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Acyclic toric sheaves

Klaus Altmann, Andreas Hochenegger, Frederik Witt

TL;DR

The paper addresses the problem of when a toric sheaf on a smooth projective toric variety is acyclic, building on the Perlman-Smith criterion for toric vector bundles. It recasts this question via Weil decorations, providing explicit vanishing criteria: if certain divisors $ ext{O}_X(D_i)$ have vanishing cohomology above a degree $k_0$, then the toric sheaf $ amespace{}{ ext{E}}$ inherits the same vanishing for $H^k$ with $k eq 0$, and nef/immaculate decorations yield acyclicity. A geometric interpretation in terms of polyhedra shows how the PS inequality compares lattice-lengths of edges and facet-distances between polytopes associated to $ ext{E}$, with nefness of the decoration ensuring acyclicity. The paper also constructs a canonical resolution by totally split toric sheaves, and proves that under the extremal-ray condition of the PS inequality, the twist $ amespace{}{ ext{E}(D)}$ is acyclic, providing a practical criterion for applications in toric geometry.

Abstract

Let $\mathcal E$ be a torus-linearised reflexive sheaf over a smooth projective toric variety. Based on a theorem of Perlman-Smith, we prove an explicit sufficient condition for $\mathcal E$ to be acyclic via Weil decorations.

Acyclic toric sheaves

TL;DR

The paper addresses the problem of when a toric sheaf on a smooth projective toric variety is acyclic, building on the Perlman-Smith criterion for toric vector bundles. It recasts this question via Weil decorations, providing explicit vanishing criteria: if certain divisors have vanishing cohomology above a degree , then the toric sheaf inherits the same vanishing for with , and nef/immaculate decorations yield acyclicity. A geometric interpretation in terms of polyhedra shows how the PS inequality compares lattice-lengths of edges and facet-distances between polytopes associated to , with nefness of the decoration ensuring acyclicity. The paper also constructs a canonical resolution by totally split toric sheaves, and proves that under the extremal-ray condition of the PS inequality, the twist is acyclic, providing a practical criterion for applications in toric geometry.

Abstract

Let be a torus-linearised reflexive sheaf over a smooth projective toric variety. Based on a theorem of Perlman-Smith, we prove an explicit sufficient condition for to be acyclic via Weil decorations.

Paper Structure

This paper contains 13 sections, 7 theorems, 56 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Perlman-Smith theorem.
  3. Weil decorations
  4. Weil decorations
  5. Polytopes and Weil divisors
  6. The lattice structure
  7. Weil decorations
  8. The canonical resolution
  9. Acyclicity of toric sheaves
  10. Proof of Theorem \ref{['thm:Neflic']}
  11. Geometric interpretation of the assumption \ref{['eq:PS']}
  12. The left hand side
  13. The right hand side

Key Result

Theorem 1.2

If $k_0$ is an integer such that $\operatorname{H}^k\!(X,\mathcal{O}(D_i))=0$ for $i=1,\ldots,n$ and all $k\geq k_0$, then also $\operatorname{H}^k(X,\mathcal{E})=0$ for all $k\geq k_0$. In particular, this implies that if

Figures (2)

  • Figure 1: Left hand side: The fan of the del Pezzo surface. Right-hand side: The red dot indicates the origin in $M_\mathbb{R}$ and fixes the position of the polytopes.
  • Figure 2: Left hand side: The fan of the first Hirzebruch surface. Right-hand side: The Weil decoration of $\mathcal{O}_\mathcal{H}(\nabla)\oplus\mathcal{O}_\mathcal{H}(\nabla)'$.

Theorems & Definitions (19)

  • Remark 1.1
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 2.1
  • Proposition 2.2
  • Definition 2.3
  • ...and 9 more