Cohomology of toric line bundles via simplicial Alexander duality
Shin-Yao Jow
TL;DR
This work proves the toric line-bundle cohomology algorithm conjectured in BJRR and a faster variant by linking $H^i_*(\mathcal{O}_X)_I$ to $H^{i+1}_B(S)_I$ and then to simplicial-topological invariants. The main result expresses $H^i_*(\mathcal{O}_X)_I$ as $\widetilde{H}_{d-1-i}(\mathcal{P}_{≤ \hat{I}})$ via simplicial Alexander duality, enabling a Serre duality for Betti numbers and a practical speed-up algorithm for computing cohomology. The approach leverages the Cox ring $S$ and its local cohomology with support in the irrelevant ideal $B$, together with a cellular resolution on the moment polytope to obtain graded descriptions that depend only on the negative part of the multidegree. The corollary provides an efficient formula for $h^i(X,L)$ by summing contributions from combinatorial data $(I, \Lambda_I)$, significantly reducing computational effort for toric varieties.
Abstract
We give a rigorous mathematical proof for the validity of the toric sheaf cohomology algorithm conjectured in the recent paper by R. Blumenhagen, B. Jurke, T. Rahn, and H. Roschy (arXiv:1003.5217). We actually prove not only the original algorithm but also a speed-up version of it. Our proof is independent from (in fact appeared earlier on the arXiv than) the proof by H. Roschy and T. Rahn (arXiv:1006.2392), and has several advantages such as being shorter and cleaner and can also settle the additional conjecture on "Serre duality for Betti numbers" which was raised but unresolved in arXiv:1006.2392.