Symplectic cohomology and quantum cohomology of Fano manifolds

University Of Massachusetts Boston, Dorchester MA

Investigators

Abstract

Symplectic geometry is the study of spaces that are locally modeled on phase spaces from classical mechanics. While symplectic manifolds have no local invariants, they do have interesting global ones. A major part of this project is concerned with “Gromov-Witten” invariants, which aim to probe a symplectic manifold by studying maps from two-dimensional surfaces into the manifold that satisfy an appropriate partial differential equation. These invariants are very powerful, but it can be quite difficult to get good control over them because of their non-local nature. One key goal of the project is to prove new fundamental facts about Gromov-Witten invariants on certain symplectic manifolds that are defined by polynomial equations. The general strategy the PI will use is to “cut open” the manifold along a divisor. The corresponding invariants of the divisor complement turn out to be very tractable and also provide a stepping stone towards understanding the Gromov-Witten invariants of the original space. The PI will continue to serve as a mentor to high school students at MIT's Research Science Institute, and will organize a workshop on homological mirror symmetry for graduate students. He will also continue to work on developing a Master’s program in mathematics at the University of Massachusetts Boston. Specifically, in the main strand of the project, the PI will prove that the quantum connection on a Fano manifold with a smooth anti-canonical divisor has a singularity of unramified exponential type. The strategy is to view the quantum cohomology of the Fano manifold as a deformation of the symplectic cohomology of the complement of the divisor. The symplectic cohomology of the complement can in turn be studied via the wrapped Fukaya category, allowing one to bring tools from noncommutative geometry to bear. In a different direction, the PI will build on his previous work to relate the symplectic cohomology of an affine log Calabi-Yau variety to certain intrinsic mirror algebras recently constructed by algebraic geometers. The PI will then go on to study homological mirror symmetry for intrinsic mirror pairs using a combination of symplectic and categorical techniques also developed in previous work. 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.