Toric Equivariant Riemannn-Roch

for Pσ^∨ ={Pn

i=1λimσ,i |0≤λi <1}. In particular, if σ is smooth, then S( ˜χ(Uσ,L)) = χ^mσ

Qn

i=1(1−χ^mσ,i).

Proof. Completeness of X forces to ˜χ(X,L) be in Z[M], and from Lemma 7.2.1 shows that

S( ˜χ(X,L)) = ˜χ(X,L)) =X

p≥0

(−1)^p X

γ∈[l]_p

S( ˜χ(U_σγ,L)).

Lemma 7.2.4 (a) says that it suffices to consider the terms for dimσ_γ =n. On the other hand, in general, we use Σ_max, but completeness of X also makes us possible to use the open cover U ={U_σ}_σ∈Σ(n) for the ˇCech complex. This is because we have no cone of dimension less than nisolated in N_R'Rⁿ. Thus, the zero ˇCech complex only consists ofn-dimensional cones, i.e., Cˇ⁰(U,L) =L

σ∈Σ(n)H⁰(Uσ,L). Hence, we only have the case for p= 0, so that part (a) is done.

Part (b) is just an immediate consequence of Lemma 7.2.4 (b) and (c).

Now, we finally have the Brion’s equalities. Rather, Theorem 7.2.6 will be used in the next section.

Corollary 7.2.7 (Brion’s Equalities). Let P ⊆ M_R ' Rⁿ be a full dimensional lattice polytope, and for each vertex v∈P, let C_v = Cone(P∩M −v)⊆M_R. Then:

(a) P

m∈P∩Mχ^m =P

v:vertexχ^v· S P

m∈Cv∩Mχ^m .

(b) If P is simple and v ∈ P is a vertex, let m_v,1, . . . , m_v,n ∈ M be the minimal generators of Cv. Then

X

m∈P∩M

χ^m = X

v:vertex

χ^v·P

m∈P_v∩Mχ^m Qn

i=1(1−χ^mv,i) .

Proof. Consider the toric variety X_P and the line bundle LXP(D_P). Note that D_P = P

F facesaFDF is Cartier, where aF is from (3.2.1). Moreover, this divisor is basepoint free Cartier, so is nef. Thus, applying Demazure vanishing (theorem 6.2.3) shows that

˜

χ(X_P,L) =P

m∈P∩Mχ^m. For vertices v ofP,σ_v^∨ =C_v and (7.2.3) implies that

˜

χ(U_σv,L) =χ^v· X

m∈σ^∨_v∩M

χ^m =χ^v· X

m∈C_v∩M

χ^m

Using Theorem 7.2.6 concludes our proof.

Theorem 7.3.1(Equivariant Riemann-Roch). For a smooth complete toric varietyX= X_Σ and a line bundle L =OX(D) of a torus-invariant divisor D on X, we have

χ^T(L) = Z

X^eq

ch^T(L)Td^T(X).

In this theorem, χ^T(L),R

X^eq, ch^T(L), and Td^T(X) are equivariant versions of the corresponding objects appearing in Theorem 7.0.1.

The Equivariant Euler Characteristic. Let T = T_N is the torus associated to M.

Since we will deal with power series, we need the completion of (Λ_T)_Q=H_T^•({pt},Q) = L∞

k=0H_T^k({pt},Q), say

Λ =b Hb_T^•({pt},Q) =

∞

Y

k=0

H_T^k({pt},Q).

From (6.3.1), using symmetric algebra withs: Sym_Q(M)'(ΛT)_Q, ifm∈M, thens(m) is a degree 2 element of Λ so that the exponentialb

e^s(m)= 1 +s(m) +1

2s(m)²+ 1

6s(m)³+· · · ∈Λ.b

For a complete toric variety X = XΣ and the line bundle L = OX(D) of a torus- invariant divisorDon X, theequivariant Euler characteristic of L is defined by

χ^T(L) = X

m∈M n

X

i=0

(−1)ⁱdimHⁱ(X,L)me^s(m)∈Λ.b (7.3.1) The Nonequivariant Limit.Leti^∗_pt:Λb !Qbe the map that sends elements of posi- tive degree to zero, andi^∗_X :H_T^•(X,Q)!H^•(X,Q).i^∗_pt andi^∗_X are our “nonequivariant limit” for passing equivariant version of HRR to ordinary HRR we desired. In particular, i^∗_pt takes the equivariant Euler characteristic to the ordinary Euler characteristic

i^∗_pt(χ^T(F)) =χ(F) =

n

X

i=0

(−1)ⁱdimHⁱ(X,F). (7.3.2) Equivariant Chern Characters and Todd Classes. In this time we use another completion of the equivariant cohomologyH_T^•(X,Q). The completion is

Hb_T^•(X,Q) =

∞

Y

k=0

H_T^k(X,Q).

The nonequivariant limit mapi^∗_X :H_T^•(X,Q) !H^•(X,Q) extends to a ring homomor- phism

i^∗_X :Hb_T^•(X,Q)!H^•(X,Q) since elements of degree >2dimX map to zero.

By Proposition 6.3.1, a torus-invariant divisor D = P

ρaρDρ gives the equivariant cohomology class [D]_T =P

ρa_ρ[D_ρ]_T ∈H_T²(X,Q). Then we define theequivariant Chern

character ch^T(L) ofL =OX(D) to be

ch^T(L) =e^[D]T = 1 + [D]_T + 1

2![D]²_T +· · · ∈Hb_T^•(X,Q).

On the other hand, we have a nice form of Todd class of X from Theorem 7.1.4, it is desirable to define theequivariant Todd class ofX to be

Td^T(X) =

r

Y

i=1

[Dρ]T

1−e^−[Dρ]T.

Considering this with Bernoulli numbers shows that Td^T(X)∈Hb_T^•(X,Q). Sincei^∗_X[D_ρ]_T = [Dρ] by Proposition 6.3.1, we have

i^∗_Xch^T(L) = ch(L) and i^∗_XTd^T(L) = Td(L). (7.3.3) The Equivariant Integral.Note that the map R

X :H^•(X,Q)!Qis the generalized Gysin map of the constant function p :X ! {pt}. It also gives the equivariant Gysin map

p_!:H_T^•(X,Q)!H_T^•({pt},Q) which maps H_T^k(X,Q) to H_T^k−2dimX({pt},Q). We writeR

X^eq instead ofp!, so that Z

X^eq

:Hb_T^•(X,Q)!Hb_T^•({pt},Q) =Λ.b This is called theequivariant integral. Note thatR

X^eq andR

X are compatible with taking the nonequivariant limit (Proposition A.1.3 ). This will make it easy to derive HRR from equivariant Riemann-Roch.

Delta Map as an Isomorphism.The delta map from (6.3.2) induced by the inclusion of the fixed point set X^T ⊆X can be extended to the completion

δ :Hb_T^•(X,Q)−! M

σ∈Σ(n)

Hb_T^•({x_σ},Q) = M

σ∈Σ(n)

Λ.b

IfSbe the multiplicative set consisting of all finite products of nonzero degree 2 elements of Λ with the empty product, then the localized mapb

δS :Hb_T^•(X,Q)S −! M

σ∈Σ(n)

ΛbS. (7.3.4)

becomes an isomorphism.

Local Decomposition of Equivariant Euler Characteristics. We decompose the equivariant Euler chracteristicχ^T(L)∈Λ by local terms with respect tob n-dimensional cones, and this decomposition will appear in ΛbS. First, define the ring homomorphism

Φ :Z[M]−!Λ,b χ^m7−!e^s(m).

Φ extends a well-defined map between to two localizations: Q[M]_S and ΛbS, where S is the multiplicative set consisting of all finite products of nonzero degree 2 elements ofΛ.b In Q[M]_S, all forms of the type 1−χ^m are invertible, and in Λb_S, the image of 1−χ^m under Φ is s(m)^1−e_s(m)^s(m), where ^1−e_s(m)^s(m) =−1−¹₂s(m)− · · · is invertible inΛ. Thus, web have well-defined ring homomorphism

Φ :Q[M]_S −!Λb_S. Then, by definitions, it easy to see that

Φ( ˜χ(X,L)) =χ^T(L)

when X=XΣ is complete. Using Σ, we now define the local version of χ^T(L).

Definition 7.3.2. Let X=X_Σ be complete of dimension nand set L =OX(D) for a torus-invariant Cartier divisorD. Given σ∈Σ(n), define

χ^T_σ(L) = Φ(S( ˜χ(U_σ,L))).

Applying Φ to Theorem 7.2.6 directly gives the local decomposition of equivariant Euler characteristic.

Theorem 7.3.3. Let X = X_Σ be complete of dimension n and set L = OX(D) for a torus-invariant Cartier divisor D with Cartier data {m_σ}_σ∈Σ(n). Then:

(a) χ^T(L) =P

σ∈Σ(n)χ^T_σ(L) in Λb_S.

(b) If σ ∈ Σ(n) is simplicial and m_σ,1, . . . , m_σ,n ∈ M are the minimal generators of σ^∨ ⊆M_R, then

χ^T_σ(L) = e^s(mσ)·P

m∈P_σ∨∩Me^s(m) Qn

i=1(1−e^s(mσ,i)) for P_σ^∨ ={Pn

i=1λimσ,i |0≤λi <1}. In particular, if σ is smooth, then χ^T_σ(L) = e^s(mσ)

Qn

i=1(1−e^s(mσ,i)).

Fixed Points.The next step is to decompose the equivariant integral. LetX=XΣ, and assume thatXis complete and simplicial. Given σ∈Σ(n),V(σ) ={x_σ}is a fixed point of the torus action. Consider the mapp◦i_σfromxσ to pt, whereiσ :V(σ) ={x_σ},−!X and p:X!{pt}, induces a commutative diagram

Hb_T^•(X,Q)

i^∗_σ

((

Hb_T^•({pt},Q) =Λb

p^∗

OO

∼ //Hb_T^•({x_σ},Q)

(7.3.5)

where the isomorphism on the bottom is (p ◦iσ)^∗ = i^∗_σ ◦p^∗. From now on, for α ∈ Hb_T^•(X,Q), we can regardi^∗_σ(α)∈as an element of Λ.b

The following lemma is useful when we prove later propositions.

Lemma 7.3.4. Let X = X_Σ be complete and simplicial of dimension n. Also let σ ∈ Σ(n) and ρ∈Σ(1). Then:

(a) If ρ /∈σ(1), then i^∗_σ([D_ρ]_T) = 0.

(b) If ρ∈σ(1), theni^∗_σ([D_ρ]_T) = ⁻¹_l s(m), wherel be the smallest positive integer such thatlD_ρ is Cartier and m∈M satisfies lD_ρ∩U_σ = div(χ^m).

Proof. Using Proposition 6.3.1 (c), for any ρ ∈Σ(1), we have j_σ^∗([D_ρ]_T) = [D_ρ∩U_σ]_T, when we factor iσ as {x_σ} ,! Uσ

jσ

,! X. [Dρ∩Uσ]T is zero, because Dρ∩Uσ = ∅ if ρ6∈σ(1).

For ρ∈σ(1), using Proposition 6.3.1 (b) and (c) and our hypothesis, we obtain j_σ^∗([lDρ]T) = [lDρ∩Uσ]T = [div(χ^m)]T =−s(m)·1∈Hb_T^•(Uσ,Q).

Note that the constant map to a point makes all asΛ-modules, andb s(m) is inΛ. Thus,b considering s(m) as a scalar implies thati^∗_σ([lDρ]T) =−s(m)·1.

We also have the equivariant Gysin mapiσ! :Hb_T^•({pt},Q)!Hb_T^•(X,Q) from Propo- sition A.1.1. Here is an important property of this map.

Proposition 7.3.5. Let X =XΣ be complete and simplicial. If σ ∈Σ(n), then iσ!(1) = mult(σ) Y

ρ∈σ(1)

[Dρ]T. Proof. Note that Q

ρ∈σ(1)[Dρ]T and iσ!(1) live in H_T²ⁿ(X,Q), see Proposition A.1.1. We show the following two things:

Z

Xeq

i_σ!(1) = 1 and Z

X^eq

mult(σ) Y

ρ∈σ(1)

[D_ρ]_T = 1. (7.3.6)

i^∗_σ⁰(iσ!(1)) = 0 and i^∗_σ⁰( Y

ρ∈σ(1)

[Dρ]T) = when σ⁰∈Σ(n), σ⁰ 6=σ. (7.3.7) For (7.3.6), by Proposition A.1.1 (b) and R

X^eq = p_!, p ◦ i_σ : {x_σ} ! {pt} im- plies that p_!(i_σ!(1)) = p_!◦i_σ!(1) = 1. On the other hand, from Proposition A.1.1 (a), R

X^eq

Q

ρ∈σ(1)[Dρ]T is a degree zero element in bλ. Using the commutative diagram in Proposition A.1.3 and Proposition 6.3.1 (d),

i^∗_pt Z

X^eq

Y

ρ∈σ(1)

[Dρ]T = Z

X

i^∗_X( Y

ρ∈σ(1)

[Dρ]T) = Z

X

Y

ρ∈σ(1)

i^∗_X[Dρ]T = Z

X

Y

ρ∈σ(1)

[Dρ].

By Lemma 6.3.4, and the fact theR

X[x_σ] = 1, we are done with desired result.

To achieve (7.3.7), for different cones σ6=σ⁰ in Σ(n), Proposition A.1.2 gives 0 ^//Hb_T^•({x_σ⁰},Q)'Λb

Hb_T^•({x_σ},Q)

OO

iσ! //Hb_T^•(X,Q)

i^∗

σ0

OO

Thus we havei^∗_σ0◦i_σ!= 0. Also, Lemma 7.3.4 implies that the second equality in (7.3.7).

(7.3.6) and (7.3.7) imply that iσ!(1) and mult(σ)Q

ρ∈σ(1)[Dρ]T are equal inH_T²ⁿ(X,Q).

Decomposing the Equivariant Integral.Now we decompose the equivariant integral, for smooth complete toric varietyX =X_Σ. This is important because when we do real computation for explicit varieties as examples, applying nonequivariant limit is just the process of extracting some information from this equivariant integral.

From the constant mapp:X!{pt},Hb_T^•(X,Q) is aΛ-module viab p^∗in the commu- tative diagram (7.3.5). In addition, we have the Gysin map R

X^eq =p_!:Hb_T^•(X,Q) !Λ,b being aΛ-module homomorphism, because of Proposition A.1.1 (c). These maps extendsb to localizations at S, where S is the multiplicative set of all finite products of nonzero degree 2 elements of Λ. Note that we identifiedb Λ byb Hb_T^•({x_σ},Q) through (p◦i_σ)^∗, which is the inverse of the map (p◦i_σ)_!.

In the next theorem, we will work on the localization Hb_T^•(X,Q)S, which is a ΛbS- module.

Theorem 7.3.6. If X = X_Σ is an n-dimensional smooth complete toric variety and α∈Hb_T^•(X,Q), then

Z

X^eq

α= (−1)ⁿ X

σ∈Σ(n)

i^∗_σ(α) Qn

i=1s(m_σ,i)

in ΛbS, where mσ,1, . . . , mσ,n are the minimal generators of σ^∨ ⊆ M_R for σ ∈ Σ(n). In particular, when X is simplicial, we have

Z

X^eq

α= (−1)ⁿ X

σ∈Σ(n)

mult(σ^∨)i^∗_σ(α) Q_n

i=1s(m_σ,i)

Proof. As a first step, we calculate without summation over n-dimensional cones Z

X^eq

α· (−1)ⁿi^∗_σ(α) Qn

i=1s(mσ,i) =p!

(−1)ⁿ(α·i^∗_σ(α)) Qn

i=1s(mσ,i)

=p_!

(−1)ⁿi_σ!(i^∗_σ(α)) Qn

i=1s(mσ,i)

by Proposition A.1.1 (c)

=

(−1)ⁿ Qn

i=1s(m_σ,i)

p_!(i_σ!(i^∗_σ(α)))

=

(−1)ⁿ Q_n

i=1s(m_σ,i)

i^∗_σ(α) ∵identification via (p◦iσ)^∗. Next, we will use the map δS :Hb_T^•(X,Q)S ' L

σ∈Σ(n)ΛbS from (7.3.4). Denote the identity in Λb_S with respect to σ ∈ Σ(n) by e_σ. Let u_i be the minimal generator of ρi ∈σ(1) for 1≤i≤n. Then smoothness ofσ gives the minimal generators mσ,i of σ^∨ as the dual basis, mult(σ) = 1, and Cartier divisors Dρi with Dρi ∩Uσ = div(χ^mσ,i).

Thus,

δ(i_σ!(1)) =X

σ⁰

i^∗_σ0(i_σ!(1))·e_σ⁰

=i^∗_σ(i_σ!(1))·e_σ ∵(7.3.7)

=

n

Y

i=1

i^∗_σ([Dρi]T)·eσ by Proposition 7.3.5

=

n

Y

i=1

(−s(m_σ,i))·e_σ by Lemma 7.3.4 (b) with l= 1.

Now, consider that injectivity ofδ gives that δ

X

σ∈Σ(n)

(−1)ⁿi_σ!(1) Qn

i=1s(m_σ,i)

= X

σ∈Σ(n)

e_σ =δ(1) =⇒ X

σ∈Σ(n)

(−1)ⁿi_σ!(1) Qn

i=1s(m_σ,i) = 1∈Hb_T^•(X,Q)_S Hence, using the above, we can deduce the desired result for the smooth case.

For the simplicial case, using the fact that mult(σ)mult(σ^∨) = Qn

i=1l_i, where l_i is the smallest positive integer such thatmi=lim⁰_i ∈M for the dual basism⁰_i, after simple computation, we are done.

Proof of Equivariant RR.Now, we prove the equivariant Riemann-Roch theorem for smooth complete toric varieties.

Proof of Theorem 7.3.1. Denoteα= ch^T(L)Td^T(X).D is Cartier with the Cartier data {m_σ}_σ∈Σ(n), satisfying D|_Uσ= div(χ^mσ). Set m_i = m_σ,i, which are the minimal generatorsni ofσ, and Di =Dρi forρi ∈Σ(1) withDi|_Uσ= div(χ^mi). Then,

i^∗_σ(ch^T(L)) =i^∗_σ(e^[D]T) by Definition

=e^i∗σ([D]T)

=e^s(mσ) by the proof of Lemma 7.3.4.

i^∗_σ(Td^T(X)) =i^∗_σ( Y

ρ∈Σ(1)

[Dρ]T

1−e^−[Dρ]T) = Y

ρ∈Σ(1)

i^∗_σ

[Dρ]T

1−e^−[Dρ]T

= Y

ρ∈σ(1)

i^∗_σ

[Dρ]T

1−e^−[Dρ]T

by Lemma 7.3.4 (a)

= Y

ρ∈σ(1)

−s(m_i)

1−e^s(mi) by Lemma 7.3.4 (b)

Thus, we have

χ^T(L) = X

σ∈Σ(n)

e^s(mσ) Qn

i=1(1−e^s(mi)) by Theorem 7.3.3 (b)

= X

σ∈Σ(n)

(−1)ⁿ Qn

i=1s(mi) ·e^s(mσ)· Qn

i=1s(m_i) Qn

i=1(1−e^s(mi))

= X

σ∈Σ(n)

(−1)ⁿ Qn

i=1s(mi) ·i^∗_σ(ch^T(L))·i^∗_σ(Td^T(X))

= X

σ∈Σ(n)

(−1)ⁿ Qn

i=1s(mi) ·i^∗_σ(α)

∵i^∗_σ is a ring homomorphisim

= Z

X^eq

α by Theorem 7.3.6.

Passing from Equivariant RR to HRR.Using nonequivariant limits, we conclude our goal.

Theorem 7.3.7. For an invertible sheafL on a smooth complete toric varietyX=X_Σ, we have

χ(L) = Z

X

ch(L)Td(X).

Proof. SinceXis smooth,L =OX(D) for some Cartier divisorDonX. We may assume

that this is a torus-invariant divisor since DivTN(X)!Cl(X) is surjective. Then χ(L) =i^∗_pt(χ^T(L)) Definition of nonequivariant limit

=i^∗_pt( Z

X^eq

ch^T(L)Td^T(X)) by equivariant RR

= Z

X

i^∗_X(ch^T(L)Td^T(X)) by Proposition A.1.3

= Z

X

i^∗_X(ch^T(L))i^∗_X(Td^T(X)) sincei^∗_X is a ring homomorphism

= Z

X

ch(L)Td(X)

In fact, if we modify the the equivariant Todd class of a simplicial toric variety X and use the quotient construction ofX, then we can obtain a simplicial version of HRR.

The Simplicial Case.We introduce the equivariant Riemann-Roch theorem and HRR whenX=XΣ is complete and simplicial. In this case, Todd class cannot be represented by [Dρ] forρ∈Σ(1) as before. Thus it need to be modified.

Let X = X_Σ be a complete simplicial toric variety. Consider the group G in the quotient construction ofX, namely

G=

(tρ)∈(C^∗)^Σ(1) |Y

ρ

t^hm,uρ ^ρi= 1 for all m∈M . Each ρ∈Σ(1) gives the character

χρ: (C^∗)^Σ(1)−!C^∗

defined by projection on the ρth factor. Then for any coneσ ∈Σ, let G_σ =

g∈G|χ_ρ(g) = 1 for all ρ6∈σ(1)

=

(t_ρ)∈(C^∗)^Σ(1) |t_ρ= 1 for ρ6∈σ(1), Y

ρ∈σ(1)

t^hm,u_ρ ^ρi= 1 for m∈M . One can show that

G_σ 'N_σ/(Σ_ρ∈σ(1)Zu_ρ), so that|G_σ|= mult(σ).

Then we set

G_Σ= [

σ∈Σ

G_σ ⊆G

and define theequivariant Todd class ofX to be Td^T(X) = X

g∈G_Σ

Y

ρ∈Σ(1)

[Dρ]T

1−χρ(g)e^−[Dρ]T (7.3.8)

whenX =XΣ is complete and simplicial. Here is the result proved in [12] by Brion and Vergne. This result has also been proved by Edididin and Graham, [13], using different methods.

Theorem 7.3.8. Let X=X_Σ be a complete simplicial toric variety. If L =OX(D) is the line bundle of a torus-invariant Cartier divisor D onX, then

χ^T(L) = Z

X^eq

ch^T(L)Td^T(X), where Td^T(X) is defined in (7.3.8).

Once we have a simplicial version of equivariant Riemann-Roch, the proof of Theorem 7.3.7 gives HRR in the simplicial case as follows.

Corollary 7.3.9. Let X = X_Σ be a complete simplicial toric variety. If L is a line bundle on X, then

χ(X) = Z

X

ch(L)Td(X), where

Td(X) = X

g∈G_Σ

Y

ρ∈Σ(1)

[D_ρ] 1−χ_ρ(g)e^−[Dρ].