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Cohomology of Line Bundles: Proof of the Algorithm

Helmut Roschy, Thorsten Rahn

TL;DR

The paper delivers a rigorous proof of the conjectured algorithm for computing line bundle cohomology on toric varieties by reframing sheaf cohomology as local cohomology of the Cox ring with support on the irrelevant ideal $B_{\Sigma}$ and exploiting a $\mathbb{Z}^{n}$-grading together with the Taylor resolution. It establishes a concrete formula for $h^{i}(\alpha)$ in terms of Betti numbers of the Stanley–Reisner quotient $S/I_{\Sigma}$, via a direct Tor–Hochster–Alexander duality bridge to relative simplicial homology $\widetilde{H}_{*}(\Gamma^{\sigma})$, and parallels Jow’s simplicial/topological approach. The work identifies a potential Serre duality analogue for Betti numbers in experiments and emphasizes the computational advantages of focusing on the SR ideal's combinatorics, offering a path toward faster cohomology computations in string theory applications. Overall, it provides a solid mathematical foundation and a scalable combinatorial method for line bundle cohomology on toric ambient spaces with broad implications for phenomenology and algebraic geometry.

Abstract

We present a proof of the algorithm for computing line bundle valued cohomology classes over toric varieties conjectured by R.~Blumenhagen, B.~Jurke and the authors (arXiv:1003.5217) and suggest a kind of Serre duality for combinatorial Betti numbers that we observed when computing examples.

Cohomology of Line Bundles: Proof of the Algorithm

TL;DR

The paper delivers a rigorous proof of the conjectured algorithm for computing line bundle cohomology on toric varieties by reframing sheaf cohomology as local cohomology of the Cox ring with support on the irrelevant ideal and exploiting a -grading together with the Taylor resolution. It establishes a concrete formula for in terms of Betti numbers of the Stanley–Reisner quotient , via a direct Tor–Hochster–Alexander duality bridge to relative simplicial homology , and parallels Jow’s simplicial/topological approach. The work identifies a potential Serre duality analogue for Betti numbers in experiments and emphasizes the computational advantages of focusing on the SR ideal's combinatorics, offering a path toward faster cohomology computations in string theory applications. Overall, it provides a solid mathematical foundation and a scalable combinatorial method for line bundle cohomology on toric ambient spaces with broad implications for phenomenology and algebraic geometry.

Abstract

We present a proof of the algorithm for computing line bundle valued cohomology classes over toric varieties conjectured by R.~Blumenhagen, B.~Jurke and the authors (arXiv:1003.5217) and suggest a kind of Serre duality for combinatorial Betti numbers that we observed when computing examples.

Paper Structure

This paper contains 8 sections, 2 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Mathematical Preliminaries
  3. Normal Toric Varieties and Alexander duality
  4. Sheaf-Module-Correspondence and Local Coho-mo-logy
  5. Simplicial Complexes and Free Resolutions
  6. Local Cohomology and Betti Numbers
  7. Proof of the Conjecture
  8. Conclusions and Open Questions

Key Result

Theorem 1

Let $\alpha\in\mathop{\mathrm{\mathrm{Cl}}}\nolimits(X)$ and $h^{i}(\alpha )\mathrel{\mathop:}= \dim_{\mathbb{C}}\, H^{i}(X,\mathcal{O}_{X}(\alpha ))$. Then we have:

Theorems & Definitions (4)

  • Theorem
  • proof : Proof of the Theorem:
  • Lemma
  • proof