# Vanishing theorems on toric DM stacks

In chapter 9 of the book Toric varieties by Cox-Little-Schenck several cohomology vanishing theorems for toric varieties are proved or mentioned.

In this question I am interested in references for versions (if existing) of the analogous vanishing theorems for toric Deligne-Mumford stacks (especially the smooth ones, e.g. in the sense of Fantechi-Mann-Nironi). I am happy with assuming we are over $\mathbb{C}$ and the stacks have trivial generic stabilizer.

Edit. To answer the question of Jason Starr in the comments, I think I am mostly interested in a stacky version of Batyrev-Borisov vanishing theorem, which I am reporting below. Nevertheless, I would also be glad to receive answers about the other vanishing theorems that I mentioned or alluded to in my question.

Theorem (Batyrev-Borisov). Let $X=X_{\Sigma}$ be a complete toric variety, and $D=\sum_{\rho\in\Sigma(1)} a_\rho D_\rho$ a nef $\mathbb{Q}$-Cartier divisor on $X$. Then

$$H^p(X,\mathcal{O}(-D))=0\;\;\;\;\forall\; p\neq \dim P_D,$$ $$H^{\dim P_D}(X,\mathcal{O}(-D))=\bigoplus_{m\in \mathrm{relint}(P_D)\cap M}\mathbb{C}\cdot \chi^{-m},$$ where $M$ is the lattice of $\Sigma$ and $P_D$ is a polytope defined by

$$P_D=\{ x \in M \otimes\mathbb{R}\;|\; \langle x,u_{\rho} \rangle \leq -a_{\rho} \; \forall \; \rho\in\Sigma (1)\}$$ $u_{\rho}$ being the minimal generator of the ray $\rho$.

• Could you state one of the vanishing theorems you would like for a toric Deligne-Mumford stack? For instance, Olsson-Matsuki have an extension of Kodaira vanishing to stacks (and they use this to reprove many case of Kawamata-Viehweg vanishing for usual varieties). – Jason Starr Nov 9 '15 at 22:11
• I think I do not understand what you mean. For the toric boundary $\Delta$ on $\mathbb{P}^1$, $h^1(\mathbb{P}^1,\mathcal{O}(-\Delta))$ is nonzero. – Jason Starr Nov 10 '15 at 0:56
• Right. Just deleted my previous comment. – Qfwfq Nov 10 '15 at 0:58
• Could you please state the vanishing theorem that you would like to extend to toric Deligne-Mumford stacks? – Jason Starr Nov 10 '15 at 13:53
• In characteristic $0$, pushforward from the toric Deligne-Mumford stack to the coarse moduli space toric variety is an exact functor that has vanishing higher direct images. Thus, it seems to me that, via Leray-Serre, Batyrev-Borisov vanishing for the toric Deligne-Mumford stack reduces to Batyrev-Borisov vanishing for the coarse moduli space toric variety. – Jason Starr Nov 12 '15 at 14:01