Reduced Čech complexes and computing higher direct images under toric fibrations.

Abstract: Let $X$ be a topological space, $F$ a sheaf of abelian groups on $X$, and $\{ U_{\alpha}\}_{\alpha\in I\}$ an open cover of $X$. Then one can form a Čech complex, a complex of groups built from the values of $F$ on the open sets and their intersections.

If the higher cohomology of $F$ vanishes on all these open sets, then it is a well-known theorem of Leray that this complex computes the cohomology of $F$ on $X$. For instance, if $X$ is a manifold and the $U_{\alpha}$ form a `good cover’ (all the $U_{\alpha}$ and their intersections are homeomorphic to $\mathbb{R}^n$), then the Čech complex can be used to compute the topological cohomology of $X$.

For special kinds of toric varieties — those whose fans are `simplicial’ -- it is known how to construct smaller (“reduced”) complexes which still correctly compute cohomology of sheaves.

This talk has three main goals : (1) To give an axiomatization of ‘reduced Čech complexes’, valid for any topological space; (2) To extend the previous construction of reduced Čech complexes to all compact toric varieties (not just simplicial ones), and more generally to ’semi-proper’ toric varieties; (3) To use the previous method to give an algorithm for computing higher direct images (roughly the `cohomology along the fibres’) of line bundles for toric fibrations between smooth toric varieties.

No previous knowledge of toric varieties is required. This is joint work with Sasha Zotine.

algebraic geometryanalysis of PDEsalgebraic topologycomplex variablesdifferential geometrygeneral topologygeometric topologyK-theory and homologymetric geometrysymplectic geometry

Audience: researchers in the topic

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CRM - Séminaire du CIRGET / Géométrie et Topologie

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