Stably Folded Hamiltonian Manifolds and the Singular Symplectic Topology of Moduli Spaces of Pseudoholomorphic Curves

A basic object in the modern study of symplectic topology is the space 𝑀𝑔(𝐴,𝐽)

of 𝐽-holomorphic curves of genus 𝑔 representing the homology class 𝐴

in a symplectic manifold (𝑀,𝑤). In a wide variety of situations, this space is a compact manifold (or orbifold) that admits a natural stable almost complex structure. The study of this space — and its marked variants — up to complex bordism gives rise to a rich source of deformation invariants of the symplectic structure, of which the Gromov-Witten invariants are perhaps the best-known examples. In this talk I will explain how to see that 𝑀𝑔(𝐴,𝐽) in fact carries another natural structure: a closed 2-form that interacts in interesting ways with the stable almost complex structure. I'll introduce the notion of "stably folded Hamiltonian manifolds" and their cobordisms, which captures the sense in which these structures are compatible, and which allows us to study 𝑀𝑔(𝐴,𝐽)

up to a more refined notion of cobordism. Time permitting, I will discuss some of the natural topology of 𝑀𝑔(𝐴,𝐽) that emerges in studying the degeneracy loci of Ω, and how this can allow us to show that regular moduli spaces of holomorphic curves in Kähler manifolds are themselves Kähler (when g=0 and (M,J) is a convex projective manifold, this follows from classical work of Fulton-Pandharipande).