Moduli Motives and bundles. New trends in algebraic geometry

Nonabelian Hodge theory can be seen as a vast generalization of the theory of harmonic forms on smooth and projective complex varieties. It is due to Hitchin, Simpson, Donaldson and Corlette. The fundamental result of the theory is the existence of a correspondence between the “Betti cohomology” and the “Dolbeault cohomology” of the variety. In his 1992 paper ‘Higgs bundles and local systems’, Carlos Simpson observed that the Betti side generalises to non-constant coefficients, and asked about the corresponding Dolbeault space in that case. The purpose of this course is to explain one possible answer to Simpson’s question. 

Higher tangent spaces to moduli (in particular so called obstruction spaces) have been known since the beginning of infinitesimal deformation theory in the fifties. They reflect the local structure of the moduli at a point. One way to globalize them is the introduction of obstruction theories; a more general geometric interpretation is based on higher geometry. In this series of lectures we will give a gentle introduction to this circle of ideas, focusing on selected examples from classical moduli problems in complex projective geometry rather than abstract foundations. In particular, we will review deformation theory of Quot schemes and of moduli of coherent sheaves and discuss their derived structures.