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Finiteness for self-dual classes in integral variations of Hodge structure

Benjamin Bakker, Thomas W. Grimm, Christian Schnell, Jacob Tsimerman

TL;DR

The paper proves a finiteness result for the locus of self-dual integral classes in polarized integral variations of Hodge structure, extending the classical finiteness theorem for Hodge classes. It achieves this by recasting the problem in the framework of definability within the o-minimal structure $\mathbb{R}_{\mathrm{an},\exp}$, leveraging the definability of period maps and the reduction theory of Siegel sets. Central to the argument is showing that the self-dual locus, defined by the Weil operator condition $C_x v=v$ and fixed self-intersection $Q(v,v)=q$, forms a definable subspace with finite fibers, a fact underpinned by Kneser finiteness of integral orbits. The paper also derives two variants by tensoring with auxiliary Hodge structures and discusses motivations from string theory, where finiteness of self-dual fluxes corresponds to physically viable vacua.

Abstract

We generalize the finiteness theorem for the locus of Hodge classes with fixed self-intersection number, due to Cattani, Deligne, and Kaplan, from Hodge classes to self-dual classes. The proof uses the definability of period mappings in the o-minimal structure $\mathbb{R}_{\mathrm{an},\exp}$.

Finiteness for self-dual classes in integral variations of Hodge structure

TL;DR

The paper proves a finiteness result for the locus of self-dual integral classes in polarized integral variations of Hodge structure, extending the classical finiteness theorem for Hodge classes. It achieves this by recasting the problem in the framework of definability within the o-minimal structure , leveraging the definability of period maps and the reduction theory of Siegel sets. Central to the argument is showing that the self-dual locus, defined by the Weil operator condition and fixed self-intersection , forms a definable subspace with finite fibers, a fact underpinned by Kneser finiteness of integral orbits. The paper also derives two variants by tensoring with auxiliary Hodge structures and discusses motivations from string theory, where finiteness of self-dual fluxes corresponds to physically viable vacua.

Abstract

We generalize the finiteness theorem for the locus of Hodge classes with fixed self-intersection number, due to Cattani, Deligne, and Kaplan, from Hodge classes to self-dual classes. The proof uses the definability of period mappings in the o-minimal structure .
Paper Structure (7 sections, 15 theorems, 117 equations)

This paper contains 7 sections, 15 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Background on definability
  3. Background on Siegel sets
  4. Definable structures on flat bundles
  5. Proof of the main theorem
  6. Additional results
  7. Motivation from string theory

Key Result

Theorem 1.1

Let $\mathcal{H}$ be a polarized integral variation of Hodge structure of even weight on a nonsingular complex algebraic variety $X$. For each $q \geq 1$, the set is a definable, closed, real-analytic subspace of $E$, and the restriction of $p \colon E \to X$ to this set is proper with finite fibers.

Theorems & Definitions (34)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Example 1.1
  • Example 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.1
  • Definition 2.4
  • ...and 24 more