A note on finite abelian gerbes over toric Deligne-Mumford stacks

Any toric Deligne-Mumford stack is a $\mu$-gerbe over the underlying toric orbifold for a finite abelian group $\mu$. In this paper we give a sufficient condition so that certain kinds of gerbes over a toric Deligne-Mumford stack are again toric Deligne-Mumford stacks.


Introduction
Let Σ := (N, Σ, β) be a stacky fan of rank(N ) = d as defined in [4]. If there are n one-dimensional cones in the fan Σ, then modelling the construction of toric varieties [5], [6], the toric Deligne-Mumford stack X (Σ) = [Z/G] is a quotient stack, where Z = C n − V , the close subvariety V ⊂ C n is determined by the ideal J Σ generated by { ρi σ z i : σ ∈ Σ} and G acts on Z through the map α : G −→ (C × ) n in the following exact sequence determined by the stacky fan (see [4]): Let G = Im(α), then [Z/G] is the underlying toric orbifold X (Σ red ). The toric Deligne-Mumford stack X (Σ) is a µ-gerbe over X (Σ red ). Let X (Σ) be a toric Deligne-Mumford stack associated to the stacky fan Σ. Let ν be a finite abelian group, and let G be a ν-gerbe over X (Σ). We give a sufficient condition so that G is also a toric Deligne-Mumford stack. We have the following theorem: Theorem 1.1. Let X (Σ) be a toric Deligne-Mumford stack with stacky fan Σ. Then every ν-gerbe G over X (Σ) is induced by a central extension i.e., we have a Cartesian diagram: This small note is organized as follows. In Section 2 we construct the new toric Deligne-Mumford stack from an abelian central extension and prove the main results . In Section 3 we give an example of ν-gerbe over a toric Deligne-Mumford stack.
In this paper, by an orbif old we mean a smooth Deligne-Mumford stack with trivial stabilizers at the generic points.

The Proof of Main Results
We refer the reader to [4] for the construction and notation of toric Deligne-Mumford stacks. For the general theory of stacks, see [2].
Let Σ := (N, Σ, β) be a stacky fan. From Proposition 2.2 in [4], we have the following exact sequences: where β ∨ is the Gale dual of β. As a Z-module, C × is divisible, so it is an injective Z-module and hence the functor Hom Z (−, C × ) is exact. We get the exact sequence: Let µ := Hom Z (Coker(β ∨ ), C × ), we have the exact sequence (1.1). Let Σ(1) = n be the set of one dimensional cones in Σ and V ⊂ C n the closed subvariety defined by the ideal generated by Let Z := C n \ V . From [5], the complex codimension of V in C n is at least 2. The toric Deligne-Mumford stack X (Σ) = [Z/G] is the quotient stack where the action of G is through the map α in (1.1).
Proof. Consider the following exact sequence: Since Codim C (V, C n ) ≥ 2, so the real codimension is at least 4 and H i V (C n , ν) = 0 for i = 1, 2, 3, so from the exact sequence and H i (C n , ν) = 0 for all i > 0 we prove the lemma.
where G is an abelian group. So the pullback gerbe over Z under the map Z −→ [Z/G] is trivial. So we have The stack [Z/ G] is this ν-gerbe G over [Z/G]. Consider the commutative diagram: where α is the map in (1.1). So we have the following exact sequences: where T is the torus of the simplicial toric variety X(Σ). Since the abelian groups G, G and (C × ) n are all locally compact topological groups, taking Pontryagin duality and Gale dual, we have the following diagrams: where p ϕ is induced by ϕ in (2.1) under the Pontryagin duality. Suppose β : Remark 2.2. From Proposition 4.6 in [3], any Deligne-Mumford stack is a ν-gerbe over an orbifold for a finite group ν. Our results are the toric case of that general result.
In particular, from a stacky fan Σ = (N, Σ, β). Let Σ red = (N , Σ, β) be the reduced stacky fan, where N is the abelian group N modulo torsion, and β : Z n −→ N is given by {b 1 , · · · , b n } which are the images of {b 1 , · · · , b n } under the natural projection N −→ N . Then the toric orbifold X (Σ red ) = [Z/G]. From (1.1), let G = Im(α), then we have the following exact sequences: So G is an abelian central extension of G by µ. X (Σ) is a µ-gerbe over the toric orbifold X (Σ red ). Any µ-gerbe over the toric orbifold coming from an abelian central extension is a toric Deligne-Mumford stack. This is a special case of the main results and is the toric case of rigidification construction in [1]. Remark 2.3. From the proof of Corollary 1.2 we see that if a ν-gerbe over X (Σ) comes from a gerbe over BG and the central extension is abelian, then we can construct a new toric Deligne-Mumford stack.  (N, Σ, β) is a stacky fan. We compute that (β) ∨ : Z 2 −→ DG(β) = Z is given by the matrix [3,3]. So we get the following exact sequence: