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A Vanishing Theorem for Varieties with Finitely Many Solvable Group Orbits

Yiyu Wang

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.

A Vanishing Theorem for Varieties with Finitely Many Solvable Group Orbits

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.
Paper Structure (3 sections, 13 theorems, 23 equations)

This paper contains 3 sections, 13 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Twisted local system
  3. Pushforward of a twisted local system is twisted

Key Result

Theorem 1.1

For a normal toric variety $X$, let $\mathbb{T}$ denote the affine torus, and $\mathbb{T}\to X$ denote the maximal torus orbit. Suppose $\mathcal{L}$ is a rank one nontrivial local system (nontrivial means not isomorphic to the constant local system) on $\mathbb{T}$, then the intersection cohomology

Theorems & Definitions (25)

  • Theorem 1.1: Yavin1992
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Remark
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • ...and 15 more