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Degeneration of (pseudo)-holomorphic curves in symplectic geometry and mirror symmetry


Speaker: Yong-Geun Oh, University of Wisconsin at Madison
Location: 693 Kerr
Start time: Thu, Apr 14 2005, 12:00AM

I will start with discussions on how a singular degeneration, called the {\it adiabatic degeneration}, of holomorphic curves naturally arise in two prominent examples related to compactivication of the moduli space of (pseudo)-holomorphic curves in $M$. One occurs in the chain level Floer theory, which concerns the limit as $\epsilon \to 0$ of the solution spaces of the perturbed Cauchy-Riemann equation in $\mathbb{R} \times S^1 \to M$ of the type $$ \frac{\partial u}{\partial \tau}+J\Big(\frac{\partial u}{\partial t})-\epsilon X_f(u)\Big)=0 $$ where $X_f$ is the Hamiltonian vector field associated to a given Morse function $f: M\to \mathbb{R}$. The other concerns the so called {\it large complex structure limit} of the moduli space of holomorphic curves, both open and closed, in the toric fibration or in the Strominger-Yau-Zaslow fibration of Calabi-Yau manifolds. In both examples, the limiting object looks more combinatorial and more computable in relation to the counting problem than the nearby problem for $\epsilon >0$. One important problem is to understand how accuarately the limiting picture reflects the nearby problem, which I call the {\it recovering problem}. Towards this direction, in this talk I will explain one key element needed in the recovering problem. This involves a new compactification of the moduli space of holomorphic curves with prescribed singularities (e.g., that of immersed curves): this refines the well-known stable map compatification and goes deeper into higher order jets of holomorphic maps. The latter work is a ongoing joint project with K. Rukaya. If time permits, I will discuss some possible applications.