The topology and Hodge theory of algebraic maps

The award supports research in the field of algebraic geometry, the discipline devoted to the study of polynomial or algebraic equations. Algebraic equations are both beautiful and ubiquitous, as they describe many natural phenomena, from the motion of planets or the shape of leaves and flowers, to the behavior of microscopic particles. The goal of this research project is to study the deeper properties of the solutions to more complicated algebraic equations, called algebraic maps. The investigator plans to continue the long-term investigation of the topology, Hodge theory, and cycle theory of algebraic maps. The close connection between the two main threads of the research, namely the discovery of new and deep aspects of the general theory and the study of fundamental examples, is the motivating principle behind the work. It is anticipated that the results will be of use to mathematicians in algebraic geometry, combinatorics, and representation theory, and to mathematical physicists in the study of string theory. This project also provides research training opportunities for graduate students. The investigator will explore, with various teams of collaborators, the fundamental aspects of the general theory, as well as important examples, through four projects: 1) To determine the exact form of the Decomposition Theorem for the Hitchin morphism for G-Higgs bundles with log-poles over a curve over the field of complex numbers for a reductive group G. 2) To formulate and prove the Topological Mirror Symmetry Conjecture for the intersection cohomology groups of the moduli of G-Higgs bundles for the pair of Langlands dual groups SLn and PGLn with and without poles in the case when rank and degree are not coprime --the coprime case is known--; this requires new ideas, such as generically defined gerbes and a new formalism of correspondences on singular spaces for mixed Hodge modules. 3) To prove a semistable log-pole version of the non-Abelian Hodge theorem in positive characteristic, with cohomological applications. Several new ideas need to be introduced, such as log-connections for affine group schemes and their residue spaces. 4) The PI has already generalized the Alper-Hall-Rydh local structure theorem for algebraic stacks to morphisms of stacks, with geometric applications to the (intersection) cohomology of members of families of good moduli spaces; the PI plans to prove analogous geometric results for the members of families of stacks and to then apply the resulting theory to important families of moduli stacks. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.