Integration on Artin toric stacks and Euler characteristics

There is a well developed intersection theory on smooth Artin stacks with quasi-affine diagonal. However, for Artin stacks whose diagonal is not quasi-finite the notion of the degree of a Chow cycle is not defined. In this paper we propose a definition for the degree of a cycle on Artin toric stacks whose underlying toric varieties are complete. As an application we define the Euler characteristic of an Artin toric stack with complete good moduli space - extending the definition of the orbifold Euler characteristic. An explicit combinatorial formula is given for 3-dimensional Artin toric stacks.


Introduction
Let X be complete Deligne-Mumford stack. There is a well-developed intersection theory on such stacks [Vis,EG98,Kre] which includes the notion of the degree of 0cycle. Over an algebraically closed field of characteristic 0 if x is a point of X with stabilizer G x then the degree of the 0-cycle [x] ∈ A 0 (X ) is 1/|G x |. As a consequence one can compute the integral of an algebraic cohomology class on a complete Deligne-Mumford stack. This theory of integration plays a crucial role in Gromov-Witten, Donaldson-Thomas and other invariants in modern algebraic geometry.
A natural problem is to extend the notion of degree to stacks with non-finite diagonal. While such stacks are not separated (and hence cannot be proper over the ground field) there is a class of Artin stacks which in many ways behave like complete Deligne-Mumford stacks. Specifically we consider Artin stacks which are generically Deligne-Mumford and have a complete good moduli space (in the sense of [Al]). If X is such a stack then there is a well-defined pushforward of Grothendieck groups p * : K(X ) → K(pt) = Z thereby defining Euler characteristics of sheaves on such stacks. Thus it is natural to expect there to be a corresponding pushforward for algebraic cycles as well as a Riemann-Roch theorem relating the K-theoretic and the Chow-theoretic pushforward via some analogue of the Chern character.
The purpose of the present paper is to take a first step in this direction by defining the degree of a 0-cycle on a toric Artin stack in the sense of [BCS]. Our definition is based on our previous paper [EM] where we showed that if X (Σ) is a toric Artin stack with complete good moduli space X(Σ) then there is a canonical birational morphism of toric stacks X (Σ ′ ) f → X (Σ) such that X (Σ ′ ) is Deligne-Mumford and such that the induced morphism of the underlying toric varieties is proper and birational. The degree of a 0-cycle on X (Σ) is then defined as the degree of its pullback to the Deligne-Mumford stack X (Σ ′ ).
As an application we define the stacky Euler characteristic of an Artin toric stack to be the degree of the Euler class of the tangent bundle stack, and we give a combinatorial formula for this Euler characteristic in dimension 3.
Along the way we prove that the integral Chow ring of an Artin toric stack equals the Stanley-Reisner ring of the associated fan. Iwanari [Iwa] proves this for Deligne-Mumford toric stacks; our proof is different.

Toric stacks and Stanley-Reisner rings
In this section, we recall the definition of toric stacks and in the process establish the notation we will use for toric stacks. As in [Iwa], for simplicity we will restrict our attention to stacky fans for which N is torsion free. A stacky fan Σ = (N, Σ, {v 1 , . . . , v n }) consists of: 1. a finitely generated free abelian group N = Z d , 2. a (not neccesarily simplicial) fan Σ ⊂ N Q , whose rays we will denote by ρ 1 , . . . , ρ n , 3. for every 1 ≤ i ≤ n, an element v i ∈ N such that v i lies on the ray ρ i .

Chow rings of Artin toric stacks.
Our first result is a description of the Chow ring of an Artin toric stack in terms of the combinatorics of its stacky fan. (1)]. Let I Σ be the ideal of S(Σ) generated by ρ∈Σ(1) m, v ρ x ρ as m ranges over M. Let J Σ be the ideal of S(Σ) generated by monomials x ρ 1 · · · x ρs such that the rays ρ 1 , . . . , ρ s are not contained in any cone in Σ: (1) We will call J Σ the Stanley-Reisner ideal of Σ. Following [Iwa, Definition 2.1] (but not exactly the possibly more standard terminology in [CLS,Definition 12.4.10], see also [MS]), define the Stanley-Reisner ring SR(Σ) to be Suppose a group G acts on a smooth scheme X, and let X = [X/G] be the resulting quotient stack. The integral Chow ring A * (X ) of X is the G-equivariant Chow ring A * G (X) of X [EG98]. We prove that the integral Chow ring of an Artin toric stack X (Σ) is the Stanley-Reisner ring of the stacky fan Σ. Iwanari [Iwa] proved the same result for toric Deligne-Mumford stacks. Our proof is different: rather than computing the equivariant Chow groups directly from the definition, we use the excision sequence for equivariant Chow groups.
The excision sequence in equivariant Chow groups is the exact sequence: Since Spec S is an affine bundle over pt = Spec C, we have A * G (Spec S) = A * G (pt) = S(Σ)/I Σ . By the following lemma, the image of A * G (V (B(Σ))) in A * G (Spec S) is the Stanley-Reisner ideal J Σ (equation 1). Hence A * G (C(Σ)) = S(Σ)/(I Σ + J Σ ) = SR(Σ), the Stanley-Reisner ring of Σ.
Lemma 2.3. Let B(Σ) = (X σ | σ ∈ Σ)S be as in Section 2.1. Then we have a primary decomposition where the intersection ranges over sets of rays {ρ 1 , . . . , ρ s } that are not contained in some cone in Σ.
To show the opposite inclusion, let RI denote the ideal on the right hand side of equation 3. Since RI is a monomial ideal, it suffices to show that monomials in RI belong to B(Σ), i.e. we need to show if a monomial X ρ 1 · · · X ρs lies in RI, then it lies in B(Σ). We will show the contrapositive, namely that if X ρ 1 · · · X ρs / ∈ B(Σ), In the remainder of this article, by virtue of the isomorphism (1)).

Integration on complete toric Deligne-Mumford stacks
In this section, we will study intersection theory on complete toric Deligne-Mumford stacks. In this case, we have an isomorphism φ : A * (X (Σ)) Q → A * (X(Σ)) Q [EG98] and for the rest of this paper, we will work with the rational (as opposed to integral) Chow ring A * (X (Σ)) Q of toric stacks.
Definition 3.1 (D σ,Σ , stacky multiplicity of a cone). Let N be a lattice and let σ be simplicial cone N R , and let s = dim σ. Suppose v 1 , . . . , v s are elements of N such that σ = R ≥0 v 1 , . . . , v s (in particular, each v i is a lattice point on exactly one ray of σ). Let N σ = Z N ∩ σ be the lattice generated by N ∩ σ. We define the (stacky) multiplicity of the stacky cone σ = (N, σ, {v 1 , . . . , v s }) to be the order of N σ /Z v 1 , . . . , v s , and denote this positive integer by D σ . Let Σ = (N, Σ, {v 1 , v 2 , . . . , v n }) be a stacky fan and σ a simplicial cone in Σ. Label the v i so that v 1 , . . . , v s are on the rays of σ. Let σ = (N, σ, {v 1 , . . . , v s }). Define the stacky multiplicity of σ with respect to Σ to be D σ , and denote this quantity by D σ,Σ .
3.1. Self-intersections on toric Deligne-Mumford stacks. If ρ 1 , . . . , ρ s are not distinct, then to compute φ(x ρ 1 · · · x ρs ), we can express x ρ 1 · · · x ρs as a rational linear combination of monomials with no factor x ρ appearing more than once, and then we can apply Proposition 3.2 to compute φ. The following theorem gives a procedure for finding such a linear combination. (1) ) be a complete simplicial stacky fan. Let ρ 1 , . . . , ρ s be distinct rays in Σ, and let a 1 , . . . , a s be positive integers and assume For a ray ρ ∈ C, let b ρ i ,ρ be rational numbers such that (Such a relation always exists and is unique since v ρ 1 , . . . , v ρ d form a basis of R d .) Then Remark 3.4. Note that on the right-hand side of eq. 6, the exponent of x ρ i 0 is one less than on the left-hand side, but for i = i 0 (and 1 ≤ i ≤ s) the exponent of x ρ i is the same on both sides. Since there was nothing special about x ρ 1 (other than a 1 ≥ 2), we can repeatedly apply Theorem 3.3 to express any monomial s i=1 x a i ρ i as a rational linear combination of monomials with no factor x ρ appearing more than once. We can then apply Proposition 3.2 to compute φ. Hence Theorem 3.3 gives a method to compute the integral of (or more generally, φ applied to) any Chow cohomology class on X (Σ). We give an example after the proof.

Euler characteristic of toric Deligne-Mumford stacks.
Definition 3.8. If X is an n-dimensional smooth complete Deligne-Mumford stack we define the Euler characteristic of X as χ(X ) := X c n (T) where T is the tangent bundle of X . The following is well-known but we include a proof for lack of a suitable reference.
Proposition 3.10. Let Σ = (N, Σ, {v 1 , v 2 , . . . , v n }) be a complete simplicial stacky fan (i.e Σ is simplicial and complete). Let Σ max denote the set of maximal cones in Σ. For a maximal cone σ ∈ Σ max , let D σ denote the multiplicity of σ. Then . . . , X n ] is the Cox space of Σ, and G = G(Σ) is as in Section 2.1. The coarse moduli space of X is X = X(Σ), the toric variety associated to Σ. The equivariant tangent bundle T C is ⊕O(X i ) ( [EG03,p.3751]) so Proposition 2.2). Hence if we expand the product n i=1 (1 + x i ), the only monomials x i 1 · · · x is that can contribute to X φ( 1 + x i ) are ones where v i 1 , . . . , v is span a maximal cone (we require maximal cones since the integral only considers the dimension 0 part of the cycle). In this case, the image of V (X i 1 , . . . , X is ) is a point in X(Σ), and Prop. 3.2 tells us to assign a factor of 1 D σ,Σ to the class [pt] of this point. The usual identification of A 0 (X(Σ)) with Z sending [pt] ∈ A * (X(Σ)) to 1 ∈ Z gives the result.
Remark 3.11. Note that Prop. 3.10 recovers the result (e.g. [CLS,Theorem 12.3.9]) that the Euler characteristic of a smooth toric variety is the number of maximal cones.
The following result gives expresses how the Euler characteristic increases after a stacky star subdivision [EM,Definition 4.1] of a simplicial stacky fan.
. . , v s be the cone of Σ σ spanned by v 0 and all the rays of σ except for v i . Then by properties of determinants, we have D σ i ,Σσ = D σ,Σ . Then apply Prop. 3.10.

Integration on Artin toric stacks
We now come to the main definitions of the paper. Let Σ be a (not necessarily simplicial) stacky fan and let X (Σ) be the associated Artin toric stack. By [EM,Theorem 5.2] there is a simplicial stacky fan Σ ′ canonically obtained from Σ by stacky star subdivisions and a commutative diagram of stacks and toric varieties where f is birational, π ′ is finite and g is proper and birational.
Definition 4.1. If α ∈ A * (X (Σ)) define π * α := g * π ′ * f * α. In particular if the associated toric variety X(Σ) is complete we define for any α ∈ A d (X (Σ)) The Euler characteristic of an Artin toric stack with complete good moduli space. We now turn to the definition of the Euler characteristic of an Artin toric stack. If X = [X/G] is a smooth Artin stack then the tangent bundle stack to X equals the vector bundle stack TX = [T X/ Lie(G)]. The restriction of TX to the open set X DM where X is Deligne-Mumford is the usual tangent bundle to X DM . Identifying the Chow groups of X with the G-equivariant Chow groups of X we may define the Chern series of TX as the formal series c t . Moreover, when G is diagonalizable the Lie algebra of G is a trivial G-module so c t (Lie(G)) = 1 so the Chern series of TX is actually a polynomial.
We can extend the definition of the Euler characteristic to toric Artin stacks.

Formulas for Euler characteristics of 3-dimensional toric Artin stacks.
As an application of Theorem 3.3, we derive a formula for the Euler characteristic of a 3-dimensional toric Artin stack.
Lemma 4.3. Let Σ = (Z 3 , Σ, {v 1 , . . . , v n }) be a 3-dimensional complete stacky fan with only one nonsimplicial cone, call it σ, and label the v i so that σ = R ≥0 v 1 , . . . , v s . Let Σ σ = (Z 3 , Σ σ , {v 0 , v 1 , . . . , v n }) be the stacky fan formed by stacky star subdivision of Σ with respect to σ, and let f : X (Σ σ ) → X (Σ) be the induced morphism. For 0 ≤ i ≤ n, let y i ∈ A * (X (Σ σ )) be the equivariant fundamental class of the coordinate hyperplanes restricted to Cox space C(Σ σ ). Let Σ σ (3) Σ(3) be the set of maximal cones in Σ σ that are not in Σ. Then the difference between the Euler characteristic of X (Σ) and the Euler characteristic of its simplicialization X (Σ σ ) is given by Proof. With the notation as in 2.1, for 1 By Proposition 2.2 and Equations 2, 1, we have that y i 1 · · · y it = 0 in A * G (C) if v i 1 , . . . , v it are not contained in a cone. In particular, y 0 y i = 0 for any i > s. Hence when we expand ( s i=1 (1 + y 0 + y i )) i>s (1 + y i ) and collect the degree 3, we get (note that some monomials in the following equation are zero): We have ( 0<i<j<k y i y j y k ) + ( 0<j<k≤s y 0 y j y k ) = χ(X (Σ σ )).
Remark 4.4. The above proof easily extends to the case when Σ has more than one nonsimplicial cone. Suppose σ 1 and σ 2 are two different nonsimplicial cones of Σ, then the above proof shows that the equivariant fundamental class of the exceptional divisor of the blowup of V (σ 1 ) (resp. V (σ 2 )) has nonzero product only with divisors coming from rays of σ 1 (resp. σ 2 ).
By applying Theorem 3.3, we can give a combinatorial formula to the self-intersection integrals on the right hand side of equation 11.
Fix i satisfying 1 ≤ i ≤ s. We will give a formula for X (Σσ) y 2 0 y i . Let v + i and v − i be the two lattice vectors (among v 1 , . . . , v s ) such that , v i are the two maximal cones in Σ σ having γ i = R ≥0 v 0 , v i as a common face. Since τ + i and τ − i are simplicial, there is a relation (13) Finally, to compute the triple self-intersection integral X (Σσ) y 3 0 , one could use Theorem 3.3. However, because we have restricted the dimension of Σ to 3, we can instead use a relation of rational equivalence, and reduce the computation to the case of double self intersection integrals just computed. Namely, pick some m σ ∈ M such that m σ , v 0 = 0 (there exists such an m σ since v 0 = 0 since v 0 lies in the interior of σ). Then the rational equivalence relation div(χ mσ ) = 0 can be written as 0≤i≤n m σ , v i y i = 0. Hence Multiplying by y 2 0 and using the fact that y 0 y i = 0 for i > s gives X (Σσ) Using Remark 4.4, we combine the above formulas to deduce the following result: Theorem 4.5. Let Σ = (Z 3 , Σ, {v ρ } ρ∈Σ(1) ) be a 3-dimensional complete stacky fan. Let NS denote the set of nonsimplicial cones of Σ. Let Σ simp = (Z 3 , Σ simp , {v ρ } ρ∈Σ simp (1) ) be the stacky fan formed by stacky star subdivision of Σ with respect to the cones in NS, and let f : X (Σ simp ) → X (Σ) be the induced morphism.