Stratification of Joint Moduli Spaces of Higgs Bundles

Abstract

We study the marked analytic moduli space of polystable degree-zero $\mathrm{SL}_r(\C)$-Higgs bundles over Teichm\"uller space. The connected components of the orbit-type loci defined by the full determinant-one stabilizers form a complex Whitney stratification. The projection to Teichm\"uller space is a holomorphic submersion on every stratum. Each stratum carries a canonical relative holomorphic symplectic form. These forms determine a vertical holomorphic Poisson bracket on the reduced structure sheaf, whose symplectic leaves are the connected orbit-type strata in the fibers. On each fiber, the Hitchin--Kobayashi correspondence is biholomorphic on every stratum and identifies its hyperk\"ahler structure. The harmonic representatives and vertical hyperk\"ahler tensors vary smoothly along the strata of the total space.