A Cohomological Non Abelian Hodge Theorem in Positive Characteristic

We start with a curve over an algebraically closed ground field of positive characteristic $p>0$. By using specialization techniques, under suitable natural coprimality conditions, we prove a cohomological Simpson Correspondence between the moduli space of Higgs bundles and the one of connections on the curve. We also prove a new $p$-multiplicative periodicity concerning the cohomology rings of Dolbeault moduli spaces of degrees differing by a factor of $p$. By coupling this $p$-periodicity in characteristic $p$ with lifting/specialization techniques in mixed characteristic, we find, in arbitrary characteristic, cohomology ring isomorphisms between the cohomology rings of Dolbeault moduli spaces for different degrees coprime to the rank. It is interesting that this last result is proved as follows: we prove a weaker version in positive characteristic; we lift and strengthen the weaker version to the result in characteristic zero; finally, we specialize the result to positive characteristic. The moduli spaces we work with admit certain natural morphisms (Hitchin, de Rham-Hitchin, Hodge-Hitchin), and all the cohomology ring isomorphisms we find are filtered isomorphisms for the resulting perverse Leray filtrations.

Projective Completion of Moduli of $t$-Connections on Curves in Positive and Mixed Characteristic

We generalize a compactification technique due to C. Simpson in the context of $mathbb{G}_m$-actions over the ground field of complex numbers, to the case of a universally Japanese base ring. We complement this generalized compactification technique so that it can sometimes yield projectivity results for these compactifications. We apply these projectivity results to the Hodge, de Rham, and Dolbeault moduli spaces for curves, with special regards to ground fields of positive characteristic.

On the P=W conjecture for $mathrm{SL}_n$

Let $p$ be a prime number. We prove that the $P=W$ conjecture for $mathrm{SL}_p$ is equivalent to the $P=W$ conjecture for $mathrm{GL}_p$. As a consequence, we verify the $P=W$ conjecture for genus 2 and $mathrm{SL}_p$. For the proof, we compute the perverse filtration and the weight filtration for the variant cohomology associated with the $mathrm{SL}_p$-Hitchin moduli space and the $mathrm{SL}_p$-twisted character variety, relying on Grochenig-Wyss-Zieglers recent proof of the topological mirror conjecture by Hausel-Thaddeus. Finally we discuss obstructions of studying the cohomology of the $mathrm{SL}_n$-Hitchin moduli space via compact hyper-Kahler manifolds.

Hitchin fibrations, abelian surfaces, and the P=W conjecture

We study the topology of Hitchin fibrations via abelian surfaces. We establish the P=W conjecture for genus $2$ curves and arbitrary rank. In higher genus and arbitrary rank, we prove that P=W holds for the subalgebra of cohomology generated by even tautological classes. Furthermore, we show that all tautological generators lie in the correct pieces of the perverse filtration as predicted by the P=W conjecture. In combination with recent work of Mellit, this reduces the full conjecture to the multiplicativity of the perverse filtration. Our main technique is to study the Hitchin fibration as a degeneration of the Hilbert-Chow morphism associated with the moduli space of certain torsion sheaves on an abelian surface, where the symmetries induced by Markmans monodromy operators play a crucial role.

A support theorem for the Hitchin fibration: the case of $GL_n$ and $K_C$

We compute the supports of the perverse cohomology sheaves of the Hitchin fibration for $GL_n$ over the locus of reduced spectral curves. In contrast to the case of meromorphic Higgs fields we find additional supports at the loci of reducible spectral curves. Their contribution to the global cohomology is governed by a finite twist of Hitchin fibrations for Levi subgroups. The corresponding summands give non-trivial contributions to the cohomology of the moduli spaces for every $n geq 3$. A key ingredient is a restriction result for intersection cohomology sheaves that allows us to compare the fibration to the one defined over the versal deformations of spectral curves.

The Hodge numbers of OGrady 10 via Ng^o strings

We determine the Hodge numbers of the hyper-Kahler manifold known as OGrady 10 by studying some related modular Lagrangian fibrations by means of a refinement of the Ng^o Support Theorem.

Compactification of Dolbeault moduli spaces

We construct a relative compactification of Dolbeault moduli spaces of Higgs bundles for reductive algebraic groups on families of projective manifolds that is compatible with the Hitchin morphism.

The perverse filtration for the Hitchin fibration is locally constant

We prove that the perverse Leray filtration for the Hitchin morphism is locally constant in families, thus providing some evidence towards the validity of the $P=W$ conjecture due to de Cataldo, Hausel and Migliorini in non Abelian Hodge theory.

A support theorem for the Hitchin fibration: the case of $SL_n$

We prove that the direct image complex for the $D$-twisted $SL_n$ Hitchin fibration is determined by its restriction to the elliptic locus, where the spectral curves are integral. The analogous result for $GL_n$ is due to P.-H. Chaudouard and G. Laumon. Along the way, we prove that the Tate module of the relative Prym group scheme is polarizable, and we also prove $delta$-regularity results for some auxiliary weak abelian fibrations.

Perverse sheaves and the topology of algebraic varieties (2015 PCMI)

Another introduction to perverse sheaves with some exercises. Expanded version of five lectures at the 2015 PCMI.