Geodesic
Displaying the cohomology of~toric line bundles
Izvestiya. Mathematics , Tome 84 (2020) no. 4, pp. 683-693

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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, line bundle, sheaf cohomology, lattice, polytope.
Mots-clés : Cartier divisor 
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     author = {K. Altmann and D. Ploog},
     title = {Displaying the cohomology of~toric line bundles},
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K. Altmann; D. Ploog. Displaying the cohomology of~toric line bundles. Izvestiya. Mathematics , Tome 84 (2020) no. 4, pp. 683-693. http://geodesic.mathdoc.fr/item/IM2_2020_84_4_a3/