Abstract:
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 $[D_i]$, where $D_i$ 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 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.
