Invariance of microsheaves on stable Higgs bundles

TL;DR

This paper proves the invariance of microsheaves on stable Higgs bundles across deformations of the underlying curve, connecting geometric Langlands conjectures with symplectic geometry and microlocal sheaf theory.

Contribution

It establishes the invariance of microsheaves on stable Higgs bundles, a key step towards understanding automorphic categories in geometric Langlands theory.

Findings

Invariance of microsheaves on stable Higgs bundles proven.

Utilizes symplectic geometry of Higgs moduli space.

Connects microlocal sheaves with deformation invariance.

Abstract

The spectral side of the (conjectural) Betti geometric Langlands correspondence concerns sheaves on the character stack of an algebraic curve; in particular, the categories in question are manifestly invariant under deformations of the curve. By contrast the same invariance is certainly not manifest, and is presently not known, for their automorphic counterparts, in particular because the singularities of the global nilpotent cone may vary significantly with the complex structure of the curve. Here we establish the corresponding invariance statement for the category of microsheaves on the open subset of stable Higgs bundles on nonstacky components where all semistables are stable, e.g. for coprime rank and degree or for a punctured curve with generic parabolic weights. The proof uses the known global symplectic geometry of the Higgs moduli space to invoke recent results on the invariance of microlocal sheaves.