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Flow lines on the moduli space of rank $2$ twisted Higgs bundles

Graeme Wilkin

Abstract

This paper studies the gradient flow lines for the $L^2$ norm square of the Higgs field defined on the moduli space of semistable rank $2$ Higgs bundles twisted by a line bundle of positive degree over a compact Riemann surface $X$. The main result is that these spaces of flow lines have an algebro-geometric classification in terms of secant varieties for different embeddings of $X$ into the projectivisation of the negative eigenspace of the Hessian at a critical point. The Morse-theoretic compactification of spaces of flow lines given by adding broken flow lines then has a natural algebraic interpretation via a projection to Bertram's resolution of secant varieties.

Flow lines on the moduli space of rank $2$ twisted Higgs bundles

Abstract

This paper studies the gradient flow lines for the norm square of the Higgs field defined on the moduli space of semistable rank Higgs bundles twisted by a line bundle of positive degree over a compact Riemann surface . The main result is that these spaces of flow lines have an algebro-geometric classification in terms of secant varieties for different embeddings of into the projectivisation of the negative eigenspace of the Hessian at a critical point. The Morse-theoretic compactification of spaces of flow lines given by adding broken flow lines then has a natural algebraic interpretation via a projection to Bertram's resolution of secant varieties.
Paper Structure (16 sections, 22 theorems, 73 equations)

This paper contains 16 sections, 22 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Background and notational conventions
  3. Properties of the energy function
  4. Morse strata for the gradient flow
  5. The polystable locus in the rank $2$ case
  6. The unstable manifold in a neighbourhood of a critical set
  7. Secant varieties in the unstable manifold
  8. Classification of flow lines
  9. Harder-Narasimhan types in the unstable manifold
  10. Higgs pairs in the limit of the downwards flow
  11. Classification of flow lines
  12. The energy function is Morse-Bott-Smale
  13. Compactification of spaces of flow lines
  14. The Morse resolution
  15. Compactification by broken flow lines
  16. ...and 1 more sections

Key Result

Theorem 1.1

Fix $\mathop{\mathrm{rank}}\nolimits(E) = 2$, $\deg E = 0 \, \, \text{or} \, \, 1$ and let $C_\ell, C_u$ be two critical sets indexed by $0 \leq \ell < u \leq \frac{1}{2} \deg E + g-1$. Then the space $\mathcal{L}_\ell^u$ of flow lines between $C_\ell$ and $C_u$ is a circle bundle over the $(u-\ell)

Theorems & Definitions (47)

  • Theorem 1.1: Theorem \ref{['thm:space-unbroken-flow-lines']}
  • Remark 1.2
  • Theorem 1.3: Theorem \ref{['thm:Morse-resolution']}
  • Proposition 2.1: HauselHitchin22
  • Lemma 2.2
  • Definition 3.1: Secant plane of total multiplicity $N$
  • Lemma 3.2
  • proof
  • Definition 3.3: $N^{th}$ secant variety
  • Remark 3.4
  • ...and 37 more