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On intersection cohomology with torus action of complexity one, II

Marta Agustin Vicente, Narasimha Chary Bonala, Kevin Langlois

Abstract

We show that the components, appearing in the decomposition theorem for contraction maps of torus actions of complexity one, are intersection cohomology complexes of even codimensional subvarieties. As a consequence, we obtain the vanishing of the odd dimensional intersection cohomology for rational complete varieties with torus action of complexity one. The article also presents structural results on linear torus action in order to compute the intersection cohomology from the weight matrix. In particular, we determine the intersection cohomology Betti numbers of affine trinomial hypersurfaces in terms of their defining equation.

On intersection cohomology with torus action of complexity one, II

Abstract

We show that the components, appearing in the decomposition theorem for contraction maps of torus actions of complexity one, are intersection cohomology complexes of even codimensional subvarieties. As a consequence, we obtain the vanishing of the odd dimensional intersection cohomology for rational complete varieties with torus action of complexity one. The article also presents structural results on linear torus action in order to compute the intersection cohomology from the weight matrix. In particular, we determine the intersection cohomology Betti numbers of affine trinomial hypersurfaces in terms of their defining equation.

Paper Structure

This paper contains 27 sections, 53 theorems, 211 equations.

Table of Contents

  1. Introduction
  2. Convention
  3. Preliminaries
  4. Algebraic varieties
  5. Intersection cohomology
  6. Algebraic torus actions
  7. Toric varieties
  8. Complexity-one case
  9. Intersection cohomology and actions of finite groups
  10. Maps preserving intersection cohomology complexes
  11. Finite group actions on toric varieties
  12. Seifert torus bundles
  13. Linear torus actions of complexity one
  14. Linear torus actions: the affine case
  15. Curves on toric surfaces
  16. ...and 12 more sections

Key Result

Lemma 2.2

Let $D, D'$ be two $\mathbb{Q}$-divisors on a normal variety $Y$. Assume that $D$ is semi-ample and that $H^{0}(Y, \mathcal{O}_{Y}(rD))\subset H^{0}(Y, \mathcal{O}_{Y}(rD'))$ for any $r\in \mathbb{Z}_{>0}$. Then $D\leq D'$.

Theorems & Definitions (129)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Theorem 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • Lemma 2.7
  • proof
  • ...and 119 more