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A Sheaf Cohomology Restriction Formula on Toric Complete Intersections

Zhentao Lyu

Abstract

We prove a sheaf cohomology restriction (SCORE) formula for a class of vector bundles on complete intersections in toric varieties. The formula enables one to compute cohomology products on the complete intersection $X$ via computations on the ambient space $V$ and potentially compute certain quantum corrections to the classical sheaf cohomology ring. Schematically the formula reads $(s_1,...,s_r)_X = (s_1,...,s_r, g)_V$ with $g$ being an explicitly described quantity derived from the monad data of the bundle.

A Sheaf Cohomology Restriction Formula on Toric Complete Intersections

Abstract

We prove a sheaf cohomology restriction (SCORE) formula for a class of vector bundles on complete intersections in toric varieties. The formula enables one to compute cohomology products on the complete intersection via computations on the ambient space and potentially compute certain quantum corrections to the classical sheaf cohomology ring. Schematically the formula reads with being an explicitly described quantity derived from the monad data of the bundle.
Paper Structure (6 sections, 5 theorems, 36 equations)

This paper contains 6 sections, 5 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. The setup of the hypersurface case
  3. Two double complexes and consequences
  4. Two double complexes
  5. Proof of Theorem \ref{['thm:hypersuface']}
  6. SCORE: the complete intersection case

Key Result

Proposition 2.1

When $\mathcal{E}^*_V$ and $\mathcal{E}^*_X$ are small deformations of $\Omega_V$ and $\Omega_X$ respectively, $\delta: H^{n-1}( \wedge^{n-1}\mathcal{E}_X^*)\to H^n(\wedge^n\mathcal{E}_V^*)$ is an isomorphism.

Theorems & Definitions (9)

  • Proposition 2.1
  • proof
  • Theorem 2.2: SCORE formula for hypersurfaces
  • Proposition 3.1
  • proof
  • Theorem 4.1: SCORE formula for toric complete intersections
  • proof
  • Proposition 4.2
  • proof