TORIC RESIDUE AND GLOBAL DUALITY

EVGENY MATEROV

Abstract. The notion of toric residue was introduced by D. Cox. Let X be a compact n-dimensional toric variety. Then the toric residue is associated with n + 1 divisors E_{0}, E_{1}, …, E_{n} on X having empty intersection and defined by a sequence of polynomials F_{0}, F_{1}, …, F_{n} in homogeneous coordinate ring of X. In the first (algebraic) part of this paper, we prove a Global Duality Theorem for toric residues which says that a homogeneous polynomial H belongs to the ideal ⟨F_{0}, F_{1}, …, F_{n}⟩ if and only if Res_{F} (H · G) = 0 for all homogeneous polynomials G. In the second (analytic) part, using the Weil integral formula, we show that it is sufficient to verify the vanishing of the residue only for finite number of polynomials G and give an algorithm of how to choose them.

1. Introduction

We begin this section with a review of some basic properties of multidimensional residues (see [GH, Ts]). Let I(f) = ⟨f_{1}, …, f_{n}⟩ be a polynomial ideal in the ring ℂ[z] = ℂ[z_{1}, …, z_{n}] defined by a system of polynomials f = (f_{1}, …, f_{n}) with discrete set of zeros Z_{f} ⊂ ℂ^{n}. Given another polynomial h ∈ ℂ[z], consider the meromorphic n-form on ℂ^{n}

ω_{h} = \frac{h}{f_{1} ⋯ f_{n}} dz_{1} ∧ … ∧ dz_{n}.

We denote by

Res_{f}(h) = ∑_{a∈Z_{f}} res_{a}(ω_{h})

the sum of Grothendieck point residues, the so-called global residue. If h ∈ I(f), then it is well-known that Res_{f}(h) = 0. Since Z_{f} is finite, the quotient ring R(f) = ℂ[z]/I(f) is a finite dimensional ℂ-space. Therefore, the residue defines a ℂ-linear map

Res_{f} : R(f) → ℂ.

The classical Global Duality Theorem for residues says that the polynomial h ∈ ℂ[z] belongs to the ideal I(f) if and only if global residue vanishes on all possible products of h with polynomials g ∈ ℂ[z]:

h ∈ I(f) ⟺ Res_{f}(h · g) = 0, ∀g ∈ ℂ[z].

The global duality has the following algebraic interpretation. Denote by R(f)^{∗} = Hom_{ℂ}(R(f), ℂ) the dual to R(f) vector space. Then there is pairing

(1)  R(f) × R(f)^{∗} → R(f)^{∗}, (h, φ) ↦ (h.φ)

given by means of multiplication (h.φ)(g) = φ(h · g). Then the global duality is equivalent to saying that the global residue Res_{f} defines a generator of R(f)^{∗} as a R(f)-module and the pairing (1) is nondegenerate.

The aim of the first part of this paper is to show that the similar duality holds for toric residue. The notion of toric residue was introduced by D. Cox [Cox2]. It is defined for every collection of n + 1 effective divisors E_{0}, E_{1}, …, E_{n} having empty intersection on a complete toric variety X of dimension n. Let Σ be a fan defining toric variety with r one-dimensional generators e_{1}, …, e_{r}. Then there are homogeneous coordinates x_{1}, …, x_{r} of the ring S = ℂ[x_{1}, …, x_{r}] graded by A_{n−1}(X) which are associated with e_{1}, …, e_{r} [Cox1]. If F = (F_{0}, F_{1}, …, F_{n}) is a collection of homogeneous polynomials in S of degrees α_{0}, α_{1} …, α_{n} ∈ A_{n−1}(X) and defining E_{0}, E_{1}, …, E_{n}, then there is a toric residue map

(2)  Res_{F} : S_{ρ}/⟨F_{0}, F_{1}, …, F_{n}⟩_{ρ} → ℂ

on the graded part of the quotient S/⟨F_{0}, F_{1}, …, F_{n}⟩ of the so-called critical degree

ρ = ∑_{i=0}^{n} α_{i} − ∑_{j=1}^{r} deg(x_{j}).

It was recently shown in [CD] that if F_{0}, F_{1}, …, F_{n} satisfy the properties: F_{i} don’t vanish simultaneously on X and α_{i} is in Pic(X) such that O_{X}(α_{i}) is generated by global sections and each polyhedron Δ_{i} associated with α_{i} is of dimension n, then

dim_{ℂ} S_{ρ}/⟨F_{0}, F_{1}, …, F_{n}⟩_{ρ} = 1,

from which follows that the mapping (2) is an isomorphism. It is a nice open question to find the explicit description of this mapping. For example, if all degrees of F_{0}, F_{1}, …, F_{n} are equal, i.e. α = α_{0} = α_{1} = ⋯ = α_{n}, the isomorphism (2) is given by the toric Jacobian J_{F} [Cox2]:

Res_{F} (J_{F}) = Vol(Δ),

where Δ is the polyhedron associated with α. Another realizations of this mapping which gives residue one can be found in [CCD] and [AK].

We show in Section 1, that under the same conditions on F_{0}, F_{1}, …, F_{n} as above, there is a duality similar to (1):

R(F)_{α} × R(F)_{ρ−α} → R(F)_{ρ}, α ∈ Pic(X)

if the polyhedron Δ associated with α is of dimension n, which is a perfect pairing given by multiplication. The proof consists in considering a differential of some spectral sequence. This duality implies the Global Duality for toric residues: the polynomial H belongs to the ideal ⟨F_{0}, F_{1}, …, F_{n}⟩ if and only if Res_{F} (H · G) = 0 for all G ∈ S.

In Section 2, we show that it is sufficient to check the equality Res_{F} (H · G) = 0 only for finite numbers of polynomials G ∈ S and propose an algorithm of how to choose them by using the Weil integral formula. Also, this helps to find the generators of the graded piece of the quotient ring S/⟨F_{0}, F_{1}, …, F_{n}⟩ of degree α.

Again, this repeats in some sense the classical situation. Let {U_{a}}_{a∈Z_{f}} be a family of disjoint neighborhoods of the point a ∈ Z_{f}, where f = (f_{1}, …, f_{n}) is a polynomial mapping f : ℂ^{n} → ℂ^{n} with discrete set of zeros Z_{f} ⊂ ℂ^{n} as before. Choose ε ∈ ℝ_{+}^{n} so small that the polyhedron

Π_{a}^{ε} = {z ∈ U_{a} : |f_{1}(z)| < ε_{1}, …, |f_{n}(z)| < ε_{n}}

is relatively compact in U_{a}. Set Π^{ε} = ∪_{a∈Z_{f}} Π_{a}^{ε} and denote by Γ^{ε} its distinguished boundary. The Weil integral representation generalize the Cauchy integral formula. It recovers the value of the function h ∈ ℂ[z] in the polyhedron Π^{ε} from its values on the distinguished boundary Γ^{ε}:

h(z) = (\frac{1}{2πi})^{n} ∫_{Γ^{ε}} \frac{h(ξ) H(ξ, z)}{f(ξ) − f(z)} dξ_{1} ∧ … ∧ dξ_{n}, z ∈ Π^{ε}.

Here H(ξ, z) is the determinant of the matrix ||P_{jk}(ξ, z)||_{1≤j,k≤n} in the Hefer expansion

f_{j}(ξ) − f_{j}(z) = ∑_{k=1}^{n}(ξ_{k} − z_{k})P_{jk}(ξ, z), j = 1, …, n.

Since |f_{j}(z)| < |f_{j}(ξ)| for z ∈ Π^{ε}, ξ ∈ Γ^{ε}, j = 1, …, n, the integrand can be expanded into a multiple geometric progression

h(ξ) · H(ξ, z) · [f(ξ) − f(z)]^{−1} = h(ξ) · H(ξ, z) · ∑_{|α|≥0} \frac{f^{α}(z)}{f^{α+I}(ξ)}

which converges uniformly on the compact subsets of Γ^{ε} × Π^{ε}. After term-by-term integration, we obtain

h(z) = Res_{f}(h(ξ)H(ξ, z)) + ∑_{|α|≥1} (\frac{1}{2πi})^{n} [∫_{Γ^{ε}} \frac{h(ξ)H(ξ, z)dξ}{f^{α+I}(ξ)} ] f^{α}(z).

Note that the second summand is in the ideal ⟨f_{1}, …, f_{n}⟩ in O(Π^{ε}). Using the fact that Z_{f} ⊂ Π^{ε} and Res_{f}(h(ξ)H(ξ, z)) is a polynomial in z, we can write modulo the ideal I(f):

h(z) = Res_{f}(h(ξ)H(ξ, z)),

since analytic and algebraic memberships are equivalent. Now expanding the Heferian into the sum of products of functions in disjoint variables

H(ξ, z) = ∑_{j=1}^{p} φ_{j}(z) · ψ_{j}(ξ),

we obtain modulo the ideal I(f) the equality

h(z) = ∑_{j=1}^{p} Res_{f}(h(ξ) · ψ_{j}(ξ)) · φ_{j}(z)

which implies that h(z) ∈ I(f) if and only if Res_{f}(h(ξ) · ψ_{j}(ξ)) = 0 for all j = 1, …, p.

We apply the same idea for finding the polynomials G ∈ S for which it is sufficient to check the vanishing Res_{F} (H · G) = 0. To do this, we use an analytic expression of the toric residue as a sum of residues in the torus from [CCD] and Weil integral formulas for the sum of residues in the torus from [BY1, BY2].

Acknowledgments. The author was supported by grants of Ministry of Education, St. Petersburg (grant E02-1-138) and Russian Foundation for Basic Research, Leading Research Schools (grant NSh-1212.2003.1). He would like to thank Max-Planck-Institut für Mathematik for hospitality and support.

2. Global Duality Theorem

We begin with some notations and and review basic facts on toric varieties and toric residues. Let us fix a lattice N ≅ ℤ^{n} and its dual lattice M = Hom_{ℤ}(N, ℤ). Their real extensions will be denoted by N_{ℝ} and M_{ℝ} respectively. A fan Σ is a collection of convex rational polyhedral cones in N_{ℝ} such that each face of a cone in Σ belongs to Σ and the intersection of two cones in Σ is a face of each. Let X = X_{Σ} be a compact toric variety over the field ℂ of dimension n defined by the complete fan Σ. The variety X contains the torus 𝕋 = N ⊗ ℂ^{∗} ≅ (ℂ^{∗})^{n} as a dense open subset.

The set of minimal edge generators of the fan Σ will be denoted by G(Σ) = {e_{1}, …, e_{r}}. Each vector e_{j} corresponds to the torus-invariant divisor D_{i} of X and to the variable x_{j} of the homogeneous coordinate ring S = ℂ[x_{1}, …, x_{r}] introduced by Cox [Cox1]. The ring S is graded by the Chow group A_{n−1}(X) as follows: a monomial x^{a} = ∏_{i=1}^{r} x_{i}^{a_{i}} has degree α = deg(x^{a}) = [∑_{i=1}^{r} a_{i}D_{i}] ∈ A_{n−1}(X). Denote by S_{α} the α-graded part of S, i.e. the linear combinations of all monomials in S of degree α. The subring S_{α} corresponding to the class α = [D] can be identified with the space of global sections H^{0}(X, O_{X}(D)) corresponding to the invertible sheaf O_{X}(D). We define the convex polyhedron associated with the torus-invariant divisor D = ∑_{i=1}^{r} a_{i}D_{i} by the formula

Δ_{D} = {m ∈ M_{ℝ} : ⟨m, e_{i}⟩ ≥ −a_{i}, i = 1, …, r} ⊂ M_{ℝ}.

In fact, since the polyhedrons corresponding to divisors in the same class in A_{n−1}(X) are isomorphic, the polyhedron Δ_{D} depends only on the class of D, and we may assign polyhedron Δ_{α} to the class α = [D] ∈ A_{n−1}(X) by choosing any representative in α.

Given homogeneous polynomials F_{0}, F_{1}, …, F_{n} in S of degrees α_{0}, α_{1}, …, α_{n}, we denote by

ρ = ∑_{j=0}^{n} α_{j} − ∑_{i=1}^{r} deg(x_{i}) ∈ A_{n−1}(X)

their critical, or Macaulay degree. Note, that it is follows from the definition of ρ that the polyhedron Δ_{ρ} is the relative interior of the Minkowski sum of the polyhedrons Δ_{0}, Δ_{1}, …, Δ_{n} corresponding to α_{0}, α_{1}, …, α_{n}:

Δ_{ρ} = (Δ_{0} + Δ_{1} + ⋯ + Δ_{n})^{∘}.

Suppose that the divisors E_{0}, E_{1}, …, E_{n} ⊂ X of polynomials F_{0}, F_{1}, …, F_{n} are such that

E_{0} ∩ E_{1} ∩ … ∩ E_{n} = ∅.

Then, the charts U_{i} = {x ∈ X : F_{i}(x) ≠ 0} define the open covering U = {U_{i}, i = 0, …, n} of X. Denote by Ω the Euler differential form on X (see [BC]), which is a generator of H^{0}(X, Ω̂_{X}^{n}(∑_{i=1}^{r} D_{i})), where Ω̂_{X}^{n} is a sheaf of n-forms of Zariski on X. Then for any polynomial H ∈ S_{ρ}, the differential form

ω_{F} (H) = \frac{H Ω}{F_{0}F_{1} ⋯ F_{n}} ∈ Ω̂_{X}^{n}(U_{0} ∩ U_{1} ∩ … ∩ U_{n})

is a well-defined Čech cocycle with respect to the covering U. Since the sheaf Ω̂_{X}^{n} is the dualizing sheaf for X [D], there exists a trace morphism Tr_{X} : H^{n}(X, Ω̂_{X}^{n}) → ℂ.

Definition 2.1. (D. Cox, [Cox2]) The toric residue of the form ω_{F} (H) (or, of the polynomial H) is defined by the trace morphism

Res_{F} (H) := Tr_{X}([ω_{F} (H)]).

If X is simplicial toric variety, then trace map can be realized explicitly via the integration over X:

Res_{F} (H) = (\frac{−1}{2πi})^{n} ∫_{X} η_{ω},

where η_{ω} is (n, n)-form corresponding to ω_{F} (H) under the Dolbeaut isomorphism

H^{n}(X, Ω̂_{X}^{n}) ≅ Hn,n ; \overline{∂}(X) ≅ H^{2n}(X).

One of the main properties of the toric residue is: if H ∈ ⟨F_{0}, F_{1}, …, F_{n}⟩, then Res_{F} (H) = 0 [Cox2]. Indeed, if H ∈ ⟨F_{0}, F_{1}, …, F_{n}⟩, then ω_{F} (H) is a Čech coboundary and its class [ω_{F} (H)] in H^{n}(U, Ω̂_{X}^{n}) is zero. Therefore, toric residue defines the mapping

Res_{F} : S_{ρ}/⟨F_{0}, F_{1}, …, F_{n}⟩_{ρ} → ℂ, (3)

where the subscript ρ in ⟨F_{0}, F_{1}, …, F_{n}⟩_{ρ} means that we take ρ-homogeneous part of the ideal ⟨F_{0}, F_{1}, …, F_{n}⟩. Denote by R(F) the quotient ring R(F) = S/⟨F_{0}, F_{1}, …, F_{n}⟩ which is graded by the Chow group A_{n−1}(X). It was proved in [CD, Corollary 2.5], that under certain assumptions on F_{0}, F_{1}, …, F_{n} the quotient ring R(F)_{ρ} is one-dimensional, which implies that the mapping (3) is an isomorphism. We refer to the original work [CD] for the comprehensive study of the estimates of the dimension of the quotient R(F)_{ρ}.

Theorem 2.2. (D. Cox, A. Dickenstein, [CD]) Suppose that X is a compact toric variety of dimension n. Let F = (F_{0}, F_{1}, …, F_{n}) be a sequence of polynomials regular in S, where their degrees α_{0}, α_{1}, …, α_{n} ∈ Pic(X) are globally generated by sections and the corresponding polyhedra Δ_{0}, Δ_{1}, …, Δ_{n} are n-dimensional. Then

dim_{ℂ} R(F)_{ρ} = 1

and the toric residue mapping (3) is an isomorphism.

The main result of this section is the following duality theorem.

Theorem 2.3. Under assumptions of the theorem 2.2 on F = F_{0}, F_{1}, …, F_{n}, for any class α ∈ Pic(X) of a Cartier divisor on X generated by sections, such that the polyhedron Δ corresponding to α is of dimension n, the pairing

R(f)_{α} × R(f)_{ρ−α} → R(f)_{ρ}

given by multiplication is perfect.

This implies immediately the Global Duality Theorem for toric residue which answers to the question when the polynomial H belongs to the ideal ⟨F_{0}, F_{1}, …, F_{n}⟩ in terms of toric residues.

Theorem 2.4. The bilinear form

B(·, ·) : R(f)_{α} × R(f)_{ρ−α} → ℂ

defined by the formula B(\overline{H}, \overline{G}) = Res_{F} (H · G) is nondegenerate, that is, if one has Res_{F} (H · G) = 0 for all G ∈ S_{ρ−α}, then H ∈ ⟨F_{0}, F_{1}, …, F_{n}⟩_{α}.

Also, as a corollary of theorem 2.3, we get the following.

Proposition 2.5. The α-graded part R(F)_{α} of R(F) is nonzero if and only if the polyhedron Δ_{α} lies in the relative interior of the Minkowski sum of the polyhedrons Δ_{0}, Δ_{1}, …, Δ_{n}:

R(F)_{α} ≠ 0 ⟺ Δ_{α} ⊂ (Δ_{0} + Δ_{1} + ⋯ + Δ_{n})^{∘}.

Proof of the theorem 2.3. To prove the theorem, we explore a spectral sequence, similar to used in [CD]. Denote by E the sheaf E = ⊕_{i=0}^{n}O_{X}(−α_{i}), and consider the exact sequence

0 → E^{0} → E^{1} → ⋯ → E^{n+1} → 0,

where

E^{p} = ⋀^{n+1−p} E(α) ≅ ⨁_{|J|=n+1−p} O_{X}(α − ∑_{j∈J} α_{j}),

i.e.

0 → ⋀^{n+1} E(α) → ⋀^{n} E(α) → ⋯ → E(α) → O_{X}(α) → 0,

obtained by taking a tensor multiple of the Koszul complex

0 → O_{X}(− ∑_{i} α_{i}) → ⋯ → ⨁_{i} O_{X}(−α_{i}) → O_{X} → 0

by O_{X}(α) over O_{X}. Then the spectral sequence associated with the naive filtration

F^{p} : 0 → E^{p} → ⋯ → E^{n+1} → 0.

has the terms

E_{1}^{p,q} = H^{q}(X, E^{p}) ≅ ⨁_{|J|=n+1−p} H^{q}(X, O_{X}(α − ∑_{j∈J} α_{j}))

and convergent to 0. Since E_{1}^{n,0} → E_{1}^{n+1,0} → E_{1}^{n+2,0} is

⨁_{k=0}^{n} S_{α−α_{k}} → S_{α} → 0,

one has E_{2}^{n+1,0} = R(F)_{α}. Similarly, the sequence 0 → E_{1}^{0,n} → E_{1}^{1,n} gives

0 → H^{n}(X, ⋀^{n+1} E(α)) → H^{n}(X, ⋀^{n} E(α)),

from which follows that

E_{2}^{0,n} = ker (H^{n}(X, ⋀^{n+1} E(α)) → H^{n}(X, ⋀^{n} E(α))) .

Then, we note that in the conditions of the statement, E_{1}^{p,q} = 0 if q is not 0 or n, and E_{∞}^{n+1,0} = 0 that implies that the differential in

E_{∞}^{n+1,0} = coker (d_{n+1} : E_{2}^{0,n} → E_{2}^{n+1,0})

is an isomorphism. In other words, one has an isomorphism

d_{n+1} : ker (H^{n}(X, ⋀^{n+1} E(α)) → H^{n}(X, ⋀^{n} E(α))) → R(F)_{α}.

Applying the Serre duality and the isomorphisms

⋀^{n+1} E(α) = O_{X}(α − ∑_{i} α_{i}), ⋀^{n} E(α) = E^{∗}(α − ∑_{i} α_{i}),

we see that the map

H^{n}(X, ⋀^{n+1} E(α)) → H^{n}(X, ⋀^{n} E(α))

is canonically dual to the map

H^{0} (X, E (∑_{i} α_{i} − α − β_{0})) → H^{0} (X, O_{X} (∑_{i} α_{i} − α − β_{0})) ,

where we write for simplicity β_{0} = ∑_{i=1}^{r} deg(x_{i}). Since the image of the last map is

⟨F_{0}, F_{1}, …, F_{n}⟩_{∑_{i} α_{i}−α−β_{0}} = ⟨F_{0}, F_{1}, …, F_{n}⟩_{ρ−α},

the differential d_{n+1} provides an isomorphism

d_{n+1} : (R(F)_{ρ−α})^{∗} ≅ R(F)_{α}.

It is not hard to check that the isomorphism constructed above is given by the pairing

R(F)_{α} × R(F)_{ρ−α} → R(F)_{α}

defined by the multiplication. Indeed, the bilinear pairing

(·, ·) : R(F)_{α} × R(F)_{ρ−α} → R(F)_{ρ}

given by multiplication

(G, H) ↦ G · H, G ∈ R(F)_{α}, H ∈ R(F)_{ρ−α}

satisfies

(P · G, H) = (G, P · H), ∀P ∈ R(F)_{β}, G ∈ R(F)_{α}, H ∈ R(F)_{ρ−α−β}.

In other words, the diagram

R(F)_{ρ−α} ⟶^{≅} R(F)_{α}
P^{∗}↓  ↓^{P}
R(F)_{ρ−α−β} ⟶^{≅} R(F)_{α+β}

commutes. Hence, the isomorphism provided by d_{n+1} is compatible with the multiplication and provides the bilinear pairing as required. □

3. Weil Formula for Toric Residues

Throughout this section we assume that the conditions of the theorem 2.3 are fulfilled. We show that in order to check that the polynomial H is in the ideal ⟨F_{0}, F_{1}, …, F_{n}⟩, it is sufficient to show that Res_{F} (H · G) = 0 in the Global Duality theorem 2.3 only for certain finite number of polynomials G ∈ S.

The choice of the basis m_{1}, …, m_{n} of the lattice M ≅ ℤ^{n} establishes the choice of the local coordinates t_{1}, …, t_{n} on the torus 𝕋 ≅ (ℂ^{∗})^{n}:

t_{j} = x^{⟨m_{j},e⟩} = ∏_{i=1}^{r} x_{i}^{⟨m_{j},e_{i}⟩}.

Given any Laurent polynomial

f(t) = f(t_{1}, …, t_{n}) = ∑_{a∈Δ∩ℤ^{n}} c_{a}t^{a} = ∑_{a∈Δ∩ℤ^{n}} c_{a_{1}...a_{n}}t_{1}^{a_{1}} ⋯ t_{n}^{a_{n}} ∈ ℂ[t_{1}^{±1}, …, t_{n}^{±1}]

with Newton polyhedron Δ = Δ(f), we define its Σ-homogenization with respect to the fan Σ generalizing the classical homogenization of polynomial in the case X_{Σ} = ℙ^{n}:

hf = F (x_{1}, …, x_{r}) = ∑_{a∈Δ∩ℤ^{n}} c_{a} ∏_{i=1}^{r} x_{i}^{⟨a,e_{i}⟩+d_{i}} = ∏_{i=1}^{r} x_{i}^{d_{i}}f(x^{⟨m_{1},e⟩}, …, x^{⟨m_{n},e⟩}),

where d_{i} = − min_{l∈Δ}⟨l, e_{i}⟩. The obtained polynomial F (x_{1}, …, x_{r}) has degree α = [∑_{i=1}^{r} d_{i}D_{i}].

We explore the analytic properties of the toric residue. It was proved in [CCD] that toric residue with respect to the system of divisors E_{0}, E_{1}, …, E_{n} defined by polynomials F_{0}, F_{1}, …, F_{n} can be written as a sum of local residues. This theorem is analogous to the situation in ℙ^{n} considered in [Ts, Corollary 1, p. 35].

Theorem 3.1. (E. Cattani, D. Cox, A. Dickenstein, [CCD, Theorem 0.4]) Let X be a complete toric variety and F_{0}, F_{1}, …, F_{n} ∈ S are homogeneous polynomials of degrees α_{0}, α_{1}, …, α_{n} respectively whose corresponding divisors E_{0}, E_{1}, …, E_{n} defined by E_{i} = {x ∈ X : F (x) = 0} are such that

E_{0} ∩ E_{1} ∩ … ∩ E_{n} = ∅

and for some index k ∈ {0, 1, …, n}

V_{k} = {E_{0} ∩ … ∩ Ê_{k} ∩ … ∩ E_{n}} ⊂ 𝕋

is a finite set of points. Then for any polynomial H ∈ S_{ρ} the toric residue Res_{F} (H) can be written as a sum of local residues

Res_{F} (H) = (−1)^{k} ∑_{t∈V_{k}} res_{t} ( \frac{(H/F_{k})Ω}{F_{0} ⋯ F̂_{k} ⋯ F_{n}} |_{𝕋}) , (4)

where |_{𝕋} denotes restriction of the form to the torus 𝕋.

Remark that the formula in the last theorem can be considered as an analytic definition of the toric residue and, in fact, the sum of the local residues does not depend on the compactification of the torus 𝕋. This formula was first proved for simplicial toric varieties and then extended in [CDS, Theorem 3.2] to arbitrary toric variety by the following arguments: if φ : X′ → X is a birational morphism preserving the torus 𝕋 and F′_{i} = φ^{∗}(F_{i}) ∈ H^{0}(X′, φ^{∗}(O_{X}(D))), H′ = φ^{∗}(H), then

Res_{F} (H) = Tr_{X}([ω_{F} (H)]) = Tr_{X′}([ω_{F′}(H′)])

by the property of the trace map.

Definition 3.2. For any Laurent polynomials h(t), f_{1}(t), …, f_{n}(t) ∈ ℂ[t_{1}^{±1}, …, t_{n}^{±1}], where V = {t ∈ 𝕋 : f_{1}(t) = ⋯ = f_{n}(t) = 0} ⊂ 𝕋 is a discrete set of isolated points, we denote the sum of the local residues in the torus by

Res [ \begin{matrix}h dt \\ f_{1}, …, f_{n} \end{matrix} ]_{𝕋} := ∑_{t∈V} res_{t} (\frac{h}{f_{1} ⋯ f_{n}} \frac{dt_{1}}{t_{1}} ∧ … ∧ \frac{dt_{n}}{t_{n}} ) .

If Laurent polynomials h(t), f_{0}(t), …, f_{n}(t) ∈ ℂ[t_{1}^{±1}, …, t_{n}^{±1}] are such that their homogenizations with respect to the fan Σ of X are H, F_{0}, …, F_{n} respectively, then the formula (4) can be rewritten as follows:

Res_{F} (H) = (−1)^{k}Res [ \begin{matrix}h/f_{k} dt \\ f_{0}, …, f̂_{k}, …, f_{n} \end{matrix} ]_{𝕋} . (5)

To find an analogue of the Weil integral formula for toric residue, we associate with the sequence of polynomials f_{0}, f_{1}, …, f_{n} their Hefer expansions. Let us write any Laurent monomial t^{γ} ∈ ℂ[t_{1}^{±1}, …, t_{n}^{±1}] as a quotient of monomials of nonnegative degrees: t^{γ} = (t′)^{α}/(t′′)^{β}, where t′ and t′′ are disjoint sets of coordinates. Then, extending the equalities

ξ^{γ} − t^{γ} = \frac{(ξ′)^{α}(t′′)^{β} − (t′)^{α}(ξ′′)^{β}}{(ξ′′)^{β}(t′′)^{β}} = \frac{∑_{k=1}^{n} q_{k}(ξ, t)(ξ_{k} − t_{k})}{(ξ′′)^{β}(t′′)^{β}}

by linearity, we obtain for any point (ξ, t) ∈ 𝕋_{ξ} × 𝕋_{t} the Hefer expansions:

f_{j}(ξ) − f_{j}(t) = ∑_{k=1}^{n} p_{kj}(ξ, t) (ξ_{k} − t_{k}), j = 0, …, n (6)

of Laurent polynomials f_{0}(t), f_{1}(t), …, f_{n}(t) ∈ ℂ[t_{1}^{±1}, …, t_{n}^{±1}]. This expansion is of course not unique. For example:

ξ_{1}ξ_{2} − t_{1}t_{2} = t_{1}(ξ_{2} − t_{2}) + (ξ_{1} − t_{1})t_{2} + (ξ_{1} − t_{1})(ξ_{2} − t_{2})
= ξ_{2}(ξ_{1} − t_{1}) + t_{1}(ξ_{2} − t_{2})
= t_{2}(ξ_{1} − t_{1}) + ξ_{1}(ξ_{2} − t_{2}).

The sufficient geometric conditions on Δ_{j} = Δ(f_{j}) so that the Laurent polynomials p_{kj}(ξ, t) can be chosen with supports in Δ_{j} are the following.

Definition 3.3. (C. Berenstein, A. Yger, [BY1, §3.2], [BY2, Definition 4.1]) The convex polyhedron Δ_{j} ⊂ ℝ^{n} is called good polyhedron, if

∀x ∈ Δ_{j}, {y ∈ ℝ^{n} : |y_{k}| ≤ |x_{k}|, x_{k}y_{k} ≥ 0, k = 1, …, n} ⊂ Δ_{j}.

Definition 3.4. Given a Hefer expansion of the Laurent polynomials f_{0}, f_{1}, …, f_{n}, we denote by W (ξ, t) their Weil polynomial

W (ξ, t) = ξ_{1} ⋯ ξ_{n} · det ( \begin{matrix}f_{0}(ξ) & f_{1}(ξ) & ⋯ & f_{n}(ξ) \\ p_{10}(ξ, t) & p_{11}(ξ, t) & ⋯ & p_{1n}(ξ, t) \\ ⋮ & ⋮ & ... & ⋮ \\ p_{n0}(ξ, t) & p_{n1}(ξ, t) & ⋯ & p_{nn}(ξ, t) \end{matrix} ) ,

where p_{jk}(ξ, t) are from the Hefer expansion (6).

It is easy to see that if the polynomials f_{0}(t), f_{1}(t), …, f_{n}(t) have good supports Δ_{0}, Δ_{1}, …, Δ_{n}, then the Weil polynomial W (ξ, t) has support in the sum of the polyhedrons Δ_{0}+Δ_{1}+⋯+Δ_{n}+1, where 1 = (1, …, 1). We denote by W (y, x) the polynomial in homogeneous coordinates (y_{1}, …, y_{r}, x_{1}, …, x_{r}) of X × X obtained by the Σ × Σ-homogenization of W (ξ, t).

Now we are ready to formulate the main result of this section - the toric Weil formula.

Theorem 3.5. Let X be a toric variety of dimension n defined by a fan Σ. Suppose that F_{0}(x), F_{1}(x), …, F_{n}(x) are homogeneous polynomials which do not have common zeros in X and obtained as Σ-homogenizations of the Laurent polynomials f_{0}(t), f_{1}(t), …, f_{n}(t) with good supports Δ_{0}, Δ_{1}, …, Δ_{n} of dimensions n. Let W (y, x) be a Σ × Σ-homogenization of the Weil polynomial. Then,

H(x) = Res_{F} (H(y) W (y, x)) mod ⟨F_{0}(x), F_{1}(x), …, F_{n}(x)⟩,

where the toric residue is taken with respect to the variables y.

Proof. Denote by H_{k}(ξ, t) the determinant of the minor of W (ξ, t) obtained by deleting the first line and the k-th row of W (ξ, t). Then, by the torc analogue of the Weil formula (see, e.g. [BY2, Proposition 4.2]), for any index k ∈ {0, …, n} one has the equality

h(t) = Res [ \begin{matrix}h(ξ) H_{k}(ξ, t) dξ \\ f_{0}, …, f̂_{k}, …, f_{n} \end{matrix} ]_{𝕋}

+ ∑_{ν∈ℕ^{n}\\{0\}}Res [ \begin{matrix}h(ξ) H_{k}(ξ, t) dξ \\ f_{0}^{ν_{0}+1}, …, f̂_{k}, …, f_{n}^{ν_{n}+1} \end{matrix} ]_{𝕋} f_{0}(t)^{ν_{0}} ⋯ { \atop f_{k}(t)} ⋯ f_{n}(t)^{ν_{n}}.

Note that the second sum over (ν_{0}, …, ν̂_{k}, …, ν_{n}) ∈ ℕ^{n}\{0} is in fact finite under assumption that Δ_{0}, …, Δ_{n} are good by the Jacobi-Khovanskii formula [Kh]. Indeed, let Δ be a support polyhedron for h(t). Writing the Hefer expansions for any k as

H_{k}(ξ, t) = ∑_{\begin{matrix}α,β∈ℤ^{n} \\ α+β+1∈Δ[k] \end{matrix}} c_{α,β}^{k} ξ^{α}t^{β},

where Δ[k] = Δ_{0} + ⋯ + Δ̂_{k} + ⋯ + Δ_{n}, the second sum in the right hand side of integral representation of h(t) can be rewritten as

∑_{\begin{matrix}α,β∈ℤ^{n} \\ α+β+1∈Δ[k] \end{matrix}} ∑_{\begin{matrix}ν∈ℕ^{n}\\{0\} \\ Δ+α⊄(⟨ν,Δ⟩)^{∘} \end{matrix}} c_{α,β}^{k} Res [ \begin{matrix}h(ξ) ξ^{α} dξ \\ f_{0}^{ν_{0}+1}, …, f̂_{k}, …, f_{n}^{ν_{n}+1} \end{matrix} ]_{𝕋} t^{β}f_{0}(t)^{ν_{0}} ⋯ { \atop f_{k}(t)} ⋯ f_{n}(t)^{ν_{n}},

where ⟨ν, Δ⟩ = (ν_{0} + 1)Δ_{0} + ⋯ + Δ̂_{k} + ⋯ + (ν_{n} + 1)Δ_{n}. So, we have

h(t) = Res [ \begin{matrix}h(ξ) H_{k}(ξ, t) dξ \\ f_{0}, …, f̂_{k}, …, f_{n} \end{matrix} ]_{𝕋} mod ⟨f_{0}(t), f_{1}(t), …, f_{n}(t)⟩. (7)

On the other hand, expand the Weil polynomial by the first line

W (ξ, t) = ∑_{l=0}^{n}(−1)^{l} f_{l}(ξ) H_{k}(ξ, t)

and consider the residue

Res [ \begin{matrix}h(ξ) W (ξ, t) dξ \\ f_{0}, …, f̂_{k}, …, f_{n} \end{matrix} ]_{𝕋} = ∑_{l=0}^{n}(−1)^{l}Res [ \begin{matrix}h(ξ) f_{l}(ξ) H_{l}(ξ, t) dξ \\ f_{0}, …, f̂_{k}, …, f_{n} \end{matrix} ]_{𝕋} .

Taking into account the formula for toric residue (5) and the property of residue that

Res [ \begin{matrix}g(ξ) dξ \\ f_{0}, …, f̂_{k}, …, f_{n} \end{matrix} ]_{𝕋} = 0, if g(ξ) ∈ ⟨f_{0}(ξ), f_{1}(ξ), …, f_{n}(ξ)⟩,

we see that

Res [ \begin{matrix}h(ξ) W (ξ, t) dξ \\ f_{0}, …, f̂_{k}, …, f_{n} \end{matrix} ]_{𝕋} = (−1)^{k}Res [ \begin{matrix}h(ξ) f_{k}(ξ) H_{k}(ξ, t) dξ \\ f_{0}, …, f̂_{k}, …, f_{n} \end{matrix} ]_{𝕋} .

Then, after the Σ-homogenization, the last equality together with (7) and analytic property of the toric residue (5) yields

H(x) = Res_{F} (H(y) W (y, x)) mod ⟨F_{0}(x), F_{1}(x), …, F_{n}(x)⟩,

as required. □

We can work over arbitrary algebraically closed subfield k of ℂ. By theorem 3.5, after taking a specialization W (a)(x, y) of the polynomial W (x, y) at the point a ∈ X, such that F_{0}(a), F_{1}(a), …, F_{n}(a) do not have zeros at a, we get

H(a)(x) = Res_{F} (H(y)W (a)(x, y))

in the quotient ring S/⟨F_{0}(a)(x), F_{1}(a)(x), …, F_{n}(a)(x)⟩. If we expand W (x, y) in the sum

W (x, y) = ∑_{μ} φ_{μ}(x) · ψ_{μ}(y), (8)

then, for any polynomial H(x) ∈ S_{α}

H(a)(x) = ∑_{μ} Res_{F} (H(y)ψ_{μ}(y))·φ_{μ}(a)(x) mod ⟨F_{0}(a)(x), F_{1}(a)(x), …, F_{n}(a)(x)⟩,

where φ_{μ}(a)(x) is the specialization of φ_{μ}(x) at a, and the sum of residues is taken over the terms of degree α = ρ − deg(ψ_{μ}(y)). Therefore, we have the following corollary of the theorem 3.5.

Corollary 3.6. The polynomials {φ_{μ}(a)(x)} in the expansion (8) whose degrees are α = ρ − deg(ψ_{μ}(y)), generates the specialization of the graded piece of the quotient ring R(F) = S/⟨F_{0}, F_{1}, …, F_{n}⟩ of degree α at the point a ∈ X such that {F_{0}(a) = F_{1}(a) = ⋯ = F_{n}(a) = 0} = ∅.

The polynomial H(x) ∈ S belongs to the ideal ⟨F_{0}(x), F_{1}(x), …, F_{n}(x)⟩ if and only if Res_{F} (H(y)ψ_{μ}(y)) = 0 for all polynomials {ψ_{μ}(y)} in the expansion (8).

The last statement in the corollary above claims that it is sufficient to check the vanishing of the residue in the theorem 2.4 only for finite set of polynomials {ψ_{μ}(y)} which can be done, for example, using Gröbner basis algorithm.

References

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[BY1] C. A. Berenstein, A. Yger, Multidimensional residues and complexity problems, Mathematics and Computers in Simulation 42 (1996) 449–457.

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[CCD] E. Cattani, D. Cox and A. Dickenstein, Residues in toric varieties, Compositio Math. 108 (1997), 35–76.

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[Cox1] D. Cox, The homogeneous coordinate ring of a toric variety, J. Algebraic Geometry 4 (1995), 17–50.

[Cox2] D. Cox, Toric residues, Ark. Mat. 34 (1996), 73–96.

[CD] D. Cox, A. Dickenstein, A codimension theorem for complete toric varieties, preprint.

[D] V. I. Danilov, The geometry of toric varieties, Russian Math. Surveys 33 (1978), 97–154.

[GH] P. Griffiths, J. Harris, Principles of Algebraic Geometry, Wiley Interscience, New York, 1978.

[Kh] A. G. Khovanskii, Newton polyhedra and Euler-Jacobi formula, Russian Math. Surveys 33 (1978), 237–238.

[Ts] A. K. Tsikh, Multidimensional Residues and their Applications, Trans. of Math. Mono-

graphs 103, AMS 1992.

  E-mail address: materov@mpim-bonn.mpg.de, materov@atomlink.ru, materov2000@mail.ru

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Current address: Max-Planck-Institut für Mathematik, Vivatsgasse 7, Bonn, D-53111, Deutschland