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Semiprojectivity and very stability in moduli of symplectic and orthogonal parabolic Higgs bundles

Sumit Roy

Abstract

Let $X$ be a compact Riemann surface of genus $g \geq 2$, and let $D \subset X$ be a fixed finite subset. We prove the semiprojectivity of the moduli space of semistable symplectic or orthogonal parabolic Higgs bundles over $X$. We show that a stable symplectic parabolic bundle $E$ on $X$ is strongly very stable, meaning $E$ does not have any nonzero strongly parabolic nilpotent Higgs field, if and only if the symplectic parabolic Hitchin morphism induced on the affine space $$H^0(X,\mathrm{SPEnd}_\mathrm{Sp}(E) \otimes K(D))$$ is a proper morphism, where $\mathrm{SPEnd}_\mathrm{Sp}(E)$ denotes the set of symplectic strongly parabolic endomorphisms of $E$. We remark that the same criterion for very stability applies to the orthogonal case.

Semiprojectivity and very stability in moduli of symplectic and orthogonal parabolic Higgs bundles

Abstract

Let be a compact Riemann surface of genus , and let be a fixed finite subset. We prove the semiprojectivity of the moduli space of semistable symplectic or orthogonal parabolic Higgs bundles over . We show that a stable symplectic parabolic bundle on is strongly very stable, meaning does not have any nonzero strongly parabolic nilpotent Higgs field, if and only if the symplectic parabolic Hitchin morphism induced on the affine space is a proper morphism, where denotes the set of symplectic strongly parabolic endomorphisms of . We remark that the same criterion for very stability applies to the orthogonal case.
Paper Structure (14 sections, 20 theorems, 103 equations)

This paper contains 14 sections, 20 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Parabolic bundles
  4. Parabolic Higgs bundles
  5. Moduli of parabolic Higgs bundles and Hitchin morphism
  6. Parabolic Hitchin morphism
  7. Symplectic and orthogonal parabolic bundles
  8. Moduli space
  9. Semiprojectivity
  10. Semiprojectivity of the moduli space of semistable symplectic parabolic Higgs bundles
  11. Semiprojectivity of the moduli space of semistable orthogonal parabolic Higgs bundles
  12. Even orthogonal case $\mathrm{(}G=\mathrm{SO}(2m,\mathbb{C})\mathrm{)}$
  13. Odd orthogonal case $\mathrm{(}G=\mathrm{SO}(2m+1,\mathbb{C})\mathrm{)}$
  14. Very stable symplectic parabolic bundles

Key Result

Theorem 1.1

The moduli spaces and their strongly parabolic analogues are semiprojective.

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 43 more