Consider a positively monotone (Fano) closed symplectic manifold M and a symplectic simple crossings divisor D in it. Assume that the Poincare dual of the anti-canonical class is a positive
rational linear combination of the classes [Di], where Di are the components of D with their symplectic orientation. A choice of such coefficients, called the weights, (roughly speaking) equips M – D with a Liouville structure. I will start by discussing results relating the components of D with their symplectic orientation. A choice of such coefficients, called the weights, (roughly speaking) equips M – D with a Liouville structure. I will start by discussing results relating the symplectic cohomology of M – D with quantum cohomology of M. These results are particularly sharp when the weights are all at most 1 (hypothesis A). Then, I will discuss certain rigidity results (inside M) for
skeleton type subsets of M – D, which will also demonstrate the geometric meaning of hypothesis A in examples.
The talk will be mainly based on joint work with Strom Borman and Nick Sheridan.
This video is part of the 3CinG annual meeting that took place in Warwick in September 2021.
