[Paper Review] Parabolic Hitchin Maps and Their Generic Fibers

Xiaoyu Su, Bin Wang+1 more|arXiv (Cornell University)|Jun 11, 2019

Algebraic Geometry and Number Theory20 references4 citations

TL;DR

This paper establishes a parabolic BNR correspondence for moduli spaces of Higgs bundles with parabolic structures over any algebraically closed field, proving that generic fibers of the strongly parabolic Hitchin map are isomorphic to Picard varieties of normalized spectral curves. It further establishes flatness, surjectivity, and equidimensionality of the global nilpotent cone, and proves the existence of very stable parabolic bundles.

ABSTRACT

We set up a BNR correspondence for moduli spaces of Higgs bundles over a curve with a parabolic structure over any algebraically closed field. This leads to a concrete description of generic fibers of the associated strongly parabolic Hitchin map. We also show that the global nilpotent cone is equi-dimensional with half dimension of the total space. As a result, we prove the flatness and surjectivity of this map and the existence of very stable parabolic vector bundles.

Motivation & Objective

• To develop a parabolic BNR correspondence for Higgs bundles with parabolic structures over arbitrary algebraically closed fields.
• To provide a concrete geometric description of generic fibers of the strongly parabolic Hitchin map.
• To prove that the global nilpotent cone is equi-dimensional with half the dimension of the total space.
• To establish the flatness and surjectivity of the strongly parabolic Hitchin map.
• To prove the existence of very stable parabolic vector bundles via deformation theory of nilpotent Higgs bundles.

Proposed method

• Construct a parabolic BNR correspondence by analyzing spectral curves associated to strongly parabolic Higgs fields.
• Use toric resolution techniques to study singularities of generic spectral curves, particularly at parabolic points.
• Characterize the Hitchin base $ ilde{rak{H}}_P$ as an affine subspace of the full Hitchin base $rak{H}$, leveraging geometric invariant theory.
• Apply the Jacobian criterion and weak Bertini theorem to analyze smoothness and ramification of spectral curves in positive characteristic.
• Employ deformation theory within the global nilpotent cone to prove equidimensionality and flatness of the Hitchin map.
• Relate the generic fiber structure to the Picard variety of the normalization of the spectral curve, generalizing the classical BNR correspondence.

Experimental results

Research questions

• RQ1 How can the BNR correspondence be generalized to the parabolic setting over arbitrary algebraically closed fields?
• RQ2 What is the precise geometric structure of the generic fibers of the strongly parabolic Hitchin map?
• RQ3 Under what conditions is the spectral curve associated to a generic Hitchin base element integral and smooth outside the parabolic divisor?
• RQ4 Is the global nilpotent cone equi-dimensional, and what is its dimension relative to the total space?
• RQ5 Do very stable parabolic Higgs bundles exist, and can their existence be proven via deformation-theoretic methods?

Key findings

• Generic fibers of the strongly parabolic Hitchin map are isomorphic to the Picard variety of degree $d$ on the normalization of the spectral curve, where $d$ is determined by the parabolic type.
• The strongly parabolic Hitchin map $h_{P,ar{ ho}}: ext{Higgs}_{P,ar{ ho}} o ilde{rak{H}}_P$ is flat and surjective.
• The global nilpotent cone is equi-dimensional and has dimension exactly half of the total space of the moduli stack of strongly parabolic Higgs bundles.
• For generic $a o ilde{rak{H}}_P$, the spectral curve $X_a$ is integral, totally ramified over $x o D$, and smooth elsewhere.
• In characteristic $p > 0$, the results hold under the condition $r e 2$ when $p=2$, ensuring the validity of the Jacobian criterion.
• The existence of very stable parabolic vector bundles is established via deformation-theoretic analysis of the global nilpotent cone.