The $p$-adic Corlette-Simpson correspondence for abeloids

Higgs bundles on Sasakian manifolds

We extend the Donaldson-Corlette-Hitchin-Simpson correspondence between Higgs bundles and flat connections on compact Kähler manifolds to compact quasi-regular Sasakian manifolds. A particular consequence is the translation of restrictions on Kähler groups proved using the Donaldson-Corlette-Hitchin-Simpson correspondence to fundamental groups of compact Sasakian manifolds.

交换情形下主丛的 $p$-进 Simpson 对应

$p$-adic Simpson correspondences for principal bundles in abelian settings

本文研究刚性空间上主丛的 $p$-进 Simpson 对应，证明交换群情形由李群对数满射性刻画，并将 Faltings 小对应推广至普通阿贝尔簇上的一般刚性群。

Nonabelian Hodge theory for Fujiki class $\mathcal C$ manifolds

The nonabelian Hodge correspondence (Corlette-Simpson correspondence), between the polystable Higgs bundles with vanishing Chern classes on a compact Kähler manifold $X$ and the completely reducible flat connections on $X$, is extended to the Fujiki class $\mathcal C$ manifolds.

Kobayashi-Hitchin correspondence for tame harmonic bundles II

Let $X$ be a smooth projective complex variety with an ample line bundle $L$, and let $D$ be a simple normal crossing divisor. We establish the Kobayashi-Hitchin correspondence between tame harmonic bundles on $X-D$ and $μ_L$-stable parabolic $λ$-flat bundles with trivial characteristic numbers on $(X,D)$. Especially, we obtain the quasiprojective version of the Corlette-Simpson correspondence between flat bundles and Higgs bundles.

Higgs bundles and the real symplectic group

We give an overview of the work of Corlette, Donaldson, Hitchin and Simpson leading to the non-abelian Hodge theory correspondence between representations of the fundamental group of a surface and the moduli space of Higgs bundles. We then explain how this can be generalized to a correspondence between character varieties for representations of surface groups in real Lie groups G and the moduli space of G-Higgs bundles. Finally we survey recent joint work with Bradlow, García-Prada and Mundet i Riera on the moduli space of maximal Sp(2n,R)-Higgs bundles.

Wild Hodge theory and Hitchin-Kobayashi correspondence [after T. Mochizuki]

T. Mochizuki constructs a theory of variations of wild Hodge structure for which the underlying flat connection can have irregular singularities at infinity. He extends in this way the correspondence of Corlette and Simpson between irreducible flat bundles and stables Higgs bundles, taking into account objects with irregular singularities. As an application, he proves a conjecture of Kashiwara concerning a generalization of the Hard Lefschetz theorem when the coefficients are the de Rham complex of a simple holonomic D-module on smooth complex projective variety.