Local symplectic field theory and stable hypersurfaces in symplectic blow-ups
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Tue, 28/02/2012 15:45 |
Oliver Fabert (Freiburg) |
Algebraic and Symplectic Geometry Seminar  |
L3 |
| Symplectic field theory (SFT) can be viewed as TQFT approach to Gromov-Witten theory. As in Gromov-Witten theory, transversality for the Cauchy-Riemann operator is not satisfied in general, due to the presence of multiply-covered curves. When the underlying simple curve is sufficiently nice, I will outline that the transversality problem for their multiple covers can be elegantly solved using finite-dimensional obstruction bundles of constant rank. By fixing the underlying holomorphic curve, we furthermore define a local version of SFT by counting only multiple covers of this chosen curve. After introducing gravitational descendants, we use this new version of SFT to prove that a stable hypersurface intersecting an exceptional sphere (in a homologically nontrivial way) in a closed four-dimensional symplectic manifold must carry an elliptic orbit. Here we use that the local Gromov-Witten potential of the exceptional sphere factors through the local SFT invariants of the breaking orbits appearing after neck-stretching along the hypersurface. | |||