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Workshop "New advances on the Dolbeault moduli spaces"

The Dolbeault moduli spaces are fundamental objects in Non Abelian Hodge theory and representation theory. The symmetries of their cohomology play a crucial role in the proof of the Fundamental Lemma by Ngô, the P=W conjecture, and in the mathematical formulation of mirror symmetry. 
This one-day workshop at École Polytechnique is aimed to present an overview of on-going projects on the geometry of Dolbeault moduli spaces.
Salle de conférences du CMLS, École Polytechnique
Schedule: Friday 24 May 2024
10:00-11:00 Andres Fernandez Herrero, Columbia University

11:15-12:15 Roberto Fringuelli, Università La Sapienza

14:00-15:00 Mark Andrea de Cataldo, Stony Brook University


Organizers: Omid Amini, Mirko Mauri
email: omid.amini(at)polytechnique(dot)edu;  mirko.mauri(at)polytechnique(dot)edu.
The workshop is supported by PEPS 2024 "Jeunes chercheuses et jeunes chercheurs" Symmetries of Dolbeault moduli spaces.


Titles and abstracts:
Title: Introduction to the Hitchin fibration and its decomposition theorem
Speaker: Andres Fernandez Herrero


Abstract: This talk will introduce the Dolbeault moduli space and its corresponding Hitchin fibration.
We will then discuss the notion of Ngo fibration. This is a useful geometric context which puts constraints on the shape of the corresponding decomposition theorem.
In the latter part of the talk, we will explore two of the defining properties of an Ngo fibration in the context of the Hitchin morphism. 
Namely, we will explain why the Hitchin fibration is a weak abelian fibration, and why it satisfies certain cohomological bounds.
Title: A support theorem for the Hitchin fibration for reductive groups
Speaker: Roberto Fringuelli


Abstract:  A challenging task is the computation of the (intersection) cohomology of the moduli space of G-Higgs bundles with poles, with G complex connected reductive group.
One way to tackle the problem is to study the relative cohomology of the intersection complex of this moduli space along the corresponding Hitchin morphism.
Thanks to the works of Chaudouard-Laumon, de Cataldo and Maulik-Shen, we know that, when G is either GLn or SLn, the relative cohomology of the intersection complex is completely determined by its restriction to the elliptic locus, where the spectral curves are integral. In this talk, we discuss a generalization of the latter result to arbitrary reductive groups.
This is a work in progress jointly with Mark Andrea de Cataldo, Andres Fernandez Herrero and Mirko Mauri.


Title: Decomposition Theorem for the Hitchin morphism for the special linear group
Speaker: Mark Andrea de Cataldo


Abstract: I will discuss ongoing joint work with Andres Fernandez Herrero, Mirko Mauri and Roberto Fringuelli on the rational intersection cohomology groups of the moduli spaces of Higgs bundles with poles for the special linear group over a compact Riemann surface of genus at least two.