Rigid holomorphic curves are generically super-rigid

Princeton/IAS Symplectic Geometry Seminar
Topic:Rigid holomorphic curves are generically super-rigid
Speaker:Chris Wendl
Affiliation:Humboldt-Universit├Ąt zu Berlin
Date:Friday, March 31
Time/Room:1:30pm - 2:30pm/S-101
Video Link:https://video.ias.edu/puias/2017/0331-ChrisWendl

I will explain the main ideas of a proof that for generic compatible almost complex structures in symplectic manifolds of dimension at least 6, closed embedded J-holomorphic curves of index 0 are always "super-rigid", implying that their multiple covers are never limits of sequences of curves with distinct images. This condition is especially interesting in Calabi-Yau 3-folds, where it follows that the Gromov-Witten invariants can be "localized" and computed in terms of Euler classes of obstruction bundles for a finite set of disjoint embedded curves. By the same techniques, we can also show that unbranched covers of simple J-holomorphic curves are generically regular. These results are based on a decomposition of the space of branched covers into smooth strata on which certain twisted Cauchy-Riemann operators have kernel and cokernel of constant dimension.