Generalized Punctual Hilbert Schemes and $\mathfrak{g}$-complex structures

TL;DR

This paper introduces generalized punctual Hilbert schemes linked to Lie algebras, leading to new geometric structures on surfaces that resemble Hitchin components and potentially offer new insights into their moduli spaces.

Contribution

It defines novel generalizations of punctual Hilbert schemes for Lie algebras and constructs related geometric structures with properties similar to Hitchin components.

Findings

New geometric structures on surfaces related to Lie algebras

Moduli spaces sharing properties with Hitchin components

Potential homeomorphism to Hitchin components

Abstract

We define and analyze various generalizations of the punctual Hilbert scheme of the plane, associated to complex or real Lie algebras. Out of these, we construct new geometric structures on surfaces whose moduli spaces share multiple properties with Hitchin components, and which are conjecturally homeomorphic to them. For simple complex Lie algebras, this generalizes the higher complex structure. For real Lie algebras, this should give an alternative description of the Hitchin-Kostant-Rallis section.