[Paper Review] Stratifications and quasi-projective coarse moduli spaces for the stack of Higgs bundles

Eloise Hamilton|arXiv (Cornell University)|Nov 29, 2019

Algebraic Geometry and Number Theory41 references4 citations

TL;DR

This paper applies Non-Reductive Geometric Invariant Theory (NR-GIT) to construct quasi-projective coarse moduli spaces for unstable strata in the moduli stack of Higgs bundles on a compact Riemann surface. It introduces two refined stratifications—based on Higgs bundle and underlying bundle instability types—providing explicit projective completions for each stratum, with a full moduli-theoretic description in the rank 2 case.

ABSTRACT

The classification problem for Higgs bundles of a fixed rank and degree on a compact Riemann surface is encoded in the moduli stack of such objects. Nitsure's GIT construction of the moduli space of semistable Higgs bundles produces a quasi-projective coarse moduli space (if the rank and degree are coprime) for the semistable stratum of this moduli stack. Nevertheless, GIT cannot be used to produce coarse moduli spaces for the complement of the semistable stratum. In this paper we use a recent generalisation of GIT, called Non-Reductive GIT, to construct two stratifications of the stack of Higgs bundles which satisfy the property that each stratum admits a quasi-projective coarse moduli space with an explicit projective completion. The first is a refinement of the Higgs Harder-Narasimhan stratification of the stack of Higgs bundles (defined by the instability type of the Higgs bundle), while the second is a refinement of the Harder-Narasimhan stratification (defined by the instability type of the underlying bundle). We provide a complete and explicit moduli-theoretic description of both refined stratifications in the rank $2$ case.

Motivation & Objective

• To address the lack of coarse moduli spaces for unstable strata in the moduli stack of Higgs bundles, beyond Nitsure's GIT construction for semistable Higgs bundles.
• To extend moduli-theoretic constructions beyond the semistable locus using a generalization of GIT.
• To provide explicit, projective completions for each stratum in refined stratifications of the Higgs bundle stack.
• To offer a complete moduli-theoretic description of the refined stratifications in the rank 2 case.
• To establish that each stratum in the two refined stratifications admits a quasi-projective coarse moduli space.

Proposed method

• Utilizes Non-Reductive GIT to construct coarse moduli spaces for unstable strata where standard GIT fails.
• Introduces two distinct refinements of existing stratifications: one based on Higgs bundle instability type, the other on underlying vector bundle instability type.
• Defines strata via the Harder-Narasimhan type of the Higgs bundle or its underlying bundle, ensuring each stratum is invariant under the group action.
• Constructs explicit projective completions for each stratum, ensuring the coarse moduli spaces are quasi-projective.
• Applies the general framework of NR-GIT to the stack of Higgs bundles, leveraging its ability to handle non-reductive group actions.
• Provides a full moduli-theoretic description of both refined stratifications in the rank 2 case, using explicit invariants and stability conditions.

Experimental results

Research questions

• RQ1 Can Non-Reductive GIT be used to construct quasi-projective coarse moduli spaces for unstable strata in the moduli stack of Higgs bundles?
• RQ2 How can the Higgs Harder-Narasimhan stratification be refined to ensure each stratum admits a quasi-projective coarse moduli space?
• RQ3 What is the moduli-theoretic structure of the refined stratifications in the rank 2 case?
• RQ4 How does the instability type of the underlying bundle compare to that of the Higgs bundle in defining strata with good moduli properties?
• RQ5 Can explicit projective completions be constructed for each stratum in the refined stratifications?

Key findings

• Non-Reductive GIT successfully constructs quasi-projective coarse moduli spaces for each stratum in the two refined stratifications of the Higgs bundle stack.
• The first refinement is based on the Higgs Harder-Narasimhan type, refining the stratification by Higgs bundle instability.
• The second refinement is based on the Harder-Narasimhan type of the underlying vector bundle, providing an alternative stratification with good moduli properties.
• Each stratum in both refined stratifications admits a quasi-projective coarse moduli space with an explicit projective completion.
• In the rank 2 case, a complete and explicit moduli-theoretic description is provided for both refined stratifications.
• The construction extends beyond semistable Higgs bundles, resolving the gap left by standard GIT in Nitsure's construction.