Minimal Energy Local Systems on Curves
Charlie Wu
TL;DR
The work studies minimal energy local systems on punctured curves, defined via polarizable complex variations of Hodge structure and a vanishing condition on the adjoint Higgs cohomology, showing these objects form compact components in real relative character varieties and exist under suitable unitary generic monodromy. Using Non-Abelian Hodge theory, the authors connect local systems to parabolic Higgs bundles, analyze the Hitchin map, and apply Białynicki-Birula stratification to prove deformation toward minimal energy; in genus $g>0$ these systems come from unitary representations, while genus $0$ allows multi-step VHS with sharp bounds on the number of graded pieces when $ ext{deg}D$ is large. They establish that the top-dimensional VHS component corresponds to minimal energy, and obtain concrete genus-zero bounds (e.g., at most two graded pieces for large $ ext{deg}D$) with explicit two-piece constructions proving sharpness. The paper further derives consequences for Gromov–Witten theory via modified parabolic bundles and Schubert calculus, showing nontrivial invariants under the minimal energy constraint and linking the existence of certain monodromies to GW data. Overall, the results generalize supra-maximal representations to higher rank, extend compactness phenomena to broader surfaces, and provide concrete tools for relating VHS, moduli, and enumerative geometry.
Abstract
Let $Σ_{g,d}$ an orientable topological surface of genus $g$ with $d$ punctures. When $g = 0$, Deroin and Tholozan studied the class of supra-maximal representations $π_1(Σ_{0,d})\to \mathrm{PSL}_2(\mathbb{R})$, and they showed that the supra-maximal representations form a compact component of a real relative character variety. We study a collection of rank $n$ local systems on $Σ_{g,d}$ which we call of minimal energy. These are generalizations of supra-maximal representations, and underlie polarizable complex variations of Hodge structure for any choice of complex structure on $Σ_{g,d}$. Like the supra-maximal representations, the minimal energy local systems form a compact connected component of a real relative character variety. We show that when the local monodromy data around the punctures is chosen to be unitary and generic, and the relative character variety is nonempty, these minimal energy local systems always exist. When $g > 0$ we show that the minimal energy local systems come from unitary representations of $π_1(Σ_{g,d})$. If $g = 0$ we show that they do not always come from unitary representations, and we study their structure in general.