A Vanishing Theorem for Varieties with Finitely Many Solvable Group Orbits

TL;DR

This paper proves a vanishing theorem for intersection cohomology groups of certain algebraic varieties with solvable group actions, extending known results for toric varieties to a broader class including spherical varieties.

Contribution

It generalizes the vanishing theorem from toric varieties to varieties with finitely many solvable group orbits, including spherical varieties.

Findings

Vanishing of intersection cohomology groups with nontrivial local systems for these varieties.

Extension of known results from toric to spherical varieties.

Applicable to a broad class of varieties with solvable group actions.

Abstract

We reprove and generalize the result that the intersection cohomology groups of a toric variety with coefficient in a nontrivial rank one local system vanish. We prove a similar vanishing result for a certain class of varieties on which a connected linear solvable group acts, including all spherical varieties.