On Fourier-Mukai transforms of upward flows for Hitchin systems
David Fang
TL;DR
This work extends the Hausel–Hitchin mirror symmetry for Hitchin systems beyond the smooth Hitchin locus by constructing and extending a generalized Poincaré kernel via Arinkin’s framework. It proves that the Fourier–Mukai transform defined by the elliptic kernel sends the structure sheaf of upward-flow Lagrangians O_{W_δ^+}(-e) to the corresponding hyperholomorphic bundles Λ_δ on the elliptic base, and further extends to larger open sets under appropriate codimension assumptions on non-integral spectral curves. The approach combines spectral geometry, étale local analysis, and determinant-line constructions to realize a robust, potentially twisted duality that aligns Lagrangian data with hyperholomorphic spectral data in the Hitchin fibration. These results contribute to the broader program of geometric Langlands and homological mirror symmetry by clarifying the kernel and mirror correspondence for Hitchin systems across the full base."
Abstract
We consider the moduli space of semistable Higgs bundles on a smooth projective curve. Motivated by mirror symmetry, Hausel and Hitchin showed that over an open of the locus of smooth Hitchin fibers, the duality of Donagi-Pantev intertwines certain Lagrangian upward flows with hyperholomorphic vector bundles constructed from universal Higgs bundles. Using Arinkin's sheaf and some codimension estimates, we show a generalization of this result over the entire Hitchin base, for Higgs bundles of arbitrary degree.