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Generalized Bogomolov Inequalities

Mihai Pavel, Julius Ross, Matei Toma

TL;DR

This work develops a unified framework to generalize Bogomolov-type inequalities via Hodge-Riemann (HR) pairs of cohomology classes. It defines BG pairs and proves that HR pairs imply BG in several settings, notably for Schur polynomials of ample or Kähler classes, and for Segre and Chern data of vector bundles, including twists by $\mathbb{R}$-line bundles. The authors establish key boundedness results for families of semistable torsion-free sheaves under HR/BG data, and extend the BG theory to Higgs bundles using Hermitian-Einstein and non-abelian Hodge techniques. The paper also develops a robust analytic-geometry apparatus, including positivity cones in Kähler geometry, to support these generalized inequalities and their applications to moduli problems. Overall, the results provide new avenues to control semistable sheaves and their moduli via HR/BG pairs and derived positivity structures.

Abstract

We introduce the notion of a Hodge-Riemann pair of cohomology classes that generalizes the classical Hodge-Riemann bilinear relations, and the notion of a Bogomolov pair of cohomology classes that generalizes the Bogomolov inequality for semistable sheaves. We conjecture that every Hodge-Riemann pair is a Bogomolov pair, and prove various cases of this conjecture. As an application we get new results concerning boundedness of semistable sheaves.

Generalized Bogomolov Inequalities

TL;DR

This work develops a unified framework to generalize Bogomolov-type inequalities via Hodge-Riemann (HR) pairs of cohomology classes. It defines BG pairs and proves that HR pairs imply BG in several settings, notably for Schur polynomials of ample or Kähler classes, and for Segre and Chern data of vector bundles, including twists by -line bundles. The authors establish key boundedness results for families of semistable torsion-free sheaves under HR/BG data, and extend the BG theory to Higgs bundles using Hermitian-Einstein and non-abelian Hodge techniques. The paper also develops a robust analytic-geometry apparatus, including positivity cones in Kähler geometry, to support these generalized inequalities and their applications to moduli problems. Overall, the results provide new avenues to control semistable sheaves and their moduli via HR/BG pairs and derived positivity structures.

Abstract

We introduce the notion of a Hodge-Riemann pair of cohomology classes that generalizes the classical Hodge-Riemann bilinear relations, and the notion of a Bogomolov pair of cohomology classes that generalizes the Bogomolov inequality for semistable sheaves. We conjecture that every Hodge-Riemann pair is a Bogomolov pair, and prove various cases of this conjecture. As an application we get new results concerning boundedness of semistable sheaves.

Paper Structure

This paper contains 17 sections, 38 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Set-up and notation
  3. Algebraic set-up
  4. Analytic set-up
  5. $\mathbb R$-twisted vector bundles
  6. Further Notation and Conventions
  7. Hodge-Riemann pairs
  8. Bogomolov Pairs
  9. Schur polynomials of Kähler classes
  10. Segre classes of Ample Vector Bundles
  11. Globally Generated Ample Bundles
  12. Boundedness of Semistable Sheaves
  13. Bogomolov Pairs for Higgs sheaves
  14. Appendix: positive cones in Kähler geometry
  15. $d_{p,p}$ has closed range
  16. ...and 2 more sections

Key Result

Proposition 1.2

Let $X$ be a complex projective manifold of dimension $d\ge 2$, let $\lambda$ be a partition of $d-1$ and assume $A$ is an ample vector bundle of rank $e\ge d-1$. Then is a Hodge-Riemann pair. In particular, the pair $(c_{d-1}(A),c_{d-2}(A))$ of Chern classes as well as the pair $(s_{d-1}(A),s_{d-2}(A))$ of Segre classes are Hodge-Riemann pairs.

Theorems & Definitions (101)

  • Definition 1.1: Hodge-Riemann pairs of cohomology classes
  • Proposition 1.2: = Proposition \ref{['prop:ShurAmpleHRpair']}
  • Proposition 1.3: $\subset$ Proposition \ref{['prop:ShurHRpair']}
  • Conjecture 1.4
  • Theorem 1.5: = Theorem \ref{['thm:HRimpliesBG']}
  • Theorem 1.6: $\subset$ Corollary \ref{['cor:segrebogomolovpair']}
  • Theorem 1.7: = \ref{['thm:globallygenerated_chern']}
  • Theorem 1.8: $\subset$ Theorem \ref{['thm:MainBound']}
  • Corollary 1.9: = Corollary \ref{['cor:boundednessschurkahler']}
  • Corollary 1.10: = \ref{['cor:boundednessegre']}
  • ...and 91 more