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The P=W Conjecture in Non Abelian Hodge Theory

Abstract: The complex singular cohomology groups of a projective manifold can be described in at least three ways via the de Rham Theorem and the Hodge Decomposition. By taking into account integral cohomology, we obtain three different descriptions of the singular cohomology groups with coefficients in the non-zero complex numbers GL1. Now, replace GL1 with a complex algebraic reductive group G, e.g. GLn. The Non Abelian Hodge Theory of Corlette, Simpson et al. establishes a natural homeomorphism between three distinct complex algebraic varieties parametrizing three different kinds of structures on the projective manifold associated with the reductive group: representations of the fundamental group into G, flat algebraic G-connections, G-Higgs bundles. The case G=GL1, which is Abelian, recaptures the de Rham and Hodge Decomposition. The three complex algebraic varieties on Non Abelian Hodge Theory have naturally isomorphic cohomology groups. However, by taking into account their distinct structures as algebraic varieties, the cohomology groups carry additional distinct structures. The P=W Conjecture seeks to relate two of these structures, at least in the case of compact Riemann surfaces. This talk is devoted to introducing the audience to this circle of ideas and related developments.