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