Displaying the cohomology of toric line bundles

Abstract

There is a standard method to calculate the cohomology of torus-invariant sheaves $L$ on a toric variety via the simplicial cohomology of associated subsets $V(L)$ of the space $N_{\mathbb{R}}$ of 1-parameter subgroups of the torus. For a line bundle $L$ represented by a formal difference ${\Delta }^{+}-{\Delta }^{-}$ of polyhedra in the character space $M_{\mathbb{R}}$, [ABKW18] contains a simpler formula for the cohomology of $L$, replacing $V(L)$ by the set-theoretic difference ${\Delta }^{-}\setminus {\Delta }^{+}$. Here, we provide a short and direct proof of this formula.

10 pages, many color figures. v2 minor changes, accepted by Izvestiya Math