1-6【Jia Choon Lee】Relative spectral correspondence for parabolic Higgs bundles and its applications

It has been well-known since the work of Hitchin, Beauville-Narasimhan-Ramanan, Simpson that Higgs bundles on a curve correspond to pure dimension-one sheaves on the total space of the cotangent bundle of the curve. In this talk, I will explain how to extend this spectral correspondence to a relative correspondence between (weakly) parabolic Higgs bundles and pure dimension-one sheaves on a family of holomorphic symplectic surfaces. As applications, I will also explain how this surface perspective can be used to (1) study the Deligne-Simpson problem which concerns the existence of irreducible local systems on a punctured sphere with prescribed monodromy and (2) to determine the generators of the cohomology of the moduli space of parabolic Higgs bundles. This is joint work with Sukjoo Lee.