## Conference Talk.

Camilla Felisetti

University of Bologna

The intersection cohomology of the moduli space of Higgs bundles on a smooth projective curve

**Abstract:**Let $X$ be a smooth projective curve of genus $g$ over $\mathbb{C}$. The

*character variety*$\mathcal{M}_B$ parametrizing conjugacy classes of representations from the fundamental group of X into $SL(2,\mathbb{C})$ is an affine irreducible singular projective variety. The

*Non Abelian Hodge theorem*, states that there is a real analytic isomorphism between $\mathcal{M}_B$ and the quasi projective singular variety $\mathcal{M}_{Dol}$ which parametrizes semistable Higgs bundles of rank $2$ and degree $0$ on $X$. During the seminar I will present a desingularization of these moduli spaces and compute several informations about their intersection cohomology using the famous

*Decomposition theorem*by Beilinson, Bernstein, Deligne and Gabber. Time permitting, I will explain some possible applications to the so called $P=W$ conjecture, stated and proved by De Cataldo, Hausel and Migliorini in the case of rank 2 and degree 1 Higgs bundles.

The talk is part of the 2-day meeting "Current trends on spectral data for Higgs bundles III"

Tuesday November 14, 2017 at 9:30 AM in SEO 636