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

Graeme Wilkin

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arXiv · 2408.13098

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

Graeme Wilkin

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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.

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Graeme Wilkin. 2024-08-23. Flow lines on the moduli space of rank $2$ twisted Higgs bundles. https://arxiv.org/abs/2408.13098

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