## \(P = W\): a strange identity for \(\mathrm{GL}(2,\mathbb C)\)

Mark deCataldo

Stony Brook University; Member, School of Mathematics

November 24, 2014

Start with a compact Riemann surface \(X\) and a complex reductive group \(G\), like \(\mathrm{GL}(n,\mathbb C)\). According to Hitchin-Simpson's ``non abelian Hodge theory", the pair \((X,G)\) comes with two new complex manifolds: the character variety \(\mathcal M_B\) and the Higgs moduli space \(\mathcal{M}_\text{Dolbeault}\). When \(G= \mathbb C^*\), these manifolds are two instances of the usual first cohomology group of \(X\) with coefficients in the abelian \(\mathbb C^*\).