Non-abelian Hodge moduli spaces and homogeneous affine Springer fibers

TL;DR

This paper constructs and compares moduli spaces related to affine Springer fibers, providing evidence for a wild version of the Simpson correspondence by analyzing their topological and cohomological properties.

Contribution

It introduces three new moduli spaces corresponding to different aspects of a wild Simpson correspondence and establishes their topological relationships and conjectured isomorphisms.

Findings

Affine Springer fiber homeomorphic to a Lagrangian fiber in Dolbeault space

Dolbeault and de Rham spaces share the same cohomology as the Springer fiber

Proposed an analytic isomorphism between de Rham and Betti spaces

Abstract

Starting from a homogeneous affine Springer fiber $Fl_{\psi}$, we construct three moduli spaces that correspond to the Dolbeault, de Rham and Betti aspects of a hypothetical Simpson correspondence with wild ramifications. We show that $Fl_{\psi}$ is homeomorphic to the central Lagrangian fiber in the Dolbeault space, prove that the Dolbeaut and de Rham spaces both have the same cohomology as $Fl_{\psi}$, and construct a map from the de Rham space to the Betti space which we conjecture to be an analytic isomorphism.