\rho}}{1-a_{\rho}(g)\cdot
e^{-F_{\rho}}}.\end{split}$

Similarly, setting $y=0$ in Corollary 3.23, one gets the equivariant Todd
class for such a smooth quasi-projective toric variety. There is also an
equivariant version of formula (56). Finally, if $X$ is projective, the
specialization at $y=1$ fits with suitable $L$-classes, i.e.,
$\widehat{T}_{1*}(X)=L(X)$ is the Thom-Milnor $L$-class of a projective toric
variety $X$.

Moreover, taking the top degree of $y$ in (126) and using (96), yields the
following.

######  Corollary 3.25.

Under the assumptions and notations of Theorem 3.22, we have:

(131) ${\rm td}_{*}^{\mathbb{T}}([\omega_{X}]_{\mathbb{T}})=\sum_{g\in
G_{\Sigma}}\prod_{\rho\in\Sigma(1)}\frac{a_{\rho}(g)\cdot F_{\rho}\cdot
e^{-F_{\rho}}}{1-a_{\rho}(g)e^{-F_{\rho}}}\in\widehat{H}^{*}_{\mathbb{T}}(X;\mbox{$\mathbb{Q}$}).$

######  Remark 3.26.

Note that formula (131) coincides with (130) upon substituting $-F_{\rho}$ for
$F_{\rho}$, for each $\rho\in\Sigma(1)$, i.e., these classes are exchanged by
the cohomological duality involution on
$\widehat{H}^{*}_{\mathbb{T}}(X;\mbox{$\mathbb{Q}$})$ given by multiplication
with $(-1)^{i}$ in degree $2i$. (Recall that under our assumptions
$\widehat{H}^{*}_{\mathbb{T}}(X;\mbox{$\mathbb{Q}$})$ is even, by (63).)

In Section 5 below, formula (131) will be used for proving Euler-Maclaurin
type formulae for lattice points in the interior of a full-dimensional
polytope, generalizing formula (46) for lattice point counting. The next
results are motivated by the fact that, instead of deleting all facets of the
polytope $P$, one can just delete some of the facets $F_{i}$, $i\in
K\subset\Sigma_{P}(1)$.

######  Lemma 3.27.

Let $X:=X_{\Sigma}$ be a simplicial quasi-projective toric variety associated
to a fan $\Sigma$ of full-dimensional convex support. Let $U\subset X$ be the
open complement of the divisor $D_{K}:=\bigcup_{\rho\in K}D_{\rho}$, for
$K\subset\Sigma(1)$. In the notations of the Cox construction, let
$W_{\rho}=\\{x_{\rho}=0\\}\subset W$ be the inverse image of $D_{\rho}$ under
the quotient map $\pi:W\to X$. Then the preimage $\widetilde{U}$ of $U$ under
$\pi$ is the complement of the $\widetilde{{\mathbb{T}}}$-invariant normal
crossing divisor $W_{K}=\bigcup_{\rho\in K}W_{\rho}$ in $W$, and

(132) ${\rm
td}_{*}^{\mathbb{T}}\big{(}(\pi_{*}mC_{y}^{\widetilde{{\mathbb{T}}}}([\widetilde{U}\hookrightarrow
W]))^{G}\big{)}=(1+y)^{r-n}\cdot{\rm
td}_{*}^{\mathbb{T}}(mC_{y}^{\mathbb{T}}([U\hookrightarrow
X]))\in\widehat{H}^{*}_{\mathbb{T}}(X;{\mathbb{Q}})[y].$

###### Proof.

By the inclusion-exclusion formula for the equivariant motivic Chern classes,
one has as in (92),

(133) $mC^{\mathbb{T}}_{y}([U\hookrightarrow X])=\sum_{I\subset
K}(-1)^{|I|}mC^{\mathbb{T}}_{y}([D_{I}\hookrightarrow X])\>,$

with $D_{I}=\bigcap_{\rho\in I}D_{\rho}$. Similarly,

(134) $mC^{\widetilde{{\mathbb{T}}}}_{y}([\widetilde{U}\hookrightarrow
W])=\sum_{I\subset
K}(-1)^{|I|}mC^{\widetilde{{\mathbb{T}}}}_{y}([W_{I}\hookrightarrow W]),$

with $W_{I}=\bigcap_{\rho\in I}W_{\rho}$. The assertion follows now by
applying formula (125) to the summands on the right-hand side of the two
identities above.∎

By formula (91), in the above notations we have:

(135) $mC^{\widetilde{{\mathbb{T}}}}_{y}([\widetilde{U}\hookrightarrow
W])=[{\mathcal{O}}_{W}(-W_{K})\otimes\Lambda_{y}\Omega^{1}_{W}(\log
W_{K})]_{\widetilde{{\mathbb{T}}}}\in
K_{0}^{\widetilde{{\mathbb{T}}}}(W)[y]\>.$

Then by the equivariant version of the Lefschetz-Riemann-Roch theorem (see
[24][Thm.3.1 and Rem.4.3]), applied to the right-hand side of formula (135),
we get the following generalization of Theorem 3.22:

######  Theorem 3.28.

Let $X:=X_{\Sigma}$ be a simplicial quasi-projective toric variety associated
to a fan $\Sigma$ of full-dimensional convex support. Let $U\subset X$ be the
open complement of the divisor $D_{K}:=\bigcup_{\rho\in K}D_{\rho}$, for
$K\subset\Sigma(1)$. The equivariant Hirzebruch classes
${T}^{\mathbb{T}}_{y*}([U\hookrightarrow X])$ and
$\widehat{T}^{\mathbb{T}}_{y*}([U\hookrightarrow X])$ are computed by:

(136) ${T}^{\mathbb{T}}_{y*}([U\hookrightarrow X])=(1+y)^{n-r}\cdot\sum_{g\in
G_{\Sigma}}\prod_{\rho\in K}\frac{F_{\rho}\cdot(1+y)\cdot a_{\rho}(g)\cdot
e^{-F_{\rho}}}{1-a_{\rho}(g)\cdot e^{-F_{\rho}}}\prod_{\rho\notin
K}\frac{F_{\rho}\cdot\big{(}1+y\cdot a_{\rho}(g)\cdot
e^{-F_{\rho}}\big{)}}{1-a_{\rho}(g)\cdot e^{-F_{\rho}}}\>,$

(137) $\widehat{T}^{\mathbb{T}}_{y*}([U\hookrightarrow X])=\sum_{g\in
G_{\Sigma}}\prod_{\rho\in K}\frac{F_{\rho}\cdot(1+y)\cdot a_{\rho}(g)\cdot
e^{-F_{\rho}(1+y)}}{1-a_{\rho}(g)\cdot e^{-F_{\rho}(1+y)}}\prod_{\rho\notin
K}\frac{F_{\rho}\cdot\big{(}1+y\cdot a_{\rho}(g)\cdot
e^{-F_{\rho}(1+y)}\big{)}}{1-a_{\rho}(g)\cdot e^{-F_{\rho}(1+y)}}\>,$

with $r=|\Sigma(1)|$, and $F_{\rho}=[D_{\rho}]_{\mathbb{T}}$ denoting the
equivariant fundamental class of the ${\mathbb{T}}$-invariant divisor
$D_{\rho}$ corresponding to the ray $\rho\in\Sigma(1)$.

###### Proof.

The proof is similar to that of Theorem 3.22 and is based on the equivariant
Lefschetz-Riemann-Roch theorem. We only indicate here the changes. Instead of
calculating the equivariant twisted Chern character
$ch^{\widetilde{{\mathbb{T}}}}(i_{g}^{*}-)(g)$, $g\in G$, of
$\Lambda_{y}T^{*}_{W}$ for $i_{g}:W^{g}\hookrightarrow W$ the fixed point set
inclusion, one now has to calculate this for
${\mathcal{O}}_{W}(-W_{K})\otimes\Lambda_{y}\Omega^{1}_{W}(\log W_{K})$. Here,
for ${\mathcal{E}}\in K^{0}_{\widetilde{{\mathbb{T}}}}(W)$, the twisted
equivariant Chern character is defined as

$ch^{\widetilde{{\mathbb{T}}}}(i_{g}^{*}{\mathcal{E}})(g)=\sum_{\chi}\chi(g)\cdot
ch^{\widetilde{{\mathbb{T}}}}({\mathcal{E}}_{\chi}),$

where ${\mathcal{E}}\simeq\oplus_{\chi}{\mathcal{E}}_{\chi}$ is the finite
decomposition of ${\mathcal{E}}$ into sheaves ${\mathcal{E}}_{\chi}$ on which
$g$ acts by a (complex-valued) character $\chi$. These sheaves
${\mathcal{E}}_{\chi}$ are also $\widetilde{{\mathbb{T}}}$-equivariant since
$\widetilde{{\mathbb{T}}}$ is an abelian group.

We now have to evaluate $ch^{\widetilde{{\mathbb{T}}}}(i_{g}^{*}-)(g)$ on

${\mathcal{O}}_{W}(-W_{K})\otimes\Lambda_{y}\Omega^{1}_{W}(\log
W_{K})\simeq\prod_{\rho\in K}\big{(}{\mathcal{O}}_{W}(-W_{\rho})\otimes
pr_{\rho}^{*}\Lambda_{y}\Omega^{1}_{\mathbb{C}}(\log\\{0\\})\big{)}\prod_{\rho\notin
K}\Lambda_{y}{\mathcal{O}}_{W}(-W_{\rho}).$

Here $pr_{\rho}:W\to{\mathbb{C}}$ is the projection to the $\rho$-th factor,
with $\widetilde{{\mathbb{T}}}\simeq({\mathbb{C}}^{*})^{r}$ acting factorwise
via the projection $a_{\rho}:\widetilde{{\mathbb{T}}}\to{\mathbb{C}}^{*}$ to
the corresponding $\rho$-th factor. So the calculation can be done factorwise,
with

$ch^{\widetilde{{\mathbb{T}}}}(i_{g}^{*}\Lambda_{y}{\mathcal{O}}_{W}(-W_{\rho}))(g)=i_{g}^{*}\left(1+y\cdot
a_{\rho}(g^{-1})\cdot e^{-z_{\rho}}\right),$

for $\rho\notin K$, and $z_{\rho}:=[W_{\rho}]_{\widetilde{{\mathbb{T}}}}\in
H^{*}_{\widetilde{{\mathbb{T}}}}(W;{\mathbb{Q}})$ the corresponding
equivariant fundamental class as in [36][eqn.(5.13)]. By the inclusion-
exclusion formula, for $\\{0\\}\subset{\mathbb{C}}$ and $\rho\in K$, we get:

$[{\mathcal{O}}_{W}(-W_{\rho})\otimes
pr_{\rho}^{*}\Lambda_{y}\Omega^{1}_{\mathbb{C}}(\log\\{0\\})]=[\Lambda_{y}{\mathcal{O}}_{W}(-W_{\rho})]-[{\mathcal{O}}_{D_{\rho}}].$

So

$\begin{split}&ch^{\widetilde{{\mathbb{T}}}}\left(i_{g}^{*}\big{(}{\mathcal{O}}_{W}(-W_{\rho})\otimes
pr_{\rho}^{*}\Lambda_{y}\Omega^{1}_{\mathbb{C}}(\log\\{0\\})\big{)}\right)(g)\\\
&=i_{g}^{*}\big{(}1+y\cdot a_{\rho}(g^{-1})\cdot
e^{-z_{\rho}}\big{)}-ch^{\widetilde{{\mathbb{T}}}}(i_{g}^{*}\left([{\mathcal{O}}_{W}]-[{\mathcal{O}}_{W}(-D_{\rho})]\right))(g)\\\
&=i_{g}^{*}\big{(}1+y\cdot a_{\rho}(g^{-1})\cdot
e^{-z_{\rho}}\big{)}-i_{g}^{*}\big{(}1-a_{\rho}(g^{-1})\cdot
e^{-z_{\rho}}\big{)}\\\ &=i_{g}^{*}\left((1+y)\cdot a_{\rho}(g^{-1})\cdot
e^{-z_{\rho}}\right).\end{split}$

Then these formulae are pushed forward via $(i_{g})_{*}$ using the projection
formula, and one applies the ring isomorphism
$\phi:H^{*}_{\widetilde{{\mathbb{T}}}}(W;{\mathbb{Q}})\simeq
H^{*}_{{{\mathbb{T}}}}(X;{\mathbb{Q}})$. Note that $z_{\rho}$ maps to
$F_{\rho}$ under the identification
$\phi:H^{2}_{\widetilde{{\mathbb{T}}}}(W;{\mathbb{Q}})\simeq
H^{2}_{{{\mathbb{T}}}}(X;{\mathbb{Q}})$. Finally, note that $g\in G_{\Sigma}$
if and only if $g^{-1}\in G_{\Sigma}$. This completes the proof for the un-
normalized equivariant Hirzebruch classes. To get the formula for the
normalized equivariant Hirzebruch classes, we just have to substitute
$(1+y)^{-n}\cdot[X]_{{\mathbb{T}}}$ for $[X]_{{\mathbb{T}}}\in
H^{\mathbb{T}}_{2n}(X;{\mathbb{Q}})$ (implicitly used in the equivariant
Poincaré duality), and $(1+y)\cdot F_{\rho}$ for $F_{\rho}\in
H^{2}(X;{\mathbb{Q}})$. ∎

For later use, and since the needed notation was already introduced in the
previous result, let us sketch a proof of the following extension of the Todd
class formula (130) to the ${\mathbb{T}}$-equivariant coherent sheaf
$\pi_{*}({\mathcal{O}}_{W}\otimes{\mathbb{C}}_{\chi^{\widetilde{m}}})^{G}$,
for $\pi:W\to X$ the quotient map of the Cox construction and
$\chi^{\widetilde{m}}$ a character of $\widetilde{{\mathbb{T}}}$.

######  Lemma 3.29.

With the above notations, we have

(138) ${\rm
td}_{*}^{\mathbb{T}}(\pi_{*}({\mathcal{O}}_{W}\otimes{\mathbb{C}}_{\chi^{\widetilde{m}}})^{G})=\sum_{g\in
G_{\Sigma}}\chi^{\widetilde{m}}(g^{-1})\prod_{\rho\in\Sigma(1)}\frac{F_{\rho}\cdot
e^{\langle\widetilde{m},e_{\rho}\rangle\cdot
F_{\rho}}}{1-a_{\rho}(g)e^{-F_{\rho}}},$

with $\\{e_{\rho}\\}_{\rho\in\Sigma(1)}$ the standard basis of the lattice
$\widetilde{N}={\mathbb{Z}}^{|\Sigma(1)|}$.

###### Proof.

Comparing to the proof of the classical Todd class formula (130), we need to
calculate
$ch^{\widetilde{{\mathbb{T}}}}(i_{g}^{*}\big{(}{\mathcal{O}}_{W}\otimes{\mathbb{C}}_{\chi^{\widetilde{m}}}\big{)})(g)$
instead of $ch^{\widetilde{{\mathbb{T}}}}(i_{g}^{*}({\mathcal{O}}_{W}))(g)=1$.
This is given by

$ch^{\widetilde{{\mathbb{T}}}}(i_{g}^{*}\big{(}{\mathcal{O}}_{W}\otimes{\mathbb{C}}_{\chi^{\widetilde{m}}}\big{)})(g)=i_{g}^{*}\left(\chi^{\widetilde{m}}(g)e^{c(\widetilde{m})}\right),$

with $c(\widetilde{m})=\sum_{\rho}\langle\widetilde{m},e_{\rho}\rangle
z_{\rho}\in H^{2}_{\widetilde{{\mathbb{T}}}}(W;\mbox{$\mathbb{Q}$})$, e.g.,
see [16][Prop.12.4.13(b)]. ∎

######  Remark 3.30.

Note that if $K=\emptyset$, Theorem 3.28 reduces to Theorem 3.22. At the other
extreme, if $K=\Sigma(1)$, one gets for the un-normalized class first by
specializing to $y=0$ just Corollary 3.25, and using this one then recovers
for arbitrary $y$ formula (100) in the case of the zero cone.

######  Remark 3.31.

For $y=0$, both formulae of Theorem 3.28 specialize to the following Todd
class type formula, which will fit later on with the Euler-Maclaurin formulas
for polytopes with some facets removed:

(139) ${T}^{\mathbb{T}}_{0*}([U\hookrightarrow X])=\sum_{g\in
G_{\Sigma}}\prod_{\rho\in K}\frac{F_{\rho}\cdot a_{\rho}(g)\cdot
e^{-F_{\rho}}}{1-a_{\rho}(g)\cdot e^{-F_{\rho}}}\prod_{\rho\notin
K}\frac{F_{\rho}}{1-a_{\rho}(g)\cdot e^{-F_{\rho}}}\>.$

Note that in view of Remark 3.6, one also has

(140) ${T}^{\mathbb{T}}_{0*}([U\hookrightarrow
X])=td_{*}^{\mathbb{T}}(X)-td_{*}^{\mathbb{T}}(D_{K}).$

##  4\. Localization in equivariant $K$-theory and applications

In this section we apply localization techniques in ${\mathbb{T}}$-equivariant
$K$\- and homology theories of toric varieties, due to Brion-Vergne [13] and,
resp., Brylinski-Zhang [15], for the calculation of the
${\mathbb{T}}$-equivariant motivic Chern and Hirzebruch classes in this toric
context.

###  4.1. Localization in equivariant K-theory

Let $X=X_{\Sigma}$ be an $n$-dimensional toric variety with torus
${\mathbb{T}}=T_{N}$ such that the fixed-point set
$X^{\mathbb{T}}\neq\emptyset$, e.g., $X$ is projective.

By [13][Prop.1.5], the inclusion $i:X^{\mathbb{T}}\hookrightarrow X$ induces
an injective morphism of $K_{0}^{\mathbb{T}}(pt)$-modules

$i_{*}:K_{0}^{\mathbb{T}}(X^{\mathbb{T}})\hookrightarrow
K_{0}^{\mathbb{T}}(X)$

which becomes an isomorphism

$i_{*}:K_{0}^{\mathbb{T}}(X^{\mathbb{T}})_{S}\simeq K_{0}^{\mathbb{T}}(X)_{S}$

upon localization at the multiplicative subset
$S\subset{\mathbb{Z}}[M]=K_{0}^{\mathbb{T}}(pt)$ generated by the elements
$1-\chi^{m}$, for $0\neq m\in M$. Note that

$K_{0}^{\mathbb{T}}(X^{\mathbb{T}})=\bigoplus_{x\in
X^{\mathbb{T}}}K_{0}^{\mathbb{T}}(x),$

from which one gets via the isomorphism $i_{*}$ a projection map of
$K_{0}^{\mathbb{T}}(pt)$-modules, called here the $K$-theoretic localization
map at $x$,

$pr_{x}:K_{0}^{\mathbb{T}}(X)_{S}\simeq\bigoplus_{x\in
X^{\mathbb{T}}}K_{0}^{\mathbb{T}}(x)_{S}\longrightarrow
K_{0}^{\mathbb{T}}(x)_{S}={\mathbb{Z}}[M]_{S}.$

Let $x=x_{\sigma}\subset U_{\sigma}$ be the ${\mathbb{T}}$-fixed point
corresponding to a cone $\sigma\in\Sigma(n)$ of maximal dimension, as in (65),
where $U_{\sigma}\subset X$ is the unique ${\mathbb{T}}$-invariant open affine
subset containing $x_{\sigma}=O_{\sigma}$. Then the localization map at $x$
factorizes via restriction over $U_{\sigma}$ as:

$pr_{x}:K_{0}^{\mathbb{T}}(X)_{S}\longrightarrow
K_{0}^{\mathbb{T}}(U_{\sigma})_{S}\simeq
K_{0}^{\mathbb{T}}(x_{\sigma})_{S}={\mathbb{Z}}[M]_{S}.$

We now explain a different description of the isomorphism (denoted also by
$pr_{x}$)

$pr_{x}:K_{0}^{\mathbb{T}}(U_{\sigma})_{S}\simeq
K_{0}^{\mathbb{T}}(x_{\sigma})_{S}={\mathbb{Z}}[M]_{S}$

by using the eigenspace decomposition (68). As in [13][Sect.1.3] (see also
[16][Def.13.2.2]), a formal power series $f\in{\mathbb{Z}}[[M]]$ is called
summable if there is $g\in{\mathbb{Z}}[M]$ and a finite subset $I\subset
M\setminus\\{0\\}$ such that in ${\mathbb{Z}}[[M]]$ one has:
$f\cdot\prod_{m\in I}(1-\chi^{m})=g$. Let

$\mathbb{S}(f):=g\cdot\prod_{m\in I}(1-\chi^{m})^{-1}\in{\mathbb{Z}}[M]_{S}$

be the sum of $f$, which is easily seen to be independent of the
factorization. Let ${\mathbb{Z}}[[M]]_{\rm Sum}\subset{\mathbb{Z}}[[M]]$ be
the subset of summable elements in ${\mathbb{Z}}[[M]]$. This is a
${\mathbb{Z}}[M]$-submodule of ${\mathbb{Z}}[[M]]$, and the summation map
$\mathbb{S}$ induces a homomorphism of ${\mathbb{Z}}[M]$-modules

$\mathbb{S}:{\mathbb{Z}}[[M]]_{\rm Sum}\to{\mathbb{Z}}[M]_{S}.$

Let ${\mathcal{F}}$ be a ${\mathbb{T}}$-equivariant coherent sheaf on
$U_{\sigma}$. Then $W:=H^{0}(U_{\sigma};{\mathcal{F}})$ has an eigenspace
decomposition $W=\bigoplus_{m\in M}W_{\chi^{m}}$ (see [13][Sect.1.3]) with
eigenspaces $W_{\chi^{m}}$ of finite dimension as in (68) (although
$W=H^{0}(U_{\sigma};{\mathcal{F}})$ could be infinite dimensional). Then

(141) $\chi_{\sigma}^{\mathbb{T}}({\mathcal{F}}):=\sum_{m\in
M}\dim_{\mathbb{C}}W_{\chi^{m}}\cdot\chi^{m}\in{\mathbb{Z}}[[M]]$

is summable (cf. [13][Prop.1.3]). This induces a homomorphism of
${\mathbb{Z}}[M]$-modules (see [13][Rem.1.3])

(142)
$\chi_{\sigma}^{\mathbb{T}}:K_{0}^{\mathbb{T}}(U_{\sigma})\longrightarrow\mbox{$\mathbb{Z}$}[M]_{\rm
sum}$

such that the composition $\mathbb{S}\circ\chi_{\sigma}^{\mathbb{T}}$ induces
after localization the map $pr_{x}$, since

(143)
$\chi_{\sigma}^{\mathbb{T}}(i_{*}([{\mathbb{C}}_{\chi^{m}}])=\chi^{m}\in{\mathbb{Z}}[M]\subset\mbox{$\mathbb{Z}$}[M]_{\rm
sum}.$

For applications to the case of simplicial cones, we next introduce a
Lefschetz type variant of the Euler characteristic
$\chi_{\sigma}^{\mathbb{T}}$, and a corresponding summation map $\mathbb{S}$.
Let $\sigma\in\Sigma(n)$ be a simplicial cone with $u_{1},\ldots,u_{n}\in
N=N_{\sigma}$ the generators of the the rays $\rho_{j}\in\sigma(1)$,
$j=1,\ldots,n$. Let $N^{\prime}=N^{\prime}_{\sigma}$ be the finite index
sublattice of $N$ generated by $u_{1},\ldots,u_{n}$, and consider $\sigma\in
N^{\prime}_{{\mathbb{R}}}=N_{\mathbb{R}}$ so that it is smooth with respect to
the lattice $N^{\prime}$. With ${\mathbb{T}}$, ${\mathbb{T}}^{\prime}$ the
corresponding $n$-dimensional tori of the lattices $N$, resp., $N^{\prime}$,
the inclusion $N^{\prime}\hookrightarrow N$ induces a toric morphism
$\pi:U^{\prime}_{\sigma}\to U_{\sigma}$ of the associated affine toric
varieties. Let $G_{\sigma}$ be the finite kernel of the epimorphism
$\pi:{\mathbb{T}}^{\prime}\to{\mathbb{T}}$, so that
$U^{\prime}_{\sigma}/G_{\sigma}\simeq U_{\sigma}$ (e.g., see
[16][Prop.1.3.18]).

Let $m_{i}\in M=M_{\sigma}$, $1\leq i\leq n$, be the unique primitive elements
in the dual lattice $M$ of $N$ satisfying $\langle m_{i},u_{j}\rangle=0$ for
$i\neq j$ and $q_{i}:=\langle m_{i},u_{i}\rangle>0$ so that the dual lattice
$M^{\prime}=M_{\sigma}^{\prime}$ of $N^{\prime}$ is generated by the elements
$m^{\prime}_{j}:=\frac{m_{j}}{q_{j}}.$ Let
$a_{\rho_{j}}:G_{\sigma}\to{\mathbb{C}}^{*}$ be the characters of $G_{\sigma}$
as introduced in (33).

For ${\mathcal{F}}^{\prime}$ a ${\mathbb{T}}^{\prime}$-equivariant coherent
sheaf on $U^{\prime}_{\sigma}$, the vector space
$W^{\prime}:=H^{0}(U^{\prime}_{\sigma};{\mathcal{F}}^{\prime})$ has an
eigenspace decomposition $W^{\prime}=\bigoplus_{m^{\prime}\in
M^{\prime}}W^{\prime}_{\chi^{m^{\prime}}}$ as before. Since
${\mathbb{T}}^{\prime}$ is abelian, its finite subgroup $G_{\sigma}$ acts on
$W^{\prime}$ respecting this eigenspace decomposition. We can then introduce
the Lefschetz type Euler characteristic

(144)
$tr^{{\mathbb{T}}^{\prime}}_{\sigma}({\mathcal{F}}^{\prime}):=\frac{1}{|G_{\sigma}|}\bigoplus_{m^{\prime}\in
M^{\prime}}\sum_{g\in G_{\sigma}}tr(g:W^{\prime}_{\chi^{m^{\prime}}}\to
W^{\prime}_{\chi^{m^{\prime}}})\cdot\chi^{m^{\prime}}\in{\mathbb{C}}[[M^{\prime}]]_{\rm
sum}.$

In this context, the notion of summable is defined almost as above, but using
the multiplicative subset $S^{\prime}\subset{\mathbb{C}}[M^{\prime}]$
generated by elements $1-a\cdot\chi^{m^{\prime}}$, for $0\neq m^{\prime}\in
M^{\prime}$ and $a\in{\mathbb{C}}^{*}$. This induces a homomorphism of
${\mathbb{Z}}[M^{\prime}]$-modules

$tr^{{\mathbb{T}}^{\prime}}_{\sigma}:K_{0}^{\mathbb{T}}(U^{\prime}_{\sigma})\longrightarrow\mbox{$\mathbb{C}$}[M^{\prime}]_{\rm
sum}.$

The fact that $tr^{{\mathbb{T}}^{\prime}}_{\sigma}({\mathcal{F}}^{\prime})$ is
summable will be explained below for sheaves
${\mathcal{F}}^{\prime}={\mathcal{O}}_{U^{\prime}_{\sigma}}\otimes{\mathbb{C}}_{\chi^{m^{\prime}}}$,
with $m^{\prime}\in M^{\prime}$, whose classes generate
$K_{0}^{\mathbb{T}}(U^{\prime}_{\sigma})$ (as in the proof of [13][Cor.1.2]).
There is also a corresponding summation map

$\mathbb{S}^{\prime}:{\mathbb{C}}[[M^{\prime}]]_{\rm
sum}\to{\mathbb{C}}[M^{\prime}]_{S^{\prime}},$

so that the following diagram of ${\mathbb{Z}}[M]$-linear maps commutes:

(145)
$\begin{CD}K_{0}^{{\mathbb{T}}^{\prime}}(U^{\prime}_{\sigma})@>{tr^{{\mathbb{T}}^{\prime}}_{\sigma}}>{}>{\mathbb{C}}[[M^{\prime}]]_{\rm
sum}@>{\mathbb{S}^{\prime}}>{}>{\mathbb{C}}[M^{\prime}]_{S^{\prime}}\\\
@V{\pi_{*}^{G_{\sigma}}}V{}V@A{}A{}A@A{}A{}A\\\
K_{0}^{\mathbb{T}}(U_{\sigma})@>{\chi^{{\mathbb{T}}}_{\sigma}}>{}>{\mathbb{Z}}[[M]]_{\rm
sum}@>{\mathbb{S}}>{}>{\mathbb{Z}}[M]_{S}\ .\end{CD}$

Here, $\pi_{*}^{G_{\sigma}}$ is induced by the corresponding exact functor
given by taking the $G_{\sigma}$-invariant part of the pushforward for the
finite map $\pi$, which is
${\mathbb{T}}={\mathbb{T}}^{\prime}/G_{\sigma}$-equivariant. The monomorphism
${\mathbb{Z}}[[M]]_{\rm sum}\to{\mathbb{C}}[[M^{\prime}]]_{\rm sum}$ and the
algebra map ${\mathbb{Z}}[M]_{S}\to{\mathbb{C}}[M^{\prime}]_{S^{\prime}}$ are
induced by the lattice injection $M\hookrightarrow M^{\prime}$.

######  Example 4.1.

If ${\mathcal{F}}={\mathcal{O}}_{X}|_{U_{\sigma}}$, then

$\chi_{\sigma}^{\mathbb{T}}({\mathcal{O}}_{X}|_{U_{\sigma}})=\sum_{m\in\sigma^{\vee}\cap
M}\chi^{-m}\in\mbox{$\mathbb{Z}$}[M]_{\rm sum}.$

If, moreover, $\sigma$ is a smooth cone with $m_{\sigma,i}$, $i=1,\ldots,n$,
the minimal generators of $\sigma^{\vee}$, then

$\chi_{\sigma}^{\mathbb{T}}({\mathcal{O}}_{X}|_{U_{\sigma}})=\prod_{i=1}^{n}(\sum_{k\geq
0}(\chi^{-m_{\sigma,i}})^{k}),$

hence (as in [16][Lem.13.2.4])

(146)
$\mathbb{S}(\chi_{\sigma}^{\mathbb{T}}({\mathcal{O}}_{X}|_{U_{\sigma}}))=\prod_{i=1}^{n}\frac{1}{1-\chi^{-m_{\sigma,i}}}.$

If $\sigma$ is only a simplicial cone, with the same notation for the minimal
generators of $\sigma^{\vee}$, we get for
${\mathcal{F}}^{\prime}={\mathcal{O}}_{U^{\prime}_{\sigma}}$ that

$tr_{\sigma}^{{\mathbb{T}}^{\prime}}({\mathcal{O}}_{U^{\prime}_{\sigma}})=\frac{1}{|G_{\sigma}|}\sum_{g\in
G_{\sigma}}\prod_{i=1}^{n}(\sum_{k\geq
0}(a_{\rho_{i}}(g^{-1})\cdot\chi^{-m^{\prime}_{\sigma,i}})^{k}),$

hence this series is summable, and by applying diagram (145) to
${\mathcal{F}}^{\prime}={\mathcal{O}}_{U^{\prime}_{\sigma}}$, with
${\mathcal{O}}_{U_{\sigma}}\simeq\pi_{*}^{G_{\sigma}}({\mathcal{O}}_{U^{\prime}_{\sigma}})$,
we recover Molien’s formula (see [13][page 24]):

(147)
$\mathbb{S}(\chi_{\sigma}^{\mathbb{T}}({\mathcal{O}}_{X}|_{U_{\sigma}}))=\frac{1}{|G_{\sigma}|}\sum_{g\in
G_{\sigma}}\prod_{i=1}^{n}\frac{1}{1-a_{\rho_{i}}(g^{-1})\cdot\chi^{-m^{\prime}_{\sigma,i}}}\in{\mathbb{C}}[M^{\prime}]_{S^{\prime}}.$

Recall here that, by our convention (following [16]), ${\mathbb{T}}$ acts on
$\chi^{m}$ by $t\cdot\chi^{m}=\chi^{m}(t^{-1})\chi^{m}$. Note that [13] uses
Oda’s convention [39][pag.6]: $t\cdot\chi^{m}=\chi^{m}(t)\chi^{m}$, which
explains the sign difference. ∎

######  Example 4.2.

Consider the case of a simplicial cone $\sigma\in\Sigma(n)$ with minimal
generators of $\sigma^{\vee}$ as before. We get for
${\mathcal{F}}^{\prime}={\mathcal{O}}_{U^{\prime}_{\sigma}}\otimes{\mathbb{C}}_{\chi^{m^{\prime}}}$
(with $m^{\prime}\in M^{\prime}$) that

$tr_{\sigma}^{{\mathbb{T}}^{\prime}}({\mathcal{O}}_{U^{\prime}_{\sigma}}\otimes{\mathbb{C}}_{\chi^{m^{\prime}}})=\frac{1}{|G_{\sigma}|}\sum_{g\in
G_{\sigma}}\chi^{m^{\prime}}(g)\cdot\chi^{m^{\prime}}\prod_{i=1}^{n}(\sum_{k\geq
0}(a_{\rho_{i}}(g^{-1})\cdot\chi^{-m^{\prime}_{\sigma,i}})^{k}),$

hence this series is summable, and by applying diagram (145) to
${\mathcal{F}}^{\prime}={\mathcal{O}}_{U^{\prime}_{\sigma}}\otimes{\mathbb{C}}_{\chi^{m^{\prime}}}$,
with
${\mathcal{F}}:=\pi_{*}^{G_{\sigma}}({\mathcal{O}}_{U^{\prime}_{\sigma}}\otimes{\mathbb{C}}_{\chi^{m^{\prime}}})$,
we get

(148)
$\mathbb{S}(\chi_{\sigma}^{\mathbb{T}}({\mathcal{F}}))=\frac{1}{|G_{\sigma}|}\sum_{g\in
G_{\sigma}}\chi^{m^{\prime}}(g)\cdot\chi^{m^{\prime}}\prod_{i=1}^{n}\frac{1}{1-a_{\rho_{i}}(g^{-1})\cdot\chi^{-m^{\prime}_{\sigma,i}}}\in{\mathbb{C}}[M^{\prime}]_{S^{\prime}}.$

∎

######  Example 4.3.

Back to the general case, for $\sigma\in\Sigma(n)$, consider
${\mathcal{F}}={\mathcal{O}}_{X}(D)|_{U_{\sigma}}$, with $D$ a
${\mathbb{T}}$-invariant Cartier divisor on $X$. Then one gets

$\chi_{\sigma}^{\mathbb{T}}({\mathcal{O}}_{X}(D)|_{U_{\sigma}})=\chi^{-m_{\sigma}}\sum_{m\in\sigma^{\vee}\cap
M}\chi^{-m}\in\mbox{$\mathbb{Z}$}[M]_{\rm sum},$

where $m_{\sigma}\in M$ is uniquely defined so that
$D|_{U_{\sigma}}=div(\chi^{-m_{\sigma}})|_{U_{\sigma}}$. The sequence
$\\{m_{\sigma}\in M\mid\sigma\in\Sigma(n)\\}$ is the Cartier data of $D$, in
the sense of [16][Thm.4.2.8]. In other words,
${\mathcal{O}}_{X}(D)|_{U_{\sigma}}\simeq
U_{\sigma}\times{\mathbb{C}}_{\chi^{-m_{\sigma}}}$ as
${\mathbb{T}}$-equivariant line bundles (e.g., see [16][page 609]), so that
for any ${\mathbb{T}}$-equivariant coherent sheaf on $X$, we have

(149)
$\chi_{\sigma}^{\mathbb{T}}(({\mathcal{F}}\otimes{\mathcal{O}}_{X}(D))|_{U_{\sigma}})=\chi^{-m_{\sigma}}\cdot\chi_{\sigma}^{\mathbb{T}}({\mathcal{F}}|_{U_{\sigma}}).$

∎

######  Example 4.4.

Generalizing the case of ${\mathcal{O}}_{X}$, by [16][Prop.8.2.18], one has
for any $0\leq p\leq n=\dim(X)$,

$\chi_{\sigma}^{\mathbb{T}}(\widehat{\Omega}^{p}_{X}|_{U_{\sigma}})=\sum_{\tau\preceq\sigma}{\dim(O_{\tau})\choose
p}\sum_{m\in{\rm Relint}(\sigma^{\vee}\cap\tau^{\perp})\cap
M}\chi^{-m}\in\mbox{$\mathbb{Z}$}[M]_{\rm sum}.$

Hence,

(150) $\chi_{\sigma}^{\mathbb{T}}(\omega_{X}|_{U_{\sigma}})=\sum_{m\in{\rm
Relint}(\sigma^{\vee})\cap M}\chi^{-m}\in\mbox{$\mathbb{Z}$}[M]_{\rm sum}.$

In particular, if $X$ is quasi-projective, then

$\chi_{\sigma}^{\mathbb{T}}(mC^{\mathbb{T}}_{y}(X)|_{U_{\sigma}})=\sum_{\tau\preceq\sigma}(1+y)^{\dim(O_{\tau})}\sum_{m\in{\rm
Relint}(\sigma^{\vee}\cap\tau^{\perp})\cap
M}\chi^{-m}\in\mbox{$\mathbb{Z}$}[M]_{\rm
sum}\otimes_{\mathbb{Z}}{\mathbb{Z}}[y].$

If, moreover, $\sigma$ is a smooth cone with $m_{\sigma,i}$, $i=1,\ldots,n$,
the minimal generators of $\sigma^{\vee}$, then

$\chi_{\sigma}^{\mathbb{T}}(mC^{\mathbb{T}}_{y}(X)|_{U_{\sigma}})=\prod_{i=1}^{n}\left(1+(1+y)\cdot\sum_{k\geq
1}(\chi^{-m_{\sigma,i}})^{k}\right),$

hence

(151)
$\mathbb{S}(\chi_{\sigma}^{\mathbb{T}}(mC^{\mathbb{T}}_{y}(X)|_{U_{\sigma}}))=\prod_{i=1}^{n}\frac{1+y\cdot\chi^{-m_{\sigma,i}}}{1-\chi^{-m_{\sigma,i}}}\in{\mathbb{Z}}[M]_{S}.$

Consider now the case of a simplicial cone $\sigma$ with minimal generators of
$\sigma^{\vee}$ as before, and let
${\mathcal{F}}^{\prime}=\Omega^{p}_{U^{\prime}_{\sigma}}$, $p\geq 0$, resp.,
$[{\mathcal{F}}^{\prime}]=mC^{{\mathbb{T}}^{\prime}}_{y}(U^{\prime}_{\sigma})=\sum_{p=0}^{n}[\Omega^{p}_{U^{\prime}_{\sigma}}]\cdot
y^{p}\in K_{0}^{{\mathbb{T}}^{\prime}}(U^{\prime}_{\sigma})[y]$. Then

$tr_{\sigma}^{{\mathbb{T}}^{\prime}}(mC^{{\mathbb{T}}^{\prime}}_{y}(U^{\prime}_{\sigma}))=\frac{1}{|G_{\sigma}|}\sum_{g\in
G_{\sigma}}\prod_{i=1}^{n}\left(1+(1+y)\cdot\sum_{k\geq
1}(a_{\rho_{i}}(g^{-1})\cdot\chi^{-m^{\prime}_{\sigma,i}})^{k}\right).$

By applying diagram (145) to
${\mathcal{F}}^{\prime}=\Omega^{p}_{U^{\prime}_{\sigma}}$, with
${\mathcal{F}}=\pi_{*}^{G_{\sigma}}(\Omega^{p}_{U^{\prime}_{\sigma}})\simeq\widehat{\Omega}^{p}_{X}|_{U_{\sigma}}$,
resp.,
$[{\mathcal{F}}^{\prime}]=mC^{{\mathbb{T}}^{\prime}}_{y}(U^{\prime}_{\sigma})$,
with $[{\mathcal{F}}]=mC^{\mathbb{T}}_{y}(X)|_{U_{\sigma}}$, we get

(152)
$\mathbb{S}(\chi_{\sigma}^{\mathbb{T}}(mC^{\mathbb{T}}_{y}(X)|_{U_{\sigma}}))=\frac{1}{|G_{\sigma}|}\sum_{g\in
G_{\sigma}}\prod_{i=1}^{n}\frac{1+y\cdot
a_{\rho_{i}}(g^{-1})\cdot\chi^{-m^{\prime}_{\sigma,i}}}{1-a_{\rho_{i}}(g^{-1})\cdot\chi^{-m^{\prime}_{\sigma,i}}}\in{\mathbb{C}}[M^{\prime}]_{S^{\prime}}.$

Finally, one can twist the sheaves $\widehat{\Omega}^{p}_{X}$ by
${\mathcal{O}}_{X}(D)$, for $D$ a ${\mathbb{T}}$-invariant Cartier divisor,
and use (149) to get the corresponding identities. ∎

######  Remark 4.5.

If, more generally, we work with $X=X_{\Sigma^{\prime}}$ a closed
${\mathbb{T}}$-invariant algebraic subset of $X_{\Sigma}$ corresponding to a
star-closed subset $\Sigma^{\prime}\subset\Sigma$, similar localization
formulas hold for the sheaves $\widetilde{\Omega}^{p}_{X}$ of Ishida $p$-forms
of $X$ (extended by zero to $X_{\Sigma}$). More precisely, using now
[3][Sect.3], one gets for $0\leq p\leq\dim(X)$

(153)
$\chi_{\sigma}^{\mathbb{T}}(\widetilde{\Omega}^{p}_{X}|_{U_{\sigma}})=\sum_{\tau\preceq\sigma,\tau\in\Sigma^{\prime}}{\dim(O_{\tau})\choose
p}\sum_{m\in{\rm Relint}(\sigma^{\vee}\cap\tau^{\perp})\cap
M}\chi^{-m}\in\mbox{$\mathbb{Z}$}[M]_{\rm sum}.$

In particular, if $X=V_{\tau}\subset X_{\Sigma}$ is the orbit closure for
$\tau\preceq\sigma$ of dimension $d$, then
$\widetilde{\Omega}^{d}_{X}=\omega_{V_{\tau}}$, so we get by (99) that

(154) $\chi_{\sigma}^{\mathbb{T}}(mC_{0}([O_{\tau}\hookrightarrow
X_{\Sigma}])|_{U_{\sigma}})=\chi_{\sigma}^{\mathbb{T}}(\omega_{V_{\tau}}|_{U_{\sigma}})=\sum_{m\in{\rm
Relint}(\sigma^{\vee}\cap\tau^{\perp})\cap
M}\chi^{-m}\in\mbox{$\mathbb{Z}$}[M]_{\rm sum}.$

######  Example 4.6.

Let $Z=X_{\Sigma^{\prime}}$ be a closed ${\mathbb{T}}$-invariant algebraic
subset of $X=X_{\Sigma}$ corresponding to a star-closed subset
$\Sigma^{\prime}\subset\Sigma$, with open complement $V:=X\setminus Z$. Then

(155) $\chi_{\sigma}^{\mathbb{T}}(mC^{\mathbb{T}}_{y}([V\hookrightarrow
X])|_{U_{\sigma}})=\sum_{\tau\preceq\sigma;\tau\notin\Sigma^{\prime}}(1+y)^{\dim(O_{\tau})}\sum_{m\in{\rm
Relint}(\sigma^{\vee}\cap\tau^{\perp})\cap
M}\chi^{-m}\in\mbox{$\mathbb{Z}$}[M]_{\rm
sum}\otimes_{\mathbb{Z}}{\mathbb{Z}}[y].$

We now consider the case when $Z=D_{K}:=\bigcup_{\rho\in K}D_{\rho}$, for
$K\subset\Sigma(1)$, and $\sigma\in\Sigma(n)$ is a smooth, resp., simplicial
cone.
Let us first assume that $\sigma$ is smooth, with $m_{\sigma,i}$,
$i=1,\ldots,n$, the minimal generators of $\sigma^{\vee}$. Then

$\chi_{\sigma}^{\mathbb{T}}(mC^{\mathbb{T}}_{y}([V\hookrightarrow
X])|_{U_{\sigma}})\\\ =\prod_{\rho_{i}\in\sigma(1)\cap
K}\left((1+y)\cdot\sum_{k\geq
1}(\chi^{-m_{\sigma,i}})^{k}\right)\cdot\prod_{\rho_{i}\in\sigma(1)\setminus
K}\left(1+(1+y)\cdot\sum_{k\geq 1}(\chi^{-m_{\sigma,i}})^{k}\right),$

hence

(156)
$\mathbb{S}(\chi_{\sigma}^{\mathbb{T}}(mC^{\mathbb{T}}_{y}([V\hookrightarrow
X])|_{U_{\sigma}}))=\prod_{\rho_{i}\in\sigma(1)\cap
K}\frac{(1+y)\cdot\chi^{-m_{\sigma,i}}}{1-\chi^{-m_{\sigma,i}}}\prod_{\rho_{i}\in\sigma(1)\setminus
K}\frac{1+y\cdot\chi^{-m_{\sigma,i}}}{1-\chi^{-m_{\sigma,i}}}\in{\mathbb{Z}}[M]_{S}.$

Consider now the case of a simplicial cone $\sigma$ with minimal generators of
$\sigma^{\vee}$ as before, and let $V^{\prime}=U^{\prime}_{\sigma}\setminus
D_{K^{\prime}}$, with $D_{K^{\prime}}=\bigcup_{\rho\in
K^{\prime}}D^{\prime}_{\rho}$, for $K^{\prime}:=\sigma(1)\cap K$. Let
$[{\mathcal{F}}^{\prime}]=mC^{{\mathbb{T}}^{\prime}}_{y}([V^{\prime}\hookrightarrow
U^{\prime}_{\sigma}])$. Then

$tr_{\sigma}^{{\mathbb{T}}^{\prime}}(mC^{{\mathbb{T}}^{\prime}}_{y}([V\hookrightarrow
U^{\prime}_{\sigma}]))=\\\ \frac{1}{|G_{\sigma}|}\sum_{g\in
G_{\sigma}}\prod_{\rho_{i}\in K^{\prime}}\left((1+y)\sum_{k\geq
1}(a_{\rho_{i}}(g^{-1})\chi^{-m^{\prime}_{\sigma,i}})^{k}\right)\cdot\prod_{\rho_{i}\in\sigma(1)\setminus
K^{\prime}}\left(1+(1+y)\sum_{k\geq
1}(a_{\rho_{i}}(g^{-1})\chi^{-m^{\prime}_{\sigma,i}})^{k}\right).$

By applying (145) to
$[{\mathcal{F}}^{\prime}]=mC^{{\mathbb{T}}^{\prime}}_{y}([V^{\prime}\hookrightarrow
U^{\prime}_{\sigma}])$, with
$[{\mathcal{F}}]=\pi_{*}^{G_{\sigma}}([{\mathcal{F}}^{\prime}])=mC^{\mathbb{T}}_{y}([V\hookrightarrow
X])|_{U_{\sigma}}$, we get

(157)
$\mathbb{S}(\chi_{\sigma}^{\mathbb{T}}(mC^{\mathbb{T}}_{y}([V\hookrightarrow
X])|_{U_{\sigma}})=\\\ \frac{1}{|G_{\sigma}|}\sum_{g\in
G_{\sigma}}\prod_{\rho_{i}\in\sigma(1)\cap K}\frac{(1+y)\cdot
a_{\rho_{i}}(g^{-1})\cdot\chi^{-m^{\prime}_{\sigma,i}}}{1-a_{\rho_{i}}(g^{-1})\cdot\chi^{-m^{\prime}_{\sigma,i}}}\cdot\prod_{\rho_{i}\in\sigma(1)\setminus
K}\frac{1+y\cdot
a_{\rho_{i}}(g^{-1})\cdot\chi^{-m^{\prime}_{\sigma,i}}}{1-a_{\rho_{i}}(g^{-1})\cdot\chi^{-m^{\prime}_{\sigma,i}}}.$

∎

We now explain the following equality, originally due to Brion [9], where we
follow the approach of [13][Cor.1.3] (see also [16][Thm.13.2.8] for a special
case):

######  Theorem 4.7.

Let $X=X_{\Sigma}$ be a complete toric variety of complex dimension $n$, with
${\mathcal{F}}$ a ${\mathbb{T}}$-equivariant coherent sheaf on $X$. Then the
$K$-theoretic Euler characteristic of ${\mathcal{F}}$ can be calculated via
localization at the ${\mathbb{T}}$-fixed points as:

(158)
$\chi^{\mathbb{T}}(X,{\mathcal{F}})=\sum_{\sigma\in\Sigma(n)}\mathbb{S}(\chi_{\sigma}^{\mathbb{T}}({\mathcal{F}}|_{U_{\sigma}}))\in\mbox{$\mathbb{Z}$}[M]_{S}.$

###### Proof.

Let $x_{\sigma}=O_{\sigma}\subset U_{\sigma}$ be the torus fixed point
corresponding to $\sigma\in\Sigma(n)$. The assertion follows from the
commutativity of the lower right square of the following diagram:

$\begin{CD}\bigoplus_{\sigma\in\Sigma(n)}K^{\mathbb{T}}_{0}(x_{\sigma})@>{i_{*}}>{}>K_{0}^{\mathbb{T}}(X)@>{\chi^{\mathbb{T}}}>{}>{\mathbb{Z}}[M]\\\
@V{}V{}V@V{}V{}V@V{}V{}V\\\
\bigoplus_{\sigma\in\Sigma(n)}K^{\mathbb{T}}_{0}(x_{\sigma})_{S}@>{\sim}>{}>K_{0}^{\mathbb{T}}(X)_{S}@>{\chi^{\mathbb{T}}}>{}>{\mathbb{Z}}[M]_{S}\\\
\Big{\|}@V{\wr}V{}V\Big{\|}\\\
\bigoplus_{\sigma\in\Sigma(n)}K^{\mathbb{T}}_{0}(x_{\sigma})_{S}@>{\sim}>{}>\bigoplus_{\sigma\in\Sigma(n)}K^{\mathbb{T}}_{0}(U_{\sigma})_{S}@>{\sum_{\sigma}(\mathbb{S}\circ\chi_{\sigma}^{\mathbb{T}})}>{}>{\mathbb{Z}}[M]_{S}\>.\end{CD}$

The vertical maps are the natural localization maps (upon restriction to the
$U_{\sigma}$’s for the middle map). The commutativity of the outer square
follows from (143). The lower left square commutes by the functoriality of
restriction and localization. This yields the desired commutativity of the
lower right square. ∎

As a consequence, we get the following weighted version of Brion’s equality
[9][page 655], see also [16][Cor.13.2.10(a)].

######  Corollary 4.8.

Let $P$ be a full-dimensional lattice polytope with associated projective
toric variety $X=X_{P}$ and ample Cartier divisor $D=D_{P}$. For each vertex
$v$ of $P$, consider the cone $C_{v}={\rm Cone}(P\cap M-v)=\sigma_{v}^{\vee}$,
with faces $E_{v}={\rm Cone}(E\cap M-v)$ for $v\in E$. Then the following
identity holds in ${\mathbb{Z}}[M]_{S}\otimes_{\mathbb{Z}}{\mathbb{Z}}[y]$:

(159)
$\chi^{\mathbb{T}}(X,mC^{\mathbb{T}}_{y}(X)\otimes{\mathcal{O}}_{X}(D))=\sum_{v\
\text{\rm vertex}}\chi^{-v}\cdot\mathbb{S}\left(\sum_{v\in E\preceq
P}(1+y)^{\dim(E)}\cdot\sum_{m\in{\rm Relint}(E_{v})\cap M}\chi^{-m}\right).$

###### Proof.

By the definition of the inner normal fan $\Sigma_{P}$ of $P$, one has for any
vertex $v$ of $P$ that $\sigma_{v}=C_{v}^{\vee}$, with $C_{v}$ defined as
above (see [16][page 76]). By dualizing, it follows that
$C_{v}=\sigma_{v}^{\vee}$ (cf. [16][Prop.1.2.4]). By [16][Prop.1.2.10], the
faces $E_{v}$ of $C_{v}$ are exactly of the form
$\sigma_{v}^{\vee}\cap\tau^{\perp}$, with $\tau\preceq\sigma_{v}$. Note also
that the Cartier data for $D=D_{P}$ over $U_{\sigma_{v}}$ is exactly given by
$m_{\sigma_{v}}=v\in M$, see [16][(4.2.8)].

The desired identity (159) follows now from Theorem 4.7 together with Example
4.4 and formula (149). ∎

######  Remark 4.9.

A direct calculation of
$\chi^{\mathbb{T}}(X,mC^{\mathbb{T}}_{y}(X)\otimes{\mathcal{O}}_{X}(D))$,
without using localization techniques, can be obtained from [16][Lem.9.4.8 and
Thm.9.3.1], as follows:

(160)
$\chi^{\mathbb{T}}(X,mC^{\mathbb{T}}_{y}(X)\otimes{\mathcal{O}}_{X}(D))=\sum_{E\preceq
P}(1+y)^{\dim(E)}\cdot\sum_{m\in{\rm Relint}(E)\cap
M}\chi^{-m}\in{\mathbb{Z}}[M]\otimes_{\mathbb{Z}}{\mathbb{Z}}[y].$

This is a $K$-theoretic version of Corollary 3.15. For $y=0$, formula (160)
reduces to

$\chi^{\mathbb{T}}(X,{\mathcal{O}}_{X}(D))=\sum_{m\in P\cap
M}\chi^{-m}\in{\mathbb{Z}}[M],$

which can already be deduced from (38) and (39).

For $y=0$, formula (159) therefore reduces to Brion’s equality:

(161) $\sum_{m\in P\cap
M}\chi^{-m}=\chi^{\mathbb{T}}(X,{\mathcal{O}}_{X}(D))=\sum_{v\ \text{\rm
vertex}}\chi^{-v}\cdot\mathbb{S}\left(\sum_{m\in C_{v}\cap
M}\chi^{-m}\right).$

###  4.2. Localization in equivariant homology

Let $X_{\Sigma}$ be an $n$-dimensional toric variety with torus
${\mathbb{T}}=T_{N}$. Let $X=X_{\Sigma^{\prime}}$ be a
${\mathbb{T}}$-invariant closed algebraic subset of $X_{\Sigma}$, defined by a
star-closed subset $\Sigma^{\prime}\subset\Sigma$, such that the fixed-point
set $X^{\mathbb{T}}\neq\emptyset$.

By [15][Lem.8.4, Lem.8.5], the inclusion $i:X^{\mathbb{T}}\hookrightarrow X$
induces an injective morphism of $H^{*}_{\mathbb{T}}(pt;{\mathbb{Q}})$-modules

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

which becomes an isomorphism

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

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. [15][Cor.8.9]).

######  Remark 4.10.

In particular, by applying the localization isomorphism to both $X$ and
$X_{\Sigma}$, it follows that the homomorphism
$\widehat{H}_{*}^{\mathbb{T}}(X;{\mathbb{Q}})_{L}\to\widehat{H}_{*}^{\mathbb{T}}(X_{\Sigma};{\mathbb{Q}})_{L}$
induced by inclusion is injective. So in the following it suffices to work in
the localized homology
$\widehat{H}_{*}^{\mathbb{T}}(X_{\Sigma};{\mathbb{Q}})_{L}$ of the ambient
toric variety.

Let us now assume that $X=X_{\Sigma}$. 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 $i_{*}$ a projection map of
$H^{*}_{\mathbb{T}}(pt;{\mathbb{Q}})$-modules, called here the homological
localization map at $x$,

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

######  Remark 4.11.

If $X$ is a toric variety with ${H}_{*}^{\mathbb{T}}(X;{\mathbb{Q}})$ a
finitely generated free $(\Lambda_{\mathbb{T}})_{\mathbb{Q}}$-module (e.g.,
$X$ is a complete simplicial toric variety, see [16][Prop.12.4.7]), then the
localization map
$\widehat{H}_{*}^{\mathbb{T}}(X;{\mathbb{Q}})\to\widehat{H}_{*}^{\mathbb{T}}(X;{\mathbb{Q}})_{L}$
is injective.

The equivariant Todd class transformation ${\rm td}_{*}^{\mathbb{T}}$ commutes
with pushforward for closed inclusions $i_{*}$ and pullback $j^{*}$ under open
inclusions. So ${\rm td}_{*}^{\mathbb{T}}$ is compatible with the two diagrams
below:

(162)
$\setcounter{MaxMatrixCols}{11}\begin{CD}K^{\mathbb{T}}_{0}(X^{\mathbb{T}})@>{i_{*}}>{}>K_{0}^{\mathbb{T}}(X)\widehat{H}_{2*}^{\mathbb{T}}(X^{\mathbb{T}};{\mathbb{Q}})@>{i_{*}}>{}>\widehat{H}_{2*}^{\mathbb{T}}(X;{\mathbb{Q}})\\\
@V{}V{}V@V{}V{j^{*}}V@>{{\rm td}_{*}^{\mathbb{T}}}>{}>@V{}V{}V@V{}V{j^{*}}V\\\
K^{\mathbb{T}}_{0}(x_{\sigma})@>{i_{*}}>{}>K^{\mathbb{T}}_{0}(U_{\sigma})\widehat{H}_{2*}^{\mathbb{T}}(x_{\sigma};{\mathbb{Q}})@>{i_{*}}>{}>\widehat{H}_{2*}^{\mathbb{T}}(U_{\sigma};{\mathbb{Q}})\end{CD}$

To extend this property to the localized versions, we need to show that the
ring homomorphism ${\rm ch}^{\mathbb{T}}:{\mathbb{Z}}[M]\simeq
K_{\mathbb{T}}^{0}(pt)\to(\Lambda^{an}_{\mathbb{T}})_{\mathbb{Q}}\subset(\widehat{\Lambda}_{\mathbb{T}})_{\mathbb{Q}}$
given by the equivariant Chern character is compatible with the multiplicative
subsets used for localization. This fact follows as in [16][pag.644]; indeed,
if $0\neq m\in M$, then

${\rm
ch}^{\mathbb{T}}(1-\chi^{m})=1-e^{c(m)}=c(m)\frac{1-e^{c(m)}}{c(m)}\in(\Lambda^{an}_{\mathbb{T}})_{\mathbb{Q}}\subset(\widehat{\Lambda}_{\mathbb{T}})_{\mathbb{Q}}\>,$

with $c(m)\in L$ and the convergent power series
$\frac{1-e^{c(m)}}{c(m)}\in(\Lambda^{an}_{\mathbb{T}})_{\mathbb{Q}}\subset(\widehat{\Lambda}_{\mathbb{T}})_{\mathbb{Q}}$
being invertible (since its constant coefficient is a unit). So ${\rm
ch}^{\mathbb{T}}$ induces a ring homomorphism of localized rings

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

In particular, this localized version of the equivariant Chern character ${\rm
ch}^{\mathbb{T}}$ can be applied to the image of the summation map
$\mathbb{S}:{\mathbb{Z}}[[M]]_{\rm Sum}\to{\mathbb{Z}}[M]_{S}$, where by abuse
of notation we denote

$({\rm ch}^{\mathbb{T}}\circ\mathbb{S})\left(\sum_{m\in
M}a_{m}\cdot\chi^{m}\right)=:\mathbb{S}\left(\sum_{m\in M}a_{m}\cdot
e^{c(m)}\right),$

extending the corresponding formula ${\rm ch}^{\mathbb{T}}(\sum_{m\in
M}a_{m}\cdot\chi^{m})=\sum_{m\in M}a_{m}\cdot e^{c(m)}$ for finite sums in
${\mathbb{Z}}[M]\subset{\mathbb{Z}}[[M]]_{\rm Sum}$.

######  Remark 4.12.

In the context of diagram (145), we have to also consider the extension of the
Chern character to the complexified $K$-theory

${\rm ch}^{{\mathbb{T}}^{\prime}}:{\mathbb{C}}[M^{\prime}]\simeq
K_{{\mathbb{T}}^{\prime}}^{0}(pt)\otimes_{\mathbb{Z}}{\mathbb{C}}\to(\Lambda^{an}_{{\mathbb{T}}^{\prime}})_{\mathbb{C}}\subset(\widehat{\Lambda}_{{\mathbb{T}}^{\prime}})_{\mathbb{C}},$

which is compatible with localization

${\rm ch}^{{\mathbb{T}}^{\prime}}:{\mathbb{C}}[M^{\prime}]_{S^{\prime}}\simeq
K_{{\mathbb{T}}^{\prime}}^{0}(pt)_{S^{\prime}}\otimes_{\mathbb{Z}}{\mathbb{C}}\longrightarrow{L^{\prime}}^{-1}(\Lambda^{an}_{{\mathbb{T}}^{\prime}})_{\mathbb{C}}\subset{L^{\prime}}^{-1}(\widehat{\Lambda}_{{\mathbb{T}}^{\prime}})_{\mathbb{C}}\>.$

Here, $S^{\prime}\subset{\mathbb{C}}[M^{\prime}]$ is as before the
multiplicative subset generated by elements $1-a\cdot\chi^{m^{\prime}}$, for
$0\neq m^{\prime}\in M^{\prime}$ and $a\in{\mathbb{C}}^{*}$, and
$L^{\prime}\subset(\Lambda_{{\mathbb{T}}^{\prime}})_{\mathbb{C}}=H^{*}_{{\mathbb{T}}^{\prime}}(pt;{\mathbb{C}})$
is the multiplicative set generated by the elements $\pm a\cdot
c(m^{\prime})$, for $0\neq m^{\prime}\in M^{\prime}$ and
$a\in{\mathbb{C}}^{*}$. The compatibility with localization follows as above,
using in addition that

${\rm ch}^{{\mathbb{T}}^{\prime}}(1-a\cdot\chi^{m^{\prime}})=1-a\cdot
e^{c(m^{\prime})}\in(\Lambda^{an}_{{\mathbb{T}}^{\prime}})_{\mathbb{C}}\subset(\widehat{\Lambda}_{{\mathbb{T}}^{\prime}})_{\mathbb{C}}\>,$

for $a\neq 1$ and $0\neq m^{\prime}\in M^{\prime}$, is an invertible
convergent power series since its constant coefficient $1-a$ is a unit. These
two Chern characters ${\rm ch}^{{\mathbb{T}}^{\prime}}$ and ${\rm
ch}^{{\mathbb{T}}}$ fit into a commutative diagram

(163) $\begin{CD}{\mathbb{C}}[M^{\prime}]_{S^{\prime}}@>{{\rm
ch}^{{\mathbb{T}}^{\prime}}}>{}>{L^{\prime}}^{-1}(\Lambda^{an}_{{\mathbb{T}}^{\prime}})_{\mathbb{C}}@>{}>{}>{L^{\prime}}^{-1}(\widehat{\Lambda}_{{\mathbb{T}}^{\prime}})_{\mathbb{C}}\\\
@A{}A{}A@A{}A{}A@A{}A{}A\\\ {\mathbb{Z}}[M]_{S}@>{{\rm
ch}^{{\mathbb{T}}}}>{}>{L}^{-1}(\Lambda^{an}_{{\mathbb{T}}})_{\mathbb{Q}}@>{}>{}>{L}^{-1}(\widehat{\Lambda}_{{\mathbb{T}}})_{\mathbb{Q}}\
.\end{CD}$

The last two vertical arrows are injections (between convergent, resp., formal
Laurent series with rational or complex coefficients).

By the module property (81) of the equivariant Todd class transformation and
the functoriality of ${\rm ch}^{\mathbb{T}}$ under pullbacks, it follows that
${\rm td}_{*}^{\mathbb{T}}$ is compatible with the corresponding localized
versions of the diagrams appearing in (162), with the corresponding maps
labeled $i_{*}$ being now isomorphisms, as used in the definitions of the
localization maps $pr_{x}$ in $K$-theory and homology, respectively.

######  Proposition 4.13.

For a ${\mathbb{T}}$-equivariant coherent sheaf ${\mathcal{F}}$ on
$X=X_{\Sigma}$, and $x_{\sigma}\in X^{\mathbb{T}}$ a given fixed point of the
${\mathbb{T}}$-action, we have:

(164) ${\rm
td}_{*}^{\mathbb{T}}([{\mathcal{F}}])_{x_{\sigma}}:=pr_{x_{\sigma}}({\rm
td}_{*}^{\mathbb{T}}([{\mathcal{F}}])={\rm
ch}^{\mathbb{T}}((\mathbb{S}\circ\chi_{\sigma}^{\mathbb{T}})({\mathcal{F}}))\in
L^{-1}(\Lambda^{an}_{\mathbb{T}})_{\mathbb{Q}}\subset
L^{-1}(\widehat{\Lambda}_{\mathbb{T}})_{\mathbb{Q}}$

and

(165) $\begin{split}L^{-1}{\rm
td}_{*}^{\mathbb{T}}([{\mathcal{F}}])&=\bigoplus_{\sigma\in\Sigma(n)}{\rm
td}_{*}^{\mathbb{T}}([{\mathcal{F}}])_{x_{\sigma}}\cdot[x_{\sigma}]_{{\mathbb{T}}}\\\
&=\bigoplus_{\sigma\in\Sigma(n)}{\rm
ch}^{\mathbb{T}}((\mathbb{S}\circ\chi_{\sigma}^{\mathbb{T}})({\mathcal{F}}))\cdot[x_{\sigma}]_{{\mathbb{T}}}\in\widehat{H}_{*}^{\mathbb{T}}(X;{\mathbb{Q}})_{L}.\end{split}$

###### Proof.

Indeed, using the explicit calculation of the localization map in equivariant
$K$-theory (as in the previous section), we have:

$pr_{x_{\sigma}}({\rm td}_{*}^{\mathbb{T}}([{\mathcal{F}}])={\rm
td}_{*}^{\mathbb{T}}(pr_{x_{\sigma}}([{\mathcal{F}}]))={\rm
ch}^{\mathbb{T}}((\mathbb{S}\circ\chi_{\sigma}^{\mathbb{T}})({\mathcal{F}})).$

The second formula just follows from the isomorphism
$i_{*}:\widehat{H}_{*}^{\mathbb{T}}(X^{\mathbb{T}};{\mathbb{Q}})_{L}\simeq\widehat{H}_{*}^{\mathbb{T}}(X;{\mathbb{Q}})_{L}$.
∎

######  Example 4.14 (Localized equivariant Hirzebruch and Todd classes).

Assume now that $X_{\Sigma}$ is a quasi-projective toric variety. Let,
moreover, $X:=X_{\Sigma^{\prime}}$ be the ${\mathbb{T}}$-invariant closed
algebraic subset defined by a star-closed subset
$\Sigma^{\prime}\subset\Sigma$, with $x_{\sigma}\in X^{\mathbb{T}}$ a given
fixed point of the ${\mathbb{T}}$-action. By using (164) and (153), the
following identity holds in
$L^{-1}(\Lambda^{an}_{\mathbb{T}})_{\mathbb{Q}}\otimes_{\mathbb{Z}}{\mathbb{Z}}[y]\subset
L^{-1}(\widehat{\Lambda}_{\mathbb{T}})_{\mathbb{Q}}\otimes_{\mathbb{Z}}{\mathbb{Z}}[y]$:

(166) $\begin{split}T^{\mathbb{T}}_{y*}(X)_{x_{\sigma}}&:={\rm
td}_{*}^{\mathbb{T}}([mC_{y}^{\mathbb{T}}(X)])_{x_{\sigma}}=\sum_{\tau\preceq\sigma,\tau\in\Sigma^{\prime}}(1+y)^{\dim(O_{\tau})}\cdot({\rm
ch}^{\mathbb{T}}\circ\mathbb{S})\left(\sum_{m\in{\rm
Relint}(\sigma^{\vee}\cap\tau^{\perp})\cap M}\chi^{-m}\right)\\\
&=\sum_{\tau\preceq\sigma,\tau\in\Sigma^{\prime}}(1+y)^{\dim(O_{\tau})}\cdot\mathbb{S}\left(\sum_{m\in{\rm
Relint}(\sigma^{\vee}\cap\tau^{\perp})\cap M}e^{s(m)}\right)\>.\end{split}$

We thus have in
$\widehat{H}_{*}^{\mathbb{T}}(X;{\mathbb{Q}})_{L}\otimes_{\mathbb{Z}}{\mathbb{Z}}[y]$
(compare also with [42][Thm.5.3], [44][Thm.11.3]):

(167)
$L^{-1}T^{\mathbb{T}}_{y*}(X)=\bigoplus_{\sigma\in\Sigma^{\prime}(n)}\left(\sum_{\tau\preceq\sigma,\tau\in\Sigma^{\prime}}(1+y)^{\dim(O_{\tau})}\cdot\mathbb{S}\left(\sum_{m\in{\rm
Relint}(\sigma^{\vee}\cap\tau^{\perp})\cap
M}e^{s(m)}\right)\right)\cdot[x_{\sigma}]_{\mathbb{T}}.$

Specializing the above formulae to $y=0$ for $X$ quasi-projective, or applying
(164) to the structure sheaf ${\mathcal{F}}={\mathcal{O}}_{X}$ (extended by
zero to the ambient toric variety, which now can be arbitrary) we get the
following result:

(168) ${\rm
td}_{*}^{\mathbb{T}}(X)_{x_{\sigma}}=\sum_{\tau\preceq\sigma,\tau\in\Sigma^{\prime}}\mathbb{S}\left(\sum_{m\in{\rm
Relint}(\sigma^{\vee}\cap\tau^{\perp})\cap M}e^{s(m)}\right)\in
L^{-1}(\Lambda^{an}_{\mathbb{T}})_{\mathbb{Q}}\subset
L^{-1}(\widehat{\Lambda}_{\mathbb{T}})_{\mathbb{Q}}\>,$

and

(169) $L^{-1}{\rm
td}_{*}^{\mathbb{T}}(X)=\bigoplus_{\sigma\in\Sigma^{\prime}(n)}\left(\sum_{\tau\preceq\sigma,\tau\in\Sigma^{\prime}}\mathbb{S}\left(\sum_{m\in{\rm
Relint}(\sigma^{\vee}\cap\tau^{\perp})\cap
M}e^{s(m)}\right)\right)\cdot[x_{\sigma}]_{\mathbb{T}}\in\widehat{H}_{*}^{\mathbb{T}}(X;{\mathbb{Q}})_{L}.$

In the case $\Sigma^{\prime}=\Sigma$, formulas (168) and (169) reduce to a
result of Brylinski-Zhang [15][Thm.9.4]:

(170) ${\rm
td}_{*}^{\mathbb{T}}(X_{\Sigma})_{x_{\sigma}}=\mathbb{S}\left(\sum_{m\in\sigma^{\vee}\cap
M}e^{s(m)}\right)\in L^{-1}(\Lambda^{an}_{\mathbb{T}})_{\mathbb{Q}}\subset
L^{-1}(\widehat{\Lambda}_{\mathbb{T}})_{\mathbb{Q}}\>,$

and

(171) $L^{-1}{\rm
td}_{*}^{\mathbb{T}}(X_{\Sigma})=\bigoplus_{\sigma\in\Sigma(n)}\mathbb{S}\left(\sum_{m\in\sigma^{\vee}\cap
M}e^{s(m)}\right)\cdot[x_{\sigma}]_{\mathbb{T}}\in\widehat{H}_{*}^{\mathbb{T}}(X_{\Sigma};{\mathbb{Q}})_{L}.$

∎

######  Example 4.15 (Localized equivariant Hirzebruch classes for complements
of divisors).

Let $X=X_{\Sigma}$ be a toric variety with $V:=X\setminus D_{K}$, and
$D_{K}=\bigcup_{\rho\in K}D_{\rho}$ for $K\subset\Sigma(1)$. Let
$\sigma\in\Sigma(n)$ be a simplicial cone with $x_{\sigma}\in X^{\mathbb{T}}$
the corresponding ${\mathbb{T}}$-fixed point. Then we get by (157) and
Proposition 4.13 the following identity in
${L^{\prime}}^{-1}(\Lambda^{an}_{{\mathbb{T}}^{\prime}})_{\mathbb{C}}[y]\subset{L^{\prime}}^{-1}(\widehat{\Lambda}_{{\mathbb{T}}^{\prime}})_{\mathbb{C}}[y]$:

(172) $T^{\mathbb{T}}_{y*}([V\hookrightarrow X])_{x_{\sigma}}:={\rm
td}_{*}^{\mathbb{T}}([mC_{y}^{\mathbb{T}}([V\hookrightarrow
X])])_{x_{\sigma}}\\\ =\frac{1}{|G_{\sigma}|}\sum_{g\in
G_{\sigma}}\prod_{\rho_{i}\in\sigma(1)\cap K}\frac{(1+y)\cdot
a_{\rho_{i}}(g^{-1})\cdot
e^{-c(m^{\prime}_{\sigma,i})}}{1-a_{\rho_{i}}(g^{-1})\cdot
e^{-c(m^{\prime}_{\sigma,i})}}\cdot\prod_{\rho_{i}\in\sigma(1)\setminus
K}\frac{1+y\cdot a_{\rho_{i}}(g^{-1})\cdot
e^{-c(m^{\prime}_{\sigma,i})}}{1-a_{\rho_{i}}(g^{-1})\cdot
e^{-c(m^{\prime}_{\sigma,i})}}.$

In particular, if $\sigma$ is a smooth cone, then

$\begin{split}T^{\mathbb{T}}_{y*}([V\hookrightarrow
X])_{x_{\sigma}}=\prod_{\rho_{i}\in\sigma(1)\cap K}\frac{(1+y)\cdot
e^{-c(m_{\sigma,i})}}{1-e^{-c(m_{\sigma,i})}}\cdot\prod_{\rho_{i}\in\sigma(1)\setminus
K}\frac{1+y\cdot e^{-c(m_{\sigma,i})}}{1-e^{-c(m_{\sigma,i})}},\end{split}$

with $m^{\prime}_{\sigma,i}=m_{\sigma,i}$ for all $i$ (since $\sigma$ is a
smooth cone).

By specializing (172) to $y=0$, we get a local version of the Todd type
formula (139). In particular, for $y=0$ and $\sigma$ smooth, we get:

(173) $T^{\mathbb{T}}_{0*}([V\hookrightarrow
X])_{x_{\sigma}}=\prod_{\rho_{i}\in\sigma(1)\cap
K}\frac{e^{-c(m_{\sigma,i})}}{1-e^{-c(m_{\sigma,i})}}\cdot\prod_{\rho_{i}\in\sigma(1)\setminus
K}\frac{1}{1-e^{-c(m_{\sigma,i})}}.$

In the case when $K=\emptyset$, we thus obtain a more explicit expression for
the equivariant Hirzebruch and Todd classes of $X$ localized at $x_{\sigma}$.
∎

As a consequence of Proposition 4.13 and Theorem 4.7, we also have the
following.

######  Corollary 4.16.

Let $X=X_{\Sigma}$ be a complete toric variety of complex dimension $n$, with
${\mathcal{F}}$ a ${\mathbb{T}}$-coherent sheaf on $X$. Then the cohomological
Euler characteristic of ${\mathcal{F}}$ can be calculated via localization at
the ${\mathbb{T}}$-fixed points as:

(174)
$\begin{split}\chi^{\mathbb{T}}(X,{\mathcal{F}})&=\sum_{\sigma\in\Sigma(n)}({\rm
ch}^{\mathbb{T}}\circ\mathbb{S})(\chi_{\sigma}^{\mathbb{T}}({\mathcal{F}}|_{U_{\sigma}}))\\\
&=\sum_{\sigma\in\Sigma(n)}{\rm
td}_{*}^{\mathbb{T}}([{\mathcal{F}}])_{x_{\sigma}}\in
L^{-1}(\Lambda^{an}_{\mathbb{T}})_{\mathbb{Q}}\subset
L^{-1}(\widehat{\Lambda}_{\mathbb{T}})_{\mathbb{Q}}\>.\end{split}$

Before giving a more classical description of the homological localization map
$pr_{x}$ in the context of simplicial toric varieties, let us formulate the
following compatibility of
$pr_{x}:\widehat{H}_{*}^{\mathbb{T}}(X;{\mathbb{Q}})_{L}\to\widehat{H}_{*}^{\mathbb{T}}(x;{\mathbb{Q}})_{L}$
with cap products. As before, $X=X_{\Sigma}$ is a toric variety, with $x\in
X^{\mathbb{T}}$, and let $i_{x}:\\{x\\}\hookrightarrow X$ be 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:

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

This follows from the definition of $pr_{x}$, since the cap product $\cap$
commutes with pullback for open inclusions, together with the projection
formula for a closed inclusion.

For the rest of this section we assume that $X=X_{\Sigma}$ is a simplicial
toric variety, so that one has equivariant Poincaré duality with
${\mathbb{Q}}$-coefficients (recall that in the non-compact case,
${H}_{*}^{\mathbb{T}}$ denotes equivariant Borel-Moore homology):

$\cap[X]_{\mathbb{T}}:\widehat{H}^{*}_{\mathbb{T}}(X;{\mathbb{Q}})_{L}\overset{\sim}{\longrightarrow}\widehat{H}_{*}^{\mathbb{T}}(X;{\mathbb{Q}})_{L}.$

For this reason, in the following we use interchangeably homology and
cohomology, with a cap product in homology corresponding to multiplication in
cohomology.

######  Proposition 4.17.

Let $X=X_{\Sigma}$ be a simplicial toric variety of dimension $n$, with
$x_{\sigma}\in X^{\mathbb{T}}$ a torus fixed point and inclusion map
$i_{\sigma}:\\{x_{\sigma}\\}\hookrightarrow X$. Then

(176)
$pr_{x_{\sigma}}=\frac{i_{\sigma}^{*}}{Eu^{\mathbb{T}}_{X}(x_{\sigma})}:\widehat{H}^{*}_{\mathbb{T}}(X;{\mathbb{Q}})_{L}\to\widehat{H}^{*}_{\mathbb{T}}(x_{\sigma};{\mathbb{Q}})_{L},$

with the generalized Euler class of the fixed point $x_{\sigma}$ in $X$
defined by

$0\neq Eu^{\mathbb{T}}_{X}(x_{\sigma}):=i_{\sigma}^{*}\left({\rm
mult}(\sigma)\cdot\prod_{\rho\in\sigma(1)}[D_{\rho}]_{\mathbb{T}}\right)\in{\mathbb{Q}}\cdot
L.$

If $X$ is, moreover, a complete simplicial toric variety, let
$\int_{X}:\widehat{H}^{*}_{\mathbb{T}}(X;\mbox{$\mathbb{Q}$})_{L}\to\widehat{H}^{*}_{\mathbb{T}}(pt;\mbox{$\mathbb{Q}$})_{L}=L^{-1}(\widehat{\Lambda}_{\mathbb{T}})_{\mathbb{Q}}$
be the equivariant Gysin map (or, equivalently, the equivariant pushforward)
for the constant map $X\to pt$. Then

(177)
$\int_{X}=\sum_{\sigma\in\Sigma(n)}\frac{i_{\sigma}^{*}}{Eu^{\mathbb{T}}_{X}(x_{\sigma})}:\widehat{H}^{*}_{\mathbb{T}}(X;{\mathbb{Q}})_{L}\to\widehat{H}^{*}_{\mathbb{T}}(pt;{\mathbb{Q}})_{L}.$

###### Proof.

For $\sigma\in\Sigma(n)$ a cone of maximal dimension, formula (62) written in
homological terms becomes:

$\left({\rm
mult}(\sigma)\cdot\prod_{\rho\in\sigma(1)}[D_{\rho}]_{\mathbb{T}}\right)\cap[X]_{\mathbb{T}}={(i_{\sigma})}_{*}[x_{\sigma}]_{\mathbb{T}}.$

Using (175), we get

$i_{\sigma}^{*}\left({\rm
mult}(\sigma)\cdot\prod_{\rho\in\sigma(1)}[D_{\rho}]_{\mathbb{T}}\right)\cap
pr_{x_{\sigma}}([X]_{\mathbb{T}})=pr_{x_{\sigma}}({(i_{\sigma})}_{*}[x_{\sigma}]_{\mathbb{T}})=[x_{\sigma}]_{\mathbb{T}},$

where the last equality follows from the fact that the composition
$pr_{x_{\sigma}}\circ{(i_{\sigma})}_{*}$ is by definition just the identity.
The factor $i_{\sigma}^{*}\left({\rm
mult}(\sigma)\cdot\prod_{\rho\in\sigma(1)}[D_{\rho}]_{\mathbb{T}}\right)$ is
by definition the generalized Euler class $0\neq
Eu^{\mathbb{T}}_{X}(x_{\sigma})$ (this is non-zero by the formula above, since
$[x_{\sigma}]_{\mathbb{T}}\neq 0$), and it is an element in ${\mathbb{Q}}\cdot
L$ (see [16][Lem..13.3.5]) since any divisor on a simplicial toric variety is
${\mathbb{Q}}$-Cartier ([16][Prop.4.2.7]). So formula (176) follows now from
the projection formula (175), i.e., for
$a\in\widehat{H}^{*}_{\mathbb{T}}(X;{\mathbb{Q}})_{L}$, we have:

$pr_{x_{\sigma}}(a\cap[X]_{\mathbb{T}})=i_{\sigma}^{*}(a)\cap\frac{[x_{\sigma}]_{\mathbb{T}}}{Eu^{\mathbb{T}}_{X}(x_{\sigma})}.$

For the second assertion, note that the map
$\int_{X}:\widehat{H}^{*}_{\mathbb{T}}(X;\mbox{$\mathbb{Q}$})_{L}\to\widehat{H}^{*}_{\mathbb{T}}(pt;\mbox{$\mathbb{Q}$})_{L}=L^{-1}(\widehat{\Lambda}_{\mathbb{T}})_{\mathbb{Q}}$
corresponds (by the functoriality of pushforward) under the localization
isomorphism

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

to the sum of localization maps $pr_{x}$, $x\in X^{\mathbb{T}}$, i.e.,

$\int_{X}=\sum_{\sigma\in\Sigma(n)}pr_{x_{\sigma}}:\widehat{H}^{*}_{\mathbb{T}}(X;{\mathbb{Q}})_{L}\to\widehat{H}^{*}_{\mathbb{T}}(pt;{\mathbb{Q}})_{L}.$

∎

######  Remark 4.18.

If the fixed point $x_{\sigma}$ corresponds to a smooth cone
$\sigma\in\Sigma$, then ${\rm mult}(\sigma)=1$ and the divisors $D_{\rho}$
($\rho\in\sigma(1)$) are Cartier divisors. Moreover, the normal bundle
$N_{x_{\sigma}}U_{\sigma}$ of $x_{\sigma}$ in the smooth affine variety
$U_{\sigma}$ is given by

$N_{x_{\sigma}}U_{\sigma}=\bigoplus_{\rho\in\sigma(1)}{\mathcal{O}}(D_{\rho})|_{x_{\sigma}}.$

Hence
$Eu^{\mathbb{T}}_{X}(x_{\sigma})=c^{\mathbb{T}}_{n}(N_{x_{\sigma}}U_{\sigma})$
is the classical equivariant Euler class of the fixed point $x_{\sigma}$,
given by the top equivariant Chern class of the normal bundle
$N_{x_{\sigma}}U_{\sigma}$.

Translating Theorem 4.7 into the (co)homological context, using Proposition
4.13 and, in the simplicial context, Proposition 4.17, we get the following
consequence.

######  Corollary 4.19.

Let $X=X_{\Sigma}$ be a complete toric variety, $D$ a ${\mathbb{T}}$-invariant
Cartier divisor, with ${\mathcal{F}}$ a ${\mathbb{T}}$-equivariant coherent
sheaf on $X$. Then the cohomological equivariant Euler characteristic of
${\mathcal{F}}$ is computed by

(178)
$\chi^{\mathbb{T}}(X,{\mathcal{O}}_{X}(D)\otimes{\mathcal{F}})=\sum_{\sigma\in\Sigma(n)}pr_{x_{\sigma}}({\rm
td}_{*}^{\mathbb{T}}([{\mathcal{O}}_{X}(D)\otimes{\mathcal{F}}])\in
L^{-1}(\Lambda^{an}_{\mathbb{T}})_{\mathbb{Q}}.$

If, moreover, $X$ is a simplicial toric variety, then

(179)
$\chi^{\mathbb{T}}(X,{\mathcal{O}}_{X}(D)\otimes{\mathcal{F}})=\sum_{\sigma\in\Sigma(n)}\frac{i_{\sigma}^{*}(e^{[D]_{\mathbb{T}}}{\rm
td}_{*}^{\mathbb{T}}([{\mathcal{F}}]))}{Eu_{X}^{\mathbb{T}}(x_{\sigma})}\in
L^{-1}(\Lambda^{an}_{\mathbb{T}})_{\mathbb{Q}}.$

###  4.3. Equivariant Hirzebruch classes of simplicial toric varieties via
localization

In this section, we explain how to reprove Theorems 3.22, 3.28 and Lemma 3.29
by localization techniques, instead of using the equivariant Lefschetz-
Riemann-Roch theorem of [24][Thm.3.1 and Rem.4.3] (as done in Section 3.3).

Let $X:=X_{\Sigma}$ be a simplicial quasi-projective toric variety associated
to a fan $\Sigma$ of full-dimensional convex support. Then the equivariant
cohomology of $X$ is a free $(\Lambda_{\mathbb{T}})_{\mathbb{Q}}$-module (by
(63)), so that the injectivity part of the sequence (65), resp., (66), still
holds for the equivariant cohomology (as pointed out in Remark 2.8). Hence the
corresponding map for the completed equivariant cohomology rings is injective,
i.e.,

$\bigoplus_{\sigma\in\Sigma(n)}i_{\sigma}^{*}:\widehat{H}^{*}_{\mathbb{T}}(X;{\mathbb{Q}})\hookrightarrow\bigoplus_{\sigma\in\Sigma(n)}\widehat{H}^{*}_{\mathbb{T}}(x_{\sigma};{\mathbb{Q}}).$

By localizing at the multiplicative set $L$, and using the exactness of
localization, we get an injective map

$\bigoplus_{\sigma\in\Sigma(n)}i_{\sigma}^{*}:\widehat{H}^{*}_{\mathbb{T}}(X;{\mathbb{Q}})_{L}\hookrightarrow\bigoplus_{\sigma\in\Sigma(n)}\widehat{H}^{*}_{\mathbb{T}}(x_{\sigma};{\mathbb{Q}})_{L}.$

So it is enough to check these characteristic class formulae by using for each
fixed point $x_{\sigma}$, $\sigma\in\Sigma(n)$, the induced restriction map
$pr_{x_{\sigma}}$

(180)
$\widehat{H}^{*}_{\mathbb{T}}(X;{\mathbb{Q}})\to\widehat{H}^{*}_{\mathbb{T}}(X;{\mathbb{Q}})_{L}\overset{pr_{x_{\sigma}}}{\to}\widehat{H}^{*}_{\mathbb{T}}(x_{\sigma};{\mathbb{Q}})_{L}\to\widehat{H}^{*}_{{\mathbb{T}}^{\prime}}(x_{\sigma};{\mathbb{C}})_{L^{\prime}},$

with the middle arrow $pr_{x_{\sigma}}$ as in (176). Even the direct sum
$\bigoplus_{\sigma\in\Sigma(n)}pr_{x_{\sigma}}$ of these induced restriction
maps is still injective, since the localization map on the integral domain
$\widehat{H}^{*}_{\mathbb{T}}(x_{\sigma};{\mathbb{Q}})\to\widehat{H}^{*}_{\mathbb{T}}(x_{\sigma};{\mathbb{Q}})_{L}$
is injective, and
$pr_{x_{\sigma}}=\frac{i_{\sigma}^{*}}{Eu^{\mathbb{T}}_{X}(x_{\sigma})}$
differs from $i_{\sigma}^{*}$ by the unit

$Eu^{\mathbb{T}}_{X}(x_{\sigma})=i_{\sigma}^{*}\left({\rm
mult}(\sigma)\cdot\prod_{\rho\in\sigma(1)}[D_{\rho}]_{\mathbb{T}}\right)=|G_{\sigma}|\prod_{\rho\in\sigma(1)}i_{\sigma}^{*}F_{\rho}\in\widehat{H}^{*}_{\mathbb{T}}(x_{\sigma};{\mathbb{Q}})_{L}.$

Moreover, as mentioned after diagram (163), no information is lost if we
consider complex instead of rational coefficients.

####  4.3.1. Proof of Lemma 3.29 by localization

In order to reprove formula (138) by localization, we start with the local
version discussed in the following example.

######  Example 4.20.

Let $\sigma\in\Sigma(n)$ be a simplicial cone with minimal generators
$\\{m_{\sigma,i}\\}_{i}$ of $\sigma^{\vee}$ as in the context of diagram
(145). For
${\mathcal{F}}:=\pi_{*}^{G_{\sigma}}({\mathcal{O}}_{U^{\prime}_{\sigma}}\otimes{\mathbb{C}}_{\chi^{m^{\prime}}})$
with $m^{\prime}\in M^{\prime}$, we get by Proposition 4.13 and formula (148)
the identity

(181) ${\rm
td}^{\mathbb{T}}_{*}([{\mathcal{F}}])_{x_{\sigma}}=\frac{1}{|G_{\sigma}|}\sum_{g\in
G_{\sigma}}\chi^{m^{\prime}}(g)e^{c(m^{\prime})}\prod_{i=1}^{n}\frac{1}{1-a_{\rho_{i}}(g^{-1})\cdot
e^{-c(m^{\prime}_{\sigma,i})}}\in\widehat{H}^{*}_{{\mathbb{T}}^{\prime}}(x_{\sigma};{\mathbb{C}})_{L^{\prime}}.$

∎

Using Example 4.20, we can now reprove formula (138) of Lemma 3.29 by
localization techniques. Recall that the global formula

(182) ${\rm
td}_{*}^{\mathbb{T}}(\pi_{*}({\mathcal{O}}_{W}\otimes{\mathbb{C}}_{\chi^{\widetilde{m}}})^{G})=\sum_{g\in
G_{\Sigma}}\chi^{\widetilde{m}}(g^{-1})\prod_{\rho\in\Sigma(1)}\frac{F_{\rho}\cdot
e^{\langle\widetilde{m},e_{\rho}\rangle\cdot
F_{\rho}}}{1-a_{\rho}(g)e^{-F_{\rho}}}\in\widehat{H}^{*}_{\mathbb{T}}(X;{\mathbb{Q}}),$

with $\\{e_{\rho}\\}_{\rho\in\Sigma(1)}$ the standard basis of the lattice
$\widetilde{N}={\mathbb{Z}}^{|\Sigma(1)|}$ and $\widetilde{m}$ in the dual
character lattice $\widetilde{M}$, is formulated in terms of the Cox
construction $\pi:W\to X$. So let us now compare, for a fixed
$\sigma\in\Sigma(n)$ with $U_{\sigma}$ the ${\mathbb{T}}$-invariant open
affine subset of $X$ containing the corresponding fixed point $x_{\sigma}$,
the local and global formulae (181) and, resp., (182).

To simplify the notation, assume $\sigma(1)=\\{\rho_{1},\ldots,\rho_{n}\\}$.
We next adapt the arguments of [13][page 24] to our context. If $g\in
G_{\Sigma}\setminus G_{\sigma}$, by (31) there exists a $\rho\in\Sigma(1)$
with $a_{\rho}(g)\neq 1$, so that the restriction of $\frac{F_{\rho}\cdot
e^{\langle\widetilde{m},e_{\rho}\rangle\cdot
F_{\rho}}}{1-a_{\rho}(g)e^{-F_{\rho}}}$ to $U_{\sigma}$ becomes $0$. Thus, the
summation on the right-hand side of (182) reduces after restriction to
$U_{\sigma}$ to a summation over ${g\in G_{\sigma}}$. If $g\in G_{\sigma}$ and
$\rho\notin\sigma(1)$, then $a_{\rho}(g)=1$, so that the restriction of
$\frac{F_{\rho}\cdot e^{\langle\widetilde{m},e_{\rho}\rangle\cdot
F_{\rho}}}{1-a_{\rho}(g)e^{-F_{\rho}}}$ to $U_{\sigma}$ becomes $1$. So the
product on the right hand side of (182) reduces to a product over
$\rho\in\sigma(1)$. As in the proof of formula (125), we have
$\pi^{-1}(U_{\sigma})=U^{\prime}_{\sigma}\times({\mathbb{C}}^{*})^{r-n}$, with
$\widetilde{{\mathbb{T}}}\simeq{\mathbb{T}}^{\prime}\times({\mathbb{C}}^{*})^{r-n}$,
$G\simeq G_{\sigma}\times({\mathbb{C}}^{*})^{r-n}$, and
$U^{\prime}_{\sigma}/G_{\sigma}=U_{\sigma}$. Let
$\widetilde{m}=m^{\prime}+m^{\prime\prime}\in M^{\prime}\oplus
M^{\prime\prime}$ be the corresponding character decomposition. Note that
${\mathbb{C}}_{\chi^{m^{\prime\prime}}}$ is a trivial
$({\mathbb{C}}^{*})^{r-n}$-equivariant line bundle on
$({\mathbb{C}}^{*})^{r-n}$, so we can assume that $m^{\prime\prime}=0$ is the
trivial character. By the projection formula, the restriction of
$\pi_{*}({\mathcal{O}}_{W}\otimes{\mathbb{C}}_{\chi^{\widetilde{m}}})^{G}$ to
$U_{\sigma}$ is isomorphic to the ${\mathcal{F}}$ appearing in (181). Finally,

$\displaystyle pr_{x_{\sigma}}\left(\sum_{g\in
G_{\Sigma}}\chi^{\widetilde{m}}(g^{-1})\prod_{\rho\in\Sigma(1)}\frac{F_{\rho}\cdot
e^{\langle\widetilde{m},e_{\rho}\rangle\cdot
F_{\rho}}}{1-a_{\rho}(g)e^{-F_{\rho}}}\right)$
$\displaystyle=\frac{1}{Eu_{X}^{\mathbb{T}}(x_{\sigma})}\cdot{i_{\sigma}^{*}}\left(\sum_{g\in
G_{\Sigma}}\chi^{\widetilde{m}}(g^{-1})\prod_{\rho\in\Sigma(1)}\frac{F_{\rho}\cdot
e^{\langle\widetilde{m},e_{\rho}\rangle\cdot
F_{\rho}}}{1-a_{\rho}(g)e^{-F_{\rho}}}\right)$
$\displaystyle=\frac{1}{Eu_{X}^{\mathbb{T}}(x_{\sigma})}\cdot{i_{\sigma}^{*}}\left(\sum_{g\in
G_{\sigma}}\chi^{m^{\prime}}(g^{-1})\prod_{i=1}^{n}\frac{F_{\rho_{i}}\cdot
e^{\langle{m^{\prime}},e_{\rho_{i}}\rangle\cdot
F_{\rho_{i}}}}{1-a_{\rho_{i}}(g)e^{-F_{\rho_{i}}}}\right)$
$\displaystyle=\frac{1}{|G_{\sigma}|}\sum_{g\in
G_{\sigma}}\chi^{m^{\prime}}(g^{-1})\cdot
i_{\sigma}^{*}\left(\prod_{i=1}^{n}\frac{e^{\langle{m^{\prime}},e_{\rho_{i}}\rangle\cdot
F_{\rho_{i}}}}{1-a_{\rho_{i}}(g)e^{-F_{\rho_{i}}}}\right)$
$\displaystyle=\frac{1}{|G_{\sigma}|}\sum_{g\in
G_{\sigma}}\chi^{m^{\prime}}(g)e^{c(m^{\prime})}\prod_{i=1}^{n}\frac{1}{1-a_{\rho_{i}}(g^{-1})\cdot
e^{-c(m^{\prime}_{\sigma,i})}},$

where the last equality uses (67) for the torus ${\mathbb{T}}^{\prime}$ to
show that $i_{\sigma}^{*}F_{\rho_{i}}=c(m^{\prime}_{\sigma,i})$ and
$c(m^{\prime})=\sum_{i=1}^{n}\langle m^{\prime},e_{\rho_{i}}\rangle
i^{*}_{\sigma}F_{\rho_{i}}$, as well as changing $g$ by $g^{-1}$ in
$G_{\sigma}$. Also note that $a_{\rho}(g)$ for $g\in G_{\Sigma}$ and
$\rho\in\Sigma(1)$ fits with the corresponding $a_{\rho}(g)$ for $g\in
G_{\sigma}$ and $\rho\in\sigma(1)$, as in (33).

Altogether, the local formula (181) is obtained from the global formula (182)
upon applying the homological localization map $pr_{x_{\sigma}}$ of (180). ∎

####  4.3.2. Proof of Theorem 3.28 by localization

We can now explain a proof of formula (136) of Theorem 3.28 via localization.
With $X:=X_{\Sigma}$ as before, let $V\subset X$ be the open complement of the
divisor $D_{K}:=\bigcup_{\rho\in K}D_{\rho}$, for $K\subset\Sigma(1)$. Recall
that the equivariant Hirzebruch class ${T}^{\mathbb{T}}_{y*}([V\hookrightarrow
X])$ is computed by:

(183) ${T}^{\mathbb{T}}_{y*}([V\hookrightarrow X])=(1+y)^{n-r}\cdot\sum_{g\in
G_{\Sigma}}\prod_{\rho\in K}\frac{F_{\rho}\cdot(1+y)\cdot a_{\rho}(g)\cdot
e^{-F_{\rho}}}{1-a_{\rho}(g)\cdot e^{-F_{\rho}}}\prod_{\rho\notin
K}\frac{F_{\rho}\cdot\big{(}1+y\cdot a_{\rho}(g)\cdot
e^{-F_{\rho}}\big{)}}{1-a_{\rho}(g)\cdot e^{-F_{\rho}}}\>,$

with $r=|\Sigma(1)|$, and $F_{\rho}=[D_{\rho}]_{\mathbb{T}}$ the equivariant
fundamental class of the ${\mathbb{T}}$-invariant divisor $D_{\rho}$
corresponding to the ray $\rho\in\Sigma(1)$. As above, it suffices to prove
the equality of both sides of formula (183) after applying the homological
localization map $pr_{x_{\sigma}}$ of (180), with
$pr_{x_{\sigma}}(T^{\mathbb{T}}_{y*}([V\hookrightarrow
X]))=T^{\mathbb{T}}_{y*}([V\hookrightarrow X])_{x_{\sigma}}$ computed by
formula (172).

If $g\in G_{\Sigma}\setminus G_{\sigma}$, by (31) there exists a
$\rho\in\Sigma(1)$ with $a_{\rho}(g)\neq 1$, so that one factor on the
restriction of the right-hand side of (183) to $U_{\sigma}$ becomes $0$. Hence
the summation on the right-hand side of (183) reduces after restriction to
$U_{\sigma}$ to a summation over ${g\in G_{\sigma}}$. If $g\in G_{\sigma}$ and
$\rho\notin\sigma(1)$, then $a_{\rho}(g)=1$, so that the restriction of the
factor $\frac{F_{\rho}\cdot(1+y)\cdot a_{\rho}(g)\cdot
e^{-F_{\rho}}}{1-a_{\rho}(g)\cdot e^{-F_{\rho}}}$ (for $\rho\in K$) or
$\frac{F_{\rho}\cdot\big{(}1+y\cdot a_{\rho}(g)\cdot
e^{-F_{\rho}}\big{)}}{1-a_{\rho}(g)\cdot e^{-F_{\rho}}}$ (for $\rho\notin K$)
to $U_{\sigma}$ becomes $1+y$. With $r-n=|\Sigma(1)\setminus\sigma(1)|$, this
will in turn cancel the factor $(1+y)^{n-r}$ of formula (183). Therefore, we
get

$\displaystyle pr_{x_{\sigma}}\left((1+y)^{n-r}\cdot\sum_{g\in
G_{\Sigma}}\prod_{\rho\in K}\frac{F_{\rho}\cdot(1+y)\cdot a_{\rho}(g)\cdot
e^{-F_{\rho}}}{1-a_{\rho}(g)\cdot e^{-F_{\rho}}}\prod_{\rho\notin
K}\frac{F_{\rho}\cdot\big{(}1+y\cdot a_{\rho}(g)\cdot
e^{-F_{\rho}}\big{)}}{1-a_{\rho}(g)\cdot e^{-F_{\rho}}}\right)$
$\displaystyle=\frac{1}{Eu_{X}^{\mathbb{T}}(x_{\sigma})}\cdot{i_{\sigma}^{*}}\left(\sum_{g\in
G_{\sigma}}\prod_{\rho\in\sigma(1)\cap K}\frac{F_{\rho}\cdot(1+y)\cdot
a_{\rho}(g)\cdot e^{-F_{\rho}}}{1-a_{\rho}(g)\cdot
e^{-F_{\rho}}}\prod_{\rho\in\sigma(1)\setminus
K}\frac{F_{\rho}\cdot\big{(}1+y\cdot a_{\rho}(g)\cdot
e^{-F_{\rho}}\big{)}}{1-a_{\rho}(g)\cdot e^{-F_{\rho}}}\right)$
$\displaystyle=\frac{1}{|G_{\sigma}|}\sum_{g\in
G_{\sigma}}{i_{\sigma}^{*}}\left(\prod_{\rho\in\sigma(1)\cap
K}\frac{(1+y)\cdot a_{\rho}(g)\cdot e^{-F_{\rho}}}{1-a_{\rho}(g)\cdot
e^{-F_{\rho}}}\prod_{\rho\in\sigma(1)\setminus K}\frac{1+y\cdot
a_{\rho}(g)\cdot e^{-F_{\rho}}}{1-a_{\rho}(g)\cdot e^{-F_{\rho}}}\right).$

This expression reduces to formula (172), as desired, after using (67) for the
torus ${\mathbb{T}}^{\prime}$ to show that for $\rho=\rho_{i}$ the rays of
$\sigma(1)$, one gets as before
$i_{\sigma}^{*}F_{\rho_{i}}=c(m^{\prime}_{\sigma,i})$, as well as by changing
$g$ by $g^{-1}$ in $G_{\sigma}$.∎

######  Remark 4.21.

Specializing to $K=\emptyset$, the above arguments also reprove formula (126)
of Theorem 3.22.

##  5\. Euler-Maclaurin formulae via equivariant Hirzebruch-Riemann-Roch

An Euler-Maclaurin formula relates the sum $\sum_{m\in P\cap M}f(m)$ of the
values of a suitable function $f$ at the lattice points in a polyhedron
$P\subset M_{\mathbb{R}}:=M\otimes\mbox{$\mathbb{R}$}$ to integrals over
polyhedra. Here, we are interested in the case where $P$ is a polytope or a
cone.

We begin this section with a short overview of some of the literature
describing relations between the equivariant Hirzebruch-Riemann-Roch theorem
and Euler-Maclaurin type formulae. We aim to relate the equivariant geometric
context for projective simplicial toric varieties $X=X_{\Sigma}$ to a purely
combinatorial geometric context for polytopes $P\subset M_{\mathbb{R}}$. The
corresponding results for pointed cones will fit with the localization results
developed in the previous section.

We further derive an abstract Euler-Maclaurin type formula, inspired by a
corresponding proof of Brion-Vergne in the classical case. Our abstract
formula is based on the equivariant Hirzebruch-Riemann-Roch theorem and the
motivic formalism of equivariant characteristic classes developed in the
previous sections. It can be specialized to various situations, recovering
many known Euler-Maclaurin type formulae as well as obtaining several new ones
in a uniform way dictated by toric geometry.

The theory of valuations for rational polyhedra $P\subset M_{\mathbb{R}}$ was
applied by Khovanskii and Pukhlikov in [38] to obtain Euler-Maclaurin type
formulae for Delzant polytopes (corresponding to smooth projective toric
varieties) in terms of infinitesimal movements $P(h)$ of the polytope. More
precisely, for a polynomial $f$ one has the identity:

(184) $Todd(\frac{\partial}{\partial h})\left(\int_{P(h)}f(m)\
dm\right)_{|_{h=0}}=\sum_{m\in P\cap M}f(m)\>.$

Here, if the polytope $P$ is the lattice polytope defined by inequalities of
the form

$\langle m,u_{F}\rangle+c_{F}\geq 0,$

with $u_{F}$ the facet normals, the polytope $P(h)$ with shifted faces is
defined by inequalities

$\langle m,u_{F}\rangle+c_{F}+h_{F}\geq 0,$

with $h=(h_{F})_{F}$ a vector with real entries indexed by the facets $F$ of
$P$. Moreover, $Todd(\frac{\partial}{\partial h})$ is the differential
operator defined by:

$Todd(\frac{\partial}{\partial h}):=\prod_{F\ {\rm
facet}}\frac{\frac{\partial}{\partial h_{F}}}{1-e^{-\frac{\partial}{\partial
h_{F}}}}.$

The relation between formula (184) and the equivariant Hirzebruch-Riemann-Roch
theorem is only indicated in [38], but not used (but see, e.g., the proof of
[16][Thm.13.5.6]). Moreover, it is clearly pointed out that exponential
functions $f(x)=e^{\langle x,z\rangle}$ are needed to work with rational
polyhedral cones. From this, one then gets the result for polynomials $f$
(compare also with [16][Sect.13.5]).

For a direct extension of (184) to simple lattice polytopes, with a
corresponding more complicated Todd operator $Todd(\frac{\partial}{\partial
h})$, see [12] as well as the nice survey [34]. The approach to the Euler-
Maclaurin formula (184) through the equivariant Hirzebruch-Riemann-Roch
theorem for projective simplicial toric varieties corresponding to simple
lattice polytopes is due to Brion-Vergne, see [13][Thm.4.5].

In the following, we explain a uniform approach to various Euler-Maclaurin
formulae for simple lattice polytopes or pointed cones, formulated in terms of
dilatations of the polytope or cone. To have a unified language, assume $P$ is
a full-dimensional simple lattice polyhedron in $M_{\mathbb{R}}$ with
associated (semi-projective) toric variety $X=X_{P}$ and inner normal fan
$\Sigma=\Sigma_{P}$. Let $P(h)$ be the dilatation of $P$ with respect to the
vector $h=(h_{\rho})_{\rho\in\Sigma(1)}$ with real entries indexed by the rays
of $\Sigma$ (as we use the related toric geometry), or equivalently, by the
facets of $P$ (if one would like to formulate everything directly in terms of
$P$). So, if $P$ is defined by inequalities of the form

$\langle m,u_{\rho}\rangle+c_{\rho}\geq 0,$

with $u_{\rho}$ the ray generators and $c_{\rho}\in{\mathbb{Z}}$, for each
$\rho\in\Sigma(1)$, then $P(h)$ is defined by inequalities

$\langle m,u_{\rho}\rangle+c_{\rho}+h_{\rho}\geq 0,$

for each $\rho\in\Sigma(1)$. For later use in the context of weighted Euler-
Maclaurin formulae, we also define a parametrized dilatation $P_{y}(h)$ of $P$
by:

$P_{y}(h):=\\{m\in M\mid\langle m,u_{\rho}\rangle+(1+y)\cdot
c_{\rho}+h_{\rho}\geq 0,\ {\rm for\ all}\ \rho\in\Sigma(1)\\},$

with $h=(h_{\rho})_{\rho}$ as before a vector with real entries indexed by the
rays $\rho$ of $\Sigma$. Note that $P_{0}(h)=P(h)$, and $P_{y}(0)=:P_{y}$ is
the dilatation of $P$ by the factor $1+y$.

Let us first consider the case when $P$ is a full-dimensional lattice polytope
in $M_{\mathbb{R}}$. In what follows, we adapt the arguments of Brion-Vergne
[13][Thm.4.5] to our context. If $h$ is in a small enough open neighborhood
$U$ of the origin in ${\mathbb{R}}^{r}$ (with $r$ the number of facets of
$P$), then $P(h)$ is again a simple polytope of the same combinatorial type,
and similarly for $P_{y}(h)$. Let us fix $h\in U\cap{\mathbb{Q}}^{r}$ and
$y\in{\mathbb{Q}}$, and choose $k\in{\mathbb{N}}$ so that $k\cdot P_{y}(h)$ is
a lattice polytope in $M_{\mathbb{R}}$ with ample Cartier divisor

$D_{k\cdot P_{y}(h)}=k\cdot
D_{P_{y}(h)}=k\left(\sum_{\rho\in\Sigma(1)}((1+y)\cdot c_{\rho}+h_{\rho})\cdot
D_{\rho}\right),$

where $D_{P}=\sum_{\rho\in\Sigma(1)}c_{\rho}\cdot D_{\rho}$ is the ample
Cartier divisor of $P$. On a simplicial toric variety any divisor is
${\mathbb{Q}}$-Cartier ([16][Prop.4.2.7]), so that $D_{P_{y}(h)}$ has an
equivariant fundamental class with rational coefficients satisfying
$k\cdot[D_{P_{y}(h)}]_{\mathbb{T}}:=[k\cdot
D_{P_{y}(h)}]_{\mathbb{T}}=c^{\mathbb{T}}_{1}({\mathcal{O}}_{X}(k\cdot
D_{P_{y}(h)}))$, e.g., see [16][Prop.12.4.13]. Applying formula (84) for
$k\cdot P(h)$, and using the fact that the associated toric variety does not
change under polytope dilatation, we get

(185) $\sum_{m\in k\cdot P_{y}(h)\cap
M}e^{s(m)}=\int_{X}e^{k\cdot[D_{P_{y}(h)}]_{\mathbb{T}}}\cap{\rm
td}^{\mathbb{T}}_{*}(X)\in(\Lambda^{an}_{\mathbb{T}})_{\mathbb{Q}}\subset(\widehat{\Lambda}_{\mathbb{T}})_{\mathbb{Q}}\>.$

A localized version of formula (185) can be derived from (177) as:

(186) $\sum_{m\in k\cdot P_{y}(h)\cap
M}e^{s(m)}=\sum_{\sigma\in\Sigma(n)}\frac{i_{\sigma}^{*}\left(e^{k\cdot[D_{P_{y}(h)}]_{\mathbb{T}}}\right)}{Eu^{\mathbb{T}}_{X}(x_{\sigma})}\cdot
i_{\sigma}^{*}({\rm td}^{\mathbb{T}}_{*}(X))\in
L^{-1}(\Lambda^{an}_{\mathbb{T}})_{\mathbb{Q}}\subset
L^{-1}(\widehat{\Lambda}_{\mathbb{T}})_{\mathbb{Q}}\>.$

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

$\langle-,z\rangle:Sym_{\mbox{$\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 $N_{\mathbb{R}}$. Assume now that $z$ is chosen so that
$\langle Eu^{\mathbb{T}}_{X}(x_{\sigma}),z\rangle\neq 0$ for each
$\sigma\in\Sigma(n)$, i.e., $\langle i_{\sigma}^{*}F_{\rho},z\rangle\neq 0$,
for each ray $\rho\in\sigma(1)$ of $\sigma\in\Sigma(n)$. Applying the function
$\langle-,\frac{1}{k}\cdot z\rangle$ to (186), we get

(187) $\sum_{m\in k\cdot P_{y}(h)\cap M}e^{\langle m,\frac{1}{k}\cdot
z\rangle}=\sum_{\sigma\in\Sigma(n)}\frac{e^{\langle
i_{\sigma}^{*}[D_{P_{y}(h)}]_{\mathbb{T}},z\rangle}}{\langle
Eu^{\mathbb{T}}_{X}(x_{\sigma}),\frac{1}{k}\cdot z\rangle}\cdot\langle
i^{*}_{\sigma}({\rm td}^{\mathbb{T}}_{*}(X)),\frac{1}{k}\cdot z\rangle.$

Note that by the Riemann sum approximation of an integral, we have:

$\lim_{k\to\infty}\frac{1}{k^{n}}\sum_{m\in k\cdot P_{y}(h)\cap M}e^{\langle
m,\frac{1}{k}\cdot z\rangle}=\lim_{k\to\infty}\frac{1}{k^{n}}\sum_{m\in
P_{y}(h)\cap\frac{1}{k}M}e^{\langle m,z\rangle}=\int_{P_{y}(h)}e^{\langle
m,z\rangle}\ dm,$

with the Lebesgue measure $dm$ normalized so that the unit cube in $M\subset
M_{\mathbb{R}}$ has volume $1$ (which explains the use of the factor
$\frac{1}{k^{n}}$). We next study the limits on the right-hand side of (187).
Note that

${k^{n}}\cdot\langle Eu^{\mathbb{T}}_{X}(x_{\sigma}),\frac{1}{k}\cdot
z\rangle=\langle Eu^{\mathbb{T}}_{X}(x_{\sigma}),z\rangle$

since $\langle Eu^{\mathbb{T}}_{X}(x_{\sigma}),z\rangle$ is a (non-zero)
homogeneous polynomial of degree $n$ in $z$. Finally, we have that

(188) $\lim_{k\to\infty}\langle i^{*}_{\sigma}({\rm
td}^{\mathbb{T}}_{*}(X)),\frac{1}{k}\cdot z\rangle=1.$

Indeed, by formula (130), we have

$\langle i^{*}_{\sigma}({\rm td}^{\mathbb{T}}_{*}(X)),\frac{1}{k}\cdot
z\rangle=\sum_{g\in G_{\Sigma}}\prod_{\rho\in\Sigma(1)}\frac{\langle
i^{*}_{\sigma}F_{\rho},\frac{1}{k}\cdot z\rangle}{1-a_{\rho}(g)\cdot
e^{-\langle i^{*}_{\sigma}F_{\rho},\frac{1}{k}\cdot z\rangle}},$

with $F_{\rho}=[D_{\rho}]_{\mathbb{T}}$ denoting the equivariant fundamental
class of the ${\mathbb{T}}$-invariant divisor $D_{\rho}$ corresponding to the
ray $\rho\in\Sigma(1)$, and note that for $k$ fixed (or $k=1$) this is a
convergent power series function around zero in $z$. Finally,

$\lim_{k\to\infty}\frac{\langle i^{*}_{\sigma}F_{\rho},\frac{1}{k}\cdot
z\rangle}{1-a_{\rho}(g)\cdot e^{-\langle
i^{*}_{\sigma}F_{\rho},\frac{1}{k}\cdot z\rangle}}=\begin{cases}0,\ \
\text{if}\ a_{\rho}(g)\neq 1,\\\ 1,\ \ \text{if}\ a_{\rho}(g)=1.\end{cases}$

So only $g=id_{G}$ contributes a non-zero limit to (188), and this
contribution is $1$. Altogether, we get the following result.

######  Theorem 5.1.

In the above notations, we have

(189) $\begin{split}\int_{P_{y}(h)}e^{\langle m,z\rangle}\
dm&=\sum_{\sigma\in\Sigma(n)}\frac{e^{\langle
i_{\sigma}^{*}[D_{P_{y}(h)}],z\rangle}}{\langle
Eu^{\mathbb{T}}_{X}(x_{\sigma}),z\rangle}\\\
&=\sum_{\sigma\in\Sigma(n)}\frac{e^{\langle(1+y)\cdot
i_{\sigma}^{*}c^{\mathbb{T}}_{1}({\mathcal{O}}_{X}(D_{P})),z\rangle}}{\langle
Eu^{\mathbb{T}}_{X}(x_{\sigma}),z\rangle}\cdot e^{\sum_{\rho}h_{\rho}\langle
i_{\sigma}^{*}F_{\rho},z\rangle}\>.\end{split}$

######  Remark 5.2.

The left hand side of (189) is a continuous function in $h$ near zero, and for
all $z\in N_{\mathbb{C}}$, resp., in $y$, whereas the right hand side is an
analytic function in $h$ near zero, and for $z\in N_{\mathbb{C}}$ away from
the linear hyperplanes $\langle i_{\sigma}^{*}F_{\rho},z\rangle=0$ for each
ray $\rho\in\sigma(1)$ of $\sigma\in\Sigma(n)$, resp., in $y$. But then both
sides of this equality have to be analytic functions in $h$ near zero and all
$z\in N_{\mathbb{C}}$, resp., in $y\in{\mathbb{R}}$, with the corresponding
Taylor series around zero converging uniformly on small compact neighborhoods
of zero in the variables $h$ and $z$ (cf. also [34][page 27]).

For later use, we state the following consequence of Theorem 5.1 (see also
[12][Lem.3.11]).

######  Corollary 5.3.

Let $f$ be a polynomial function of degree $d$ on $M_{\mathbb{R}}$. Then
$\int_{P_{y}(h)}f(m)\ dm$ is a polynomial function in $h$ of degree bounded
above by $d+n$.

###### Proof.

We follow the idea of proof from [12][Lem.3.11], adapted to our language. By
replacing in (189) $z$ by $tz$, with $0\neq t\in{\mathbb{R}}$ small (so that
also $\langle i_{\sigma}^{*}F_{\rho},tz\rangle\neq 0$ for each ray
$\rho\in\sigma(1)$ of $\sigma\in\Sigma(n)$), and multiplying both sides by
$t^{n}$ , we get an equality of analytic functions in $t$, with each term on
the right hand side also analytic even in zero, since $\langle
Eu^{\mathbb{T}}_{X}(x_{\sigma}),z\rangle\neq 0$ is homogeneous in $z$ of order
$n$. Now taking the Taylor expansion at $t=0$ of these two analytic functions,
we get for $z$ small and away from the linear hyperplanes $\langle
i_{\sigma}^{*}F_{\rho},z\rangle=0$ for each ray $\rho\in\sigma(1)$ of
$\sigma\in\Sigma(n)$, that the assertion holds for $f(m)=\langle
m,z\rangle^{k}$, for any given non-negative integer $k$. This then implies the
statement for any polynomial $f(m)$. ∎

######  Corollary 5.4.

Let $X=X_{P}$ be the toric variety associated to a simple full-dimensional
lattice polytope $P\subset M_{\mathbb{R}}$, with $D=D_{P}$ the corresponding
ample divisor. Let ${\mathcal{F}}$ be a ${\mathbb{T}}$-equivariant coherent
sheaf on $X$. Then, for a polynomial function $f$ on $M_{\mathbb{R}}$, the
expression

(190) $\sum_{m\in
M}\left(\sum_{i=0}^{n}(-1)^{i}\cdot\dim_{\mathbb{C}}H^{i}(X;{\mathcal{O}}_{X}((1+y)D)\otimes{\mathcal{F}})_{\chi^{-m}}\right)\cdot
f(m)$

is a polynomial in $1+y$. Moreover, the value of this polynomial at $0$ (i.e.,
for $y=-1$) is given by

$\sum_{m\in
M}\left(\sum_{i=0}^{n}(-1)^{i}\cdot\dim_{\mathbb{C}}H^{i}(X;{\mathcal{F}})_{\chi^{-m}}\right)\cdot
f(m).$

###### Proof.

Using the ideas of the above corollary (as in [12][Prop.4.1]), the assertion
follows from the formula

$\begin{split}&\sum_{m\in
M}\left(\sum_{i=0}^{n}(-1)^{i}\cdot\dim_{\mathbb{C}}H^{i}(X;{\mathcal{O}}_{X}((1+y)D)\otimes{\mathcal{F}})_{\chi^{-m}}\right)\cdot
e^{\langle m,z\rangle}\\\ &=\sum_{\sigma\in\Sigma(n)}\frac{e^{\langle
i_{\sigma}^{*}[D_{P_{y}}],z\rangle}}{\langle
Eu^{\mathbb{T}}_{X}(x_{\sigma}),z\rangle}\cdot\langle
i_{\sigma}^{*}(td_{*}^{\mathbb{T}}([{\mathcal{F}}])),z\rangle\\\
&=\sum_{\sigma\in\Sigma(n)}\frac{e^{\langle(1+y)\cdot
i_{\sigma}^{*}c^{\mathbb{T}}_{1}({\mathcal{O}}_{X}(D_{P})),z\rangle}}{\langle
Eu^{\mathbb{T}}_{X}(x_{\sigma}),z\rangle}\cdot\langle
i_{\sigma}^{*}(td_{*}^{\mathbb{T}}([{\mathcal{F}}])),z\rangle\>,\end{split}$

which can be deduced from (179), upon pairing with $z\in N_{\mathbb{C}}$. ∎

######  Remark 5.5.

For ${\mathcal{F}}={\mathcal{O}}_{X}$, the expression (190) becomes

$\sum_{m\in M\cap P_{y}}f(m)$

as in [12]. For the function $f=1$, this further specializes to the classical
Ehrhart polynomial. As another special case, for
$[{\mathcal{F}}]=mC^{\mathbb{T}}_{0}([O_{\tau}\hookrightarrow X])\in
K_{0}^{\mathbb{T}}(X)$, (190) calculates

$\sum_{m\in M\cap{\rm Relint}(E_{y})}f(m)$

with $E$ the face of $P$ corresponding to a cone $\tau\in\Sigma$.

We next explain (in Theorem 5.8 below) a result analogous to Theorem 5.1 for
the dilatation

$Tan(P,v)(h):=(v+C_{v})(h)=v(h)+C_{v}$

of the tangent cone $Tan(P,v)=v+C_{v}$ of a vertex $v$ of the polytope $P$,
with $C_{v}=Cone(P\cap M-v)\subset M_{\mathbb{R}}$ a full-dimensional simple
lattice cone with vertex $0$. The arguments from the case of a full-
dimensional lattice polytope apply similarly to the tangent cone, once the
approximation of the integral $\int_{{C_{v}}(h)}e^{\langle m,z\rangle}\ dm$ by
the corresponding Riemann sum is explained. The corresponding equivariant
Hirzebruch-Riemann-Roch formula in this setting is (170), with
$\sigma^{\vee}=C_{v}$ and $v$ corresponding to the torus fixed point
$x_{\sigma}$ in the ${\mathbb{T}}$-invariant open affine neighborhood
$U_{\sigma}$ associated to $\sigma\in\Sigma(n)$.

######  Proposition 5.6.

With the above notations, one has that $\sum_{m\in C_{v}\cap M}e^{\langle
m,z\rangle}$ and $\int_{C_{v}}e^{\langle m,z\rangle}\ dm$ are convergent
(locally uniformly) to meromorphic functions in
$L^{-1}(\Lambda^{an}_{\mathbb{T}})_{\mathbb{C}}$, for $z\in N_{\mathbb{C}}$
satisfying $-z\in Int(\sigma)$. Moreover,

$\lim_{k\to\infty}\frac{1}{k^{n}}\sum_{m\in k\cdot{C_{v}}(h)\cap M}e^{\langle
m,\frac{1}{k}\cdot
z\rangle}=\lim_{k\to\infty}\frac{1}{k^{n}}\sum_{m\in{C_{v}}(h)\cap\frac{1}{k}M}e^{\langle
m,z\rangle}=\int_{{C_{v}}(h)}e^{\langle m,z\rangle}\ dm,$

with $h\in{\mathbb{Q}}^{r}$ a small enough dilatation vector, and
$k\in{\mathbb{N}}$ so that $k\cdot C_{v}(h)$ is a simple pointed lattice cone
in $M_{\mathbb{R}}$.

###### Proof.

Let $\sigma\in\Sigma(n)$ be a smooth cone with generators
$u_{\sigma,1},\ldots,u_{\sigma,n}$ of the rays
$\rho_{1},\ldots,\rho_{n}\in\sigma(1)$, and let
$m_{\sigma,1},\ldots,m_{\sigma,n}$ be the dual basis of
$M=\bigoplus_{i=1}^{n}{\mathbb{N}}\cdot m_{\sigma,i}$. Then

$\sum_{m\in C_{v}\cap M}e^{\langle
m,z\rangle}=\prod_{i=1}^{n}\left(\sum_{j_{i}=0}^{\infty}(e^{\langle
m_{\sigma,i},z\rangle})^{j_{i}}\right)=\prod_{i=1}^{n}\frac{1}{1-e^{\langle
m_{\sigma,i},z\rangle}}=\frac{\langle i_{\sigma}^{*}({\rm
td}_{*}^{\mathbb{T}}(X)),z\rangle}{\langle
Eu_{X}^{\mathbb{T}}(x_{\sigma}),z\rangle},$

with all $\langle m_{\sigma,i},z\rangle<0$ for $-z\in Int(\sigma)$, and the
corresponding geometric series locally uniformly convergent in these $z$. The
last equality follows from (173) (for $K=\emptyset$), using the identification
$s:Sym_{\mathbb{Q}}(M)\simeq(\Lambda_{\mathbb{T}})_{\mathbb{Q}}$. Similarly,

$\int_{C_{v}}e^{\langle m,z\rangle}\ dm=\prod_{i=1}^{n}\frac{-1}{\langle
m_{\sigma,i},z\rangle},$

for $-z\in Int(\sigma)$ and convergence as before, with the Lebesgue measure
$dm$ normalized so that the unit cube in $M\subset M_{\mathbb{R}}$ has volume
$1$ (see also [29][eq.(8)] and [38][Prop.1]). By the multiplicativity of the
equivariant Todd class transformation, we get as in the case of a polytope
that

$\begin{split}\lim_{k\to\infty}\frac{1}{k^{n}}\sum_{m\in k\cdot{C_{v}}(h)\cap
M}e^{\langle m,\frac{1}{k}\cdot
z\rangle}&=\lim_{k\to\infty}\frac{1}{k^{n}}\prod_{i=1}^{n}\left(\frac{e^{h_{i}\langle
m_{\sigma,i},z\rangle}}{1-e^{\langle m_{\sigma,i},\frac{1}{k}\cdot
z\rangle}}\right)\\\
&=\lim_{k\to\infty}\prod_{i=1}^{n}\left(\frac{e^{h_{i}\langle
m_{\sigma,i},z\rangle}\cdot\langle m_{\sigma,i},\frac{1}{k}\cdot
z\rangle}{(1-e^{\langle m_{\sigma,i},\frac{1}{k}\cdot z\rangle})\cdot\langle
m_{\sigma,i},z\rangle}\right)\\\ &=\prod_{i=1}^{n}\frac{-e^{h_{i}\langle
m_{\sigma,i},z\rangle}}{\langle m_{\sigma,i},z\rangle}\\\
&=e^{\sum_{i=1}^{n}h_{i}\langle
m_{\sigma,i},z\rangle}\cdot\prod_{i=1}^{n}\frac{-1}{\langle
m_{\sigma,i},z\rangle}\\\ &=e^{\sum_{i=1}^{n}h_{i}\langle
m_{\sigma,i},z\rangle}\cdot\int_{{C_{v}}}e^{\langle m,z\rangle}\ dm\\\
&=\int_{{C_{v}}(h)}e^{\langle m,z\rangle}\ dm.\end{split}$

For the proof of the case of a simplicial cone by smooth subdivisions, let us
remark that the above approximation by Riemann sums for the full cone $C_{v}$
holds similarly for the cone $C_{v}^{K}:=C_{v}\setminus\bigcup_{i\in K}F_{i}$
with some facets $F_{i}$ ($i\in K\subset\Sigma(1)$) removed. More precisely,
one gets

$\sum_{m\in C^{K}_{v}\cap M}e^{\langle m,z\rangle}=\prod_{i\in\sigma(1)\cap
K}\frac{e^{\langle m_{\sigma,i},z\rangle}}{1-e^{\langle
m_{\sigma,i},z\rangle}}\cdot\prod_{i\in\sigma(1)\setminus
K}\frac{1}{1-e^{\langle m_{\sigma,i},z\rangle}}=\frac{\langle
i_{\sigma}^{*}(T_{0*}^{\mathbb{T}}([V\hookrightarrow X])),z\rangle}{\langle
Eu_{X}^{\mathbb{T}}(x_{\sigma}),z\rangle},$

with the last equality following from (173), for $V=X\setminus\bigcup_{i\in
K}D_{\rho_{i}}$ (with the divisors $D_{\rho_{i}}$ corresponding to the
divisors of the original polytope $P$). The series on the left hand side are
locally uniformly convergent to a meromorphic function in
$L^{-1}(\Lambda^{an}_{\mathbb{T}})_{\mathbb{C}}$, for $z\in N_{\mathbb{C}}$
satisfying $-z\in Int(\sigma)$. Moreover, a similar proof yields in this case
that

$\lim_{k\to\infty}\frac{1}{k^{n}}\sum_{m\in k\cdot{C^{K}_{v}}(h)\cap
M}e^{\langle m,\frac{1}{k}\cdot z\rangle}=\int_{{C_{v}}(h)}e^{\langle
m,z\rangle}\ dm.$

In the general case, let $C_{v}$ be as before a full-dimensional (simple)
lattice cone with vertex $0$. The assumption that $C_{v}$ is simple will not
be needed in the following arguments. Consider a refinement
$C_{v}=\bigcup_{i=1}^{s}C_{i}$ of $C_{v}$ by smooth full-dimensional lattice
cones with vertex $0$, see, e.g., [16][Thm.11.1.9] in the dual context of
toric resolutions by fan refinements. Then, considering each newly introduced
facet $F$ for only one of the cones it belongs to, we get a disjoint union
$C_{v}=\bigsqcup_{i=1}^{s}C^{K_{i}}_{i}$ into smooth full-dimensional lattice
cones with vertex $0$ and with some facets removed. Let $z\in N_{\mathbb{C}}$
be so that $-z\in Int(\sigma)$, with $\sigma=C_{v}^{\vee}$. Then we also have
that $-z\in Int(\sigma_{i})$ with $\sigma_{i}=C_{i}^{\vee}$. So we can apply
the case of smooth cones to all of the $C_{i}$. Then

$\sum_{m\in C_{v}\cap M}e^{\langle m,z\rangle}=\sum_{i=1}^{s}\left(\sum_{m\in
C^{K_{i}}_{i}\cap M}e^{\langle m,z\rangle}\right)$

is convergent (locally uniformly) to a meromorphic function in
$L^{-1}(\Lambda^{an}_{\mathbb{T}})_{\mathbb{C}}$, for $z\in N_{\mathbb{C}}$
satisfying $-z\in Int(\sigma)$. Moreover,

$\lim_{k\to\infty}\frac{1}{k^{n}}\sum_{m\in k\cdot{C_{v}}(h)\cap M}e^{\langle
m,\frac{1}{k}\cdot z\rangle}=\sum_{i=1}^{s}\int_{{C_{i}}(h)}e^{\langle
m,z\rangle}\ dm=\int_{{C_{v}}(h)}e^{\langle m,z\rangle}\ dm,$

with $h\in M_{\mathbb{Q}}\simeq{\mathbb{Q}}^{r}$ a small enough dilatation
vector, and $k\in{\mathbb{N}}$ so that $k\cdot C_{v}(h)$ is a full-dimensional
pointed lattice cone in $M_{\mathbb{R}}$. ∎

######  Remark 5.7.

By the above smooth decomposition method, one can get more explicit
expressions for the summation appearing in formula (170). In the context of
the tangent cone $Tan(P,v)$ of a simple full-dimensional lattice polytope $P$
at a vertex $v$, we also get the identification

(191) $\sum_{m\in C_{v}\cap M}e^{\langle m,z\rangle}=\langle{\rm
td}_{*}^{\mathbb{T}}(X)_{x_{\sigma}},z\rangle=\langle\mathbb{S}\left(\sum_{m\in\sigma^{\vee}\cap
M}e^{s(m)}\right),z\rangle\in L^{-1}(\Lambda^{an}_{\mathbb{T}})_{\mathbb{Q}},$

for $-z\in Int(\sigma)$. The left hand side is a convergent series for such
$z$, whereas the right hand side is a corresponding meromorphic function on
$N_{\mathbb{C}}$. Furthermore, still assuming $-z\in Int(\sigma)$, we get more
generally, for $\tau\preceq\sigma$ corresponding to a face $E$ of $P$
containing the vertex $v$, a similar interpretation for the following
equality:

(192) $\sum_{m\in{\rm Relint}(C_{v}\cap E)\cap M}e^{\langle
m,z\rangle}=\langle T_{0*}^{\mathbb{T}}([O_{\tau}\hookrightarrow
X])_{x_{\sigma}},z\rangle=\langle\mathbb{S}\left(\sum_{{\rm
Relint}(\sigma^{\vee}\cap\tau^{\perp})\cap M}e^{s(m)}\right),z\rangle\in
L^{-1}(\Lambda^{an}_{\mathbb{T}})_{\mathbb{Q}}.$

As a consequence of the above discussion, we can now give a geometric meaning
to the localized summands of formula (189).

######  Theorem 5.8.

Let $v\in P$ be a vertex of the full-dimensional simple lattice polytope
$P\subset M_{\mathbb{R}}$, with tangent cone $Tan(P,v)$, for $C_{v}=Cone(P\cap
M-v)\subset M_{\mathbb{R}}$ a full-dimensional simple lattice cone with vertex
$0$. Consider its dilatation

$Tan(P,v)(h):=(v+C_{v})(h).$

Let $(X,D)$ be the projective toric variety with ample Cartier divisor
associated to $P$. Denote by $x_{\sigma}\in X$ the torus fixed point
corresponding to $v\in P$, with associated cone
$\sigma=C_{v}^{\vee}\in\Sigma(n)$. Then for $z\in N_{\mathbb{C}}$ satisfying
$-z\in Int(\sigma)$, we have

(193) $\begin{split}\int_{Tan(P,v)(h)}e^{\langle m,z\rangle}\
dm=\frac{e^{\langle v,z\rangle}}{\langle
Eu^{\mathbb{T}}_{X}(x_{\sigma}),z\rangle}\cdot
e^{\sum_{\rho\in\sigma(1)}h_{\rho}\langle
i_{\sigma}^{*}F_{\rho},z\rangle}\>.\end{split}$

###### Proof.

In light of the above proposition, one only has to note that
$i_{\sigma}^{*}c_{1}^{\mathbb{T}}({\mathcal{O}}(D))=-c(v)=s(v)\in
H^{2}_{\mathbb{T}}(pt;{\mathbb{Q}})\simeq M_{\mathbb{Q}}$. ∎

######  Remark 5.9.

The left hand side of the equality (193) is an analytic function in $h$ near
zero and $-z\in Int(\sigma)$, resp., in $y\in{\mathbb{R}}$. Moreover, this
integral is locally uniformly convergent in $z$. The right hand side of (193)
is an analytic function in $h$ near zero and $y\in{\mathbb{R}}$, as well as a
meromorphic function in $z$ which is holomorphic outside the hyperplanes
$\langle i_{\sigma}^{*}F_{\rho},z\rangle=0$ for each ray $\rho\in\sigma(1)$.

Back to the context of a projective simplicial toric variety associated to a
full-dimensional simple lattice polytope $P\subset M_{\mathbb{R}}$, in order
to relate the right-hand side of formula (189) to the equivariant Hirzebruch-
Riemann-Roch formulae we have to introduce the analytic subring

(194)
$\left(H^{*}_{\mathbb{T}}(X;{\mathbb{Q}})\right)^{an}\subset\widehat{H}^{*}_{\mathbb{T}}(X;{\mathbb{Q}})$

defined as the image of the analytic Stanley-Reisner subring (depending only
on the fan $\Sigma$)

$SR^{an}_{\mathbb{Q}}(\Sigma):={\mathbb{Q}}\\{x_{\rho}\mid\rho\in\Sigma(1)\\}/\sim_{SR}\
\ \subset\
{\mathbb{Q}}[[x_{\rho}\mid\rho\in\Sigma(1)]]/\sim_{SR}=:\widehat{SR_{\mathbb{Q}}(\Sigma)}$

under the isomorphism $\widehat{SR_{\mathbb{Q}}(\Sigma)}\
\simeq\widehat{H}^{*}_{\mathbb{T}}(X;{\mathbb{Q}})$ given on generators by
$x_{\rho}\mapsto F_{\rho}$, $\rho\in\Sigma(1)$. Given an element
$p(F_{\rho})\in\left(H^{*}_{\mathbb{T}}(X;{\mathbb{Q}})\right)^{an}$, with
$p(x_{\rho})\in{\mathbb{Q}}\\{x_{\rho}\mid\rho\in\Sigma(1)\\}$ a convergent
power series, one gets for the restriction $i_{\sigma}^{*}$ to a fixed point
$x_{\sigma}$ that

(195) $p(i_{\sigma}^{*}F_{\rho})=i_{\sigma}^{*}(p(F_{\rho})).$

Here, the convergent power series on the left side corresponds to the image of
$p(x_{\rho})$ under the evaluation homomorphism

${\mathbb{Q}}\\{x_{\rho}\mid\rho\in\Sigma(1)\\}\mapsto(\Lambda_{\mathbb{T}}^{an})_{\mathbb{Q}}:\>x_{\rho}\mapsto
i_{\sigma}^{*}F_{\rho}\in H^{2}_{\mathbb{T}}(pt;{\mathbb{Q}})\simeq
M_{\mathbb{Q}}\>.$

######  Remark 5.10.

Regarding $\sigma\in\Sigma(n)$ as a fan, we have a corresponding (analytic)
Stanley-Reisner ring

$SR^{an}_{\mathbb{Q}}(\sigma):={\mathbb{Q}}\\{x_{\rho}\mid\rho\in\sigma(1)\\}\
\ \subset\
{\mathbb{Q}}[[x_{\rho}\mid\rho\in\sigma(1)]]=:\widehat{SR_{\mathbb{Q}}(\sigma)},$

with an isomorphism $\widehat{SR_{\mathbb{Q}}(\sigma)}\
\simeq\widehat{H}^{*}_{\mathbb{T}}(U_{\sigma};{\mathbb{Q}})\overset{i_{\sigma}^{*}}{\simeq}\widehat{H}^{*}_{\mathbb{T}}(x_{\sigma};{\mathbb{Q}})$
given on generators by $x_{\rho}\mapsto i_{\sigma}^{*}F_{\rho}$,
$\rho\in\sigma(1)$. Then the restriction map
$\left(H^{*}_{\mathbb{T}}(X;{\mathbb{Q}})\right)^{an}\to\left(H^{*}_{\mathbb{T}}(U_{\sigma};{\mathbb{Q}})\right)^{an}$
is induced from a corresponding restriction map
$SR^{an}_{\mathbb{Q}}(\Sigma)\to SR^{an}_{\mathbb{Q}}(\sigma)$, sending
$x_{\rho}\mapsto 0$ if $\rho\notin\sigma(1)$. We get the following
factorization of the right hand side of (195):

$i_{\sigma}^{*}:\left(H^{*}_{\mathbb{T}}(X;{\mathbb{Q}})\right)^{an}\to\left(H^{*}_{\mathbb{T}}(U_{\sigma};{\mathbb{Q}})\right)^{an}\to\left(H^{*}_{\mathbb{T}}(x_{\sigma};{\mathbb{Q}})\right)^{an}.$

As a consequence of Theorem 5.1 and Remark 5.2, we get by differentiation and
convergence the following.

######  Corollary 5.11.

Let $p(\frac{\partial}{\partial h})\in{\mathbb{Q}}\\{\frac{\partial}{\partial
h_{\rho}}\mid\rho\in\Sigma(1)\\}\subset{\mathbb{Q}}[[\frac{\partial}{\partial
h_{\rho}}\mid\rho\in\Sigma(1)]]$ be an infinite order differential operator
with constant rational coefficients, i.e., obtained by substituting
$x_{\rho}\mapsto\frac{\partial}{\partial h_{\rho}}$ into a convergent (near
zero) power series with rational coefficients
$p(x)\in{\mathbb{Q}}\\{x_{\rho}\mid\rho\in\Sigma(1)\\}\subset{\mathbb{Q}}[[x_{\rho}\mid\rho\in\Sigma(1)]]$.
Then, in the above notations, we get for $z$ small enough and away from the
hyperplanes $\langle i_{\sigma}^{*}F_{\rho},z\rangle=0$ for each ray
$\rho\in\sigma(1)$ of $\sigma\in\Sigma(n)$ the following formula:

(196) $\begin{split}p(\frac{\partial}{\partial
h})\left(\int_{P_{y}(h)}e^{\langle m,z\rangle}\
dm\right)_{|_{h=0}}&=\sum_{\sigma\in\Sigma(n)}\frac{e^{\langle(1+y)\cdot
i_{\sigma}^{*}c^{\mathbb{T}}_{1}({\mathcal{O}}_{X}(D_{P})),z\rangle}}{\langle
Eu^{\mathbb{T}}_{X}(x_{\sigma}),z\rangle}\cdot\langle
p(i_{\sigma}^{*}F_{\rho}),z\rangle\\\
&=\langle\sum_{\sigma\in\Sigma(n)}\frac{i_{\sigma}^{*}\left(e^{(1+y)\cdot
c^{\mathbb{T}}_{1}({\mathcal{O}}_{X}(D))}p(F_{\rho})\right)}{Eu^{\mathbb{T}}_{X}(x_{\sigma})},z\rangle\\\
&=\langle\int_{X}e^{(1+y)\cdot
c^{\mathbb{T}}_{1}({\mathcal{O}}_{X}(D))}p(F_{\rho}),z\rangle\>.\end{split}$

If $p_{n}\in{\mathbb{Q}}[\frac{\partial}{\partial
h_{\rho}}\mid\rho\in\Sigma(1)]$ is the corresponding truncation of $p$ up to
order $n$, then both sides of (196) applied to $p_{n}$ converge for
$n\to\infty$ locally uniformly in these $z$ to (196) applied to $p$.

######  Remark 5.12.

By (196), the operator $p(\frac{\partial}{\partial
h})\left(\int_{P_{y}(h)}e^{\langle m,z\rangle}\ dm\right)_{|_{h=0}}$ depends
only on the equivalence class of $[p(x)]\in SR^{an}_{\mathbb{Q}}(\Sigma)$ and
not on the chosen convergent power series representative.

######  Remark 5.13.

Assume in addition in Corollary 5.11 that $\langle\int_{X}e^{(1+y)\cdot
c^{\mathbb{T}}_{1}({\mathcal{O}}_{X}(D))}p(F_{\rho}),z\rangle$ is a convergent
power series in $z$ near zero (for $y\in{\mathbb{R}}$ a fixed parameter, e.g.,
$y=0$). Then one gets as an application of Cauchy’s integral formula (see also
[34][page 27]), that both sides of (196) applied to $p_{n}$ converge for
$n\to\infty$ and $z$ small locally uniformly to (196) applied to $p$. In
particular, this limit commutes with finite order differentiations with
respect to $z$ (and $z$ small enough).

######  Corollary 5.14.

Let $p(\frac{\partial}{\partial h})\in{\mathbb{Q}}[[\frac{\partial}{\partial
h_{\rho}}\mid\rho\in\Sigma(1)]]$ be an infinite order differential operator
with constant rational coefficients, i.e., obtained by substituting
$x_{\rho}\mapsto\frac{\partial}{\partial h_{\rho}}$ into a formal power series
with rational coefficients
$p(x)\in{\mathbb{Q}}[[x_{\rho}\mid\rho\in\Sigma(1)]]$. Then for a polynomial
function $f$ on $M_{\mathbb{R}}$, we have the following formula:

(197) $p(\frac{\partial}{\partial h})\left(\int_{P_{y}(h)}f(m)\
dm\right)_{|_{h=0}}=f(\frac{\partial}{\partial
z})\left(\langle\int_{X}e^{(1+y)\cdot
c^{\mathbb{T}}_{1}({\mathcal{O}}_{X}(D))}p(F_{\rho}),z\rangle\right)_{|_{z=0}}\>,$

where on the right hand side the operator $\left(f(\frac{\partial}{\partial
z})\right)_{|_{z=0}}$ acts on a formal power series in $z$.

###### Proof.

Let first $p(\frac{\partial}{\partial
h})\in{\mathbb{Q}}[\frac{\partial}{\partial h_{\rho}}\mid\rho\in\Sigma(1)]$ be
a finite order differential operator with constant rational coefficients,
i.e., obtained by substituting $x_{\rho}\mapsto\frac{\partial}{\partial
h_{\rho}}$ into a polynomial with rational coefficients
$p(x)\in{\mathbb{Q}}[x_{\rho}\mid\rho\in\Sigma(1)]$. Then the result follows
from applying the operator $\left(f(\frac{\partial}{\partial
z})\right)_{|_{z=0}}$ to (196). Here we first need to assume that $z$ is small
enough and away from the hyperplanes $\langle
i_{\sigma}^{*}F_{\rho},z\rangle=0$ for each ray $\rho\in\sigma(1)$ of
$\sigma\in\Sigma(n)$, since the localization formula is used; however, formula
(196) then holds for all $z$ small enough (and $y\in{\mathbb{R}}$ fixed) by
continuity, since the left hand side of (196) is analytic in $z$ near zero by
Remark 5.2. Moreover, by Corollary 5.3 the left hand side and therefore also
the right hand side of (197) only depend on a truncation of $p$ up to order
$n+deg(f)$. Especially the left hand side is well defined and the stated
equality holds for a formal power series with rational coefficients
$p(x)\in{\mathbb{Q}}[[x_{\rho}\mid\rho\in\Sigma(1)]]$. ∎

######  Remark 5.15.

By (197), the operator $p(\frac{\partial}{\partial
h})\left(\int_{P_{y}(h)}f(z)\ dm\right)_{|_{h=0}}$ depends only on the
equivalence class of $[p(x)]\in\widehat{SR_{\mathbb{Q}}(\Sigma)}$ and not on
the chosen formal power series representative.

As a consequence of Theorem 5.8 and Remark 5.10, we get the following local
version of Corollary 5.11:

######  Corollary 5.16.

Let $p(\frac{\partial}{\partial h})\in{\mathbb{Q}}\\{\frac{\partial}{\partial
h_{\rho}}\mid\rho\in\Sigma(1)\\}\subset{\mathbb{Q}}[[\frac{\partial}{\partial
h_{\rho}}\mid\rho\in\Sigma(1)]]$ be an infinite order differential operator
with constant rational coefficients, i.e., obtained by substituting
$x_{\rho}\mapsto\frac{\partial}{\partial h_{\rho}}$ into a convergent (near
zero) power series with rational coefficients
$p(x)\in{\mathbb{Q}}\\{x_{\rho}\mid\rho\in\Sigma(1)\\}\subset{\mathbb{Q}}[[x_{\rho}\mid\rho\in\Sigma(1)]]$.
Fix a vertex $v\in P$ with tangent cone $Tan(P,v)$, and let
$\sigma\in\Sigma(n)$ be the corresponding cone. Then, for $z$ small enough and
$-z\in Int(\sigma)$, we have the following formula:

(198) $p(\frac{\partial}{\partial h})\left(\int_{Tan(P,v)(h)}e^{\langle
m,z\rangle}\
dm\right)_{|_{h=0}}=\langle\frac{e^{v}\cdot\left(i_{\sigma}^{*}p(F_{\rho})\right)}{Eu^{\mathbb{T}}_{X}(x_{\sigma})},z\rangle\>.$

If $p_{n}\in{\mathbb{Q}}[\frac{\partial}{\partial
h_{\rho}}\mid\rho\in\Sigma(1)]$ is the corresponding truncation of $p$ up to
order $n$, then both sides of (198) applied to $p_{n}$ converge for
$n\to\infty$ locally uniformly in these $z$ to (198) applied to $p$.

We also have the following.

######  Proposition 5.17.

Let $X=X_{\Sigma}$ be a projective simplicial toric variety. For any class
$[{\mathcal{F}}]\in K^{\mathbb{T}}_{0}(X)$, its equivariant Todd class is an
element in the analytic cohomology ring of $X$, i.e.,

${\rm
td}_{*}^{\mathbb{T}}([{\mathcal{F}}])\in\left(H^{*}_{\mathbb{T}}(X;{\mathbb{Q}})\right)^{an}\subset\widehat{H}^{*}_{\mathbb{T}}(X;{\mathbb{Q}}).$

###### Proof.

For ${\mathcal{F}}={\mathcal{O}}_{X}$, this follows from the explicit formula
given in (130). More generally, this holds for sheaves
${\mathcal{F}}=\pi_{*}({\mathcal{O}}_{W}\otimes{\mathbb{C}}_{\chi^{\widetilde{m}}})^{G}$
as in the explicit formula (138). As in the proof of [13][Cor.1.2], the
equivariant Grothendieck group $K^{\mathbb{T}}_{0}(X)$ is generated by classes
of sheaves of the form
$\pi_{*}({\mathcal{O}}_{W}\otimes{\mathbb{C}}_{\chi^{\widetilde{m}}})^{G}$. ∎

Altogether, we get the following abstract Euler-Maclaurin formula coming from
the equivariant Hirzebruch-Riemann-Roch theorem.

######  Theorem 5.18.

Let $X=X_{P}$ be the projective simplicial toric variety associated to a full-
dimensional simple lattice polytope $P\subset M_{\mathbb{R}}$. Let
$\Sigma:=\Sigma_{P}$ be the inner normal fan of $P$, and $D:=D_{P}$ the ample
Cartier divisor associated to $P$. Let $[{\mathcal{F}}]\in
K^{\mathbb{T}}_{0}(X)$ be fixed, and choose a convergent power series
$p(x_{\rho})\in{\mathbb{Q}}\\{x_{\rho}\mid\rho\in\Sigma(1)\\}$ so that
$p(F_{\rho})={\rm
td}^{\mathbb{T}}_{*}([{\mathcal{F}}])\in\left(H^{*}_{\mathbb{T}}(X;{\mathbb{Q}})\right)^{an}$.
Then

(199) $\begin{split}p(\frac{\partial}{\partial h})\left(\int_{P(h)}e^{\langle
m,z\rangle}\
dm\right)_{|_{h=0}}&=\langle\chi^{\mathbb{T}}(X,{\mathcal{O}}_{X}(D)\otimes{\mathcal{F}}),z\rangle\\\
&=\sum_{m\in
M}\left(\sum_{i=0}^{n}(-1)^{i}\cdot\dim_{\mathbb{C}}H^{i}(X;{\mathcal{O}}_{X}(D)\otimes{\mathcal{F}})_{\chi^{-m}}\right)\cdot
e^{\langle m,z\rangle}\>,\end{split}$

as analytic functions in $z$ with $z$ small enough, and with
$\chi^{\mathbb{T}}(X,{\mathcal{O}}_{X}(D)\otimes{\mathcal{F}})\in(\Lambda^{an}_{\mathbb{T}})_{\mathbb{Q}}$
the cohomological equivariant Euler characteristic of
${\mathcal{O}}_{X}(D)\otimes{\mathcal{F}}$.

###### Proof.

Equation (196) for $y=0$ can now be calculated as

$\displaystyle p(\frac{\partial}{\partial h})\left(\int_{P(h)}e^{\langle
m,z\rangle}\ dm\right)_{|_{h=0}}$ $\displaystyle=$
$\displaystyle\langle\int_{X}e^{c^{\mathbb{T}}_{1}({\mathcal{O}}_{X}(D))}{\rm
td}^{\mathbb{T}}_{*}([{\mathcal{F}}]),z\rangle$ $\displaystyle=$
$\displaystyle\langle\chi^{\mathbb{T}}(X,{\mathcal{O}}_{X}(D)\otimes{\mathcal{F}}),z\rangle,$

where the last equality follows from the equivariant Hirzebruch-Riemann-Roch
formula (83). The second equality of (199) follows from (71) which uses the
eigenspace decomposition, with the minus sign of (199) due to the appearance
of $c(m)=-s(m)$ in (71). Recall that we work with the identification
$s:Sym_{\mbox{$\mathbb{Q}$}}(M)\simeq
H_{\mathbb{T}}^{*}(pt;\mbox{$\mathbb{Q}$})=:(\Lambda_{\mathbb{T}})_{\mbox{$\mathbb{Q}$}}$.
Finally, in the proof we first need to assume that $z$ is small enough and
away from the hyperplanes $\langle i_{\sigma}^{*}F_{\rho},z\rangle=0$ for each
ray $\rho\in\sigma(1)$ of $\sigma\in\Sigma(n)$, since the localization formula
is used; however, formula (199) then holds for all $z$ small enough, by Remark
5.13. ∎

Evaluating (199) at $z=0$, we get the following generalized volume formula.

######  Corollary 5.19.

In the notations of the previous theorem, we have