Nathan Clement December 30 , 2016 Research Statement Parabolic Higgs Bundles

My main project to date ([3]) is concerned with certain moduli spaces of Higgs bundles and isomorphisms between them, which arise from natural operations on the bundles. Conventionally1, a Higgs bundle is a pair (V, φ) on a Riemann surface X. Here V is a vector bundle (algebraic) and φ is a linear map V → ΩX⊗V called the Higgs field. Why consider such pairs? If nothing else, maybe a Higgs bundle is an interesting example of a family of endomorphisms of a vector space. The full story is deeper: a standard deformation theory argument reveals that the first order deformations of the vector bundle V are parametrized by Ext1(V, V ) = H1(End(V )). A simple Serre duality calculation shows that H1(EndV )∗ = H0(EndV ⊗ ΩX). From this calculation we can see that a Higgs field is a cotangent vector at the point V in the moduli space of vector bundles on X. Taken together, the space of all Higgs bundles on X forms the cotangent space to the moduli space of stable vector bundles on X.