Cohomology of line bundles on smooth projective toric Deligne-Mumford stacks - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T21:39:18Z http://mathoverflow.net/feeds/question/95900 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/95900/cohomology-of-line-bundles-on-smooth-projective-toric-deligne-mumford-stacks Cohomology of line bundles on smooth projective toric Deligne-Mumford stacks Francesco Sala 2012-05-03T17:44:00Z 2012-05-03T23:22:11Z <p>Let $\mathcal{X}$ be a smooth projective toric Deligne-Mumford stack with coarse moduli scheme $\pi\colon \mathcal{X}\rightarrow X.$</p> <p>Let $[D]$ be a Cartier divisor on $\mathcal{X}$ such that $[D']=\pi_*(a[D])$ is an invariant Cartier divisor on $X$ (associated to some ray in the fan of $X$).</p> <p><strong>Question:</strong> For any integer $m$, is it true that $H^i(\mathcal{X}, \mathcal{O}(mD)) \cong H^i(X, \mathcal{O}(qD'))$, where $q$ is the ceiling of $m/a$ (i.e. the smallest integer number such that $m/a\leq q$)?</p> <p>P.S.: I do not know so much about (toric) stacks, hence I do not know if my question is well-posed. Please, suggest me any kind of corrections.</p>