Math Seminars.
Eleny Ionel (Univ. of Wisc.),
Gromov-Witten Invariants of Symplectic Sums and Applications
The natural cut-and-paste operation for symplectic manifolds is
the symplectic sum along a codimension 2 symplectic submanifold.
In this talk we will describe a gluing formula for Gromov-Witten
invariants of the symplectic sum in terms of the relative GW
invariants of the two pieces. This degeneration formula (which is
joint work with Tom Parker) describes what happens to
holomorphic curves as one pinches the neck and explains how to
compute the GW invariant from the limiting curves.
The degeneration formula has plenty of applications, ranging from
the construction of infinitely many exotic symplectic structures
on the same smooth manifold to solving enumerative questions in
algebraic geometry. In the second part of the talk I will
describe how to use the degeneration formula to get new relations
in the cohomology of the moduli space of complex structures on a
genus g Riemann surface with n marked points.