Reduced Čech complexes and computing higher direct images under toric maps
Mike Roth, Sasha Zotine
TL;DR
This work develops an axiomatic framework for reduced Čech complexes to compute cohomology with fewer intersections, affiliated to a cell complex $P$ that encodes toric combinatorics via the non-negative part $X_{ geq 0}$. It proves that for semi-proper toric varieties one can construct torus-stable affine covers whose associated polyhedral complex $P$ has dimension equal to the cohomological dimension, enabling cohomology calculations and spectral sequence relations. Building on this, the authors present a constructive method to compute higher direct images $R^{i}\varphi_{*}L$ for toric fibrations $\varphi:X\to Y$ by decomposing into $T_K$-eigensheaves and realizing each weight as the cohomology of a complex of monomial ideals, encoded via divisors $(D_\mu,E_\mu)$ and a complex $\check{C}_P(\mathcal{I}_{D_\mu})$. The approach reduces the computation to finitely generated Cox-ring modules, enabling practical calculation (e.g., in Macaulay2) and generalizing known results to broader toric contexts. The framework thus provides both theoretical insight and a computational pathway for higher direct images in toric geometry.
Abstract
This paper has three main goals : (1) To give an axiomatic formulation of the construction of "reduced Čech complexes", complexes using fewer than the usual number of intersections but still computing cohomology of an appropriate class of sheaves; (2) To give a construction of such a reduced Čech complex for every semi-proper toric variety $X$, where every open used in the complex is torus stable, and such that the cell complex governing the reduced Čech complex has dimension the cohomological dimension of $X$; and (3) to give an algorithm to compute the higher direct images of line bundles relative to a toric fibration between smooth proper toric varieties.