Title:
Miraculous transversality and harmless multiple covers: some low-dimensional
properties of J-holomorphic curves
Abstract:
One of the hardest problems in the field of J-holomorphic curves
involves transversality: the standard theory is very nice for curves
that are somewhere injective, but as soon as multiple covers appear,
transversality fails, moduli spaces turn out to have the wrong
dimension, and many wonderful invariants cannot easily be defined. In
some settings however, the transversality problem practically solves
itself, and the reasons are not analytical, but topological. I will
describe a few such situations in contact 3-manifolds and symplectic
4-manifolds, where intersection theory comes into play and moduli
spaces have far nicer geometric structures than one might expect.
These spaces are relevant in particular to the theory of finite energy
foliations, which one can think of as "J-holomorphic generalized open
book decompositions".