[Paper Review] Geometry of cohomology support loci II: integrability of Hitchin's map

Donu Arapura|ArXiv.org|Jan 26, 1997

Algebraic Geometry and Number Theory14 references3 citations

TL;DR

This paper establishes that the cohomology support locus—defined as the set of semistable Higgs bundles with trivial rational Chern classes and nontrivial kth cohomology on a smooth complex projective variety—is a degeneration of abelian varieties. Using Simpson's generalized Hitchin map, the authors prove that a connected component of a general fiber over the image of this map is an abelian variety, extending Green and Lazarsfeld’s generic vanishing theorem to the non-abelian setting via hyperkähler geometry and Lagrangian fibrations.

ABSTRACT

In very rough terms, the main theorem is that the set, which consists of semistable vector bundles with trivial rational Chern classes and nontrivial kth cohomology on a smooth complex projective variety, is a degeneration of a union of abelian varieties. More precisely, we consider the subset of the moduli space of Higgs bundles satisfying the analogous cohomological condition. We show that this set is Zariski closed and that if Sigma is the normalization of an irreducible component containing a stable point, then a connected component of a general fiber of the restriction of the Hitchin map to Sigma is an abelian variety. This should be interpreted as a nonabelian version of a theorem of Green and Lazarsfeld. The Hitchin map in this setting is in fact Simpson's generalization of it. The key point is to show that the general fibers of this map are lagrangian (where the target of the map is taken to be the image). This hinges on the fact that this property is essentially hereditary for hyperkaehler submanifolds. We establish various other properties of these sets including a codimension estimate which may be viewed as a generic vanishing theorem.

Motivation & Objective

• To extend Green and Lazarsfeld’s generic vanishing theorem to the non-abelian setting using Higgs bundles.
• To analyze the geometry of cohomology support loci in the moduli space of semistable Higgs bundles with trivial rational Chern classes.
• To establish that the fibers of the restricted Hitchin map over irreducible components of the support locus are Lagrangian and contain abelian varieties as connected components.
• To prove that the set of Higgs bundles with nontrivial cohomology is Zariski closed in the moduli space.
• To provide a codimension estimate for the cohomology support locus, interpreted as a generic vanishing result in the non-abelian context.

Proposed method

• Restrict the Hitchin map to the normalization Σ of an irreducible component of the cohomology support locus containing a stable Higgs bundle.
• Use the fact that the target of the Hitchin map is taken to be its image, ensuring the map is well-defined and proper.
• Leverage the hereditary property of Lagrangian fibrations under hyperkähler submanifold restriction to show that general fibers are Lagrangian.
• Apply hyperkähler geometry techniques to deduce that connected components of general fibers are abelian varieties.
• Establish the Zariski closedness of the cohomology support locus via algebraic geometry of moduli spaces.
• Derive a codimension estimate for the support locus using the structure of the Hitchin fibration and the dimension of fibers.

Experimental results

Research questions

• RQ1 Under what conditions is the cohomology support locus in the moduli space of Higgs bundles Zariski closed?
• RQ2 Can the fibers of the Hitchin map over the cohomology support locus be shown to contain abelian varieties as connected components?
• RQ3 To what extent does the Lagrangian structure of the Hitchin fibration persist under restriction to irreducible components of the support locus?
• RQ4 How does the codimension of the cohomology support locus relate to the dimension of the fibers of the Hitchin map?
• RQ5 In how far does Simpson’s generalization of the Hitchin map allow a non-abelian extension of Green and Lazarsfeld’s generic vanishing theorem?

Key findings

• The cohomology support locus of semistable Higgs bundles with trivial rational Chern classes and nontrivial kth cohomology is Zariski closed in the moduli space.
• For an irreducible component Σ of the support locus containing a stable Higgs bundle, the normalization of Σ admits a restriction of the Hitchin map whose general fiber has a connected component that is an abelian variety.
• The general fibers of the restricted Hitchin map are Lagrangian with respect to the holomorphic symplectic form on the moduli space.
• The codimension of the cohomology support locus is bounded below by the dimension of the base of the Hitchin fibration, providing a generic vanishing estimate.
• The Lagrangian property of the fibers is preserved under restriction to hyperkähler submanifolds, which is essential for the proof.
• The result provides a non-abelian generalization of Green and Lazarsfeld’s theorem on generic vanishing for line bundles to the setting of Higgs bundles and moduli spaces of representations.