Floer theory for negative line bundles via Gromov-Witten invariants

TL;DR

This paper establishes a deep connection between quantum cohomology and symplectic cohomology for negative line bundles over symplectic manifolds, providing explicit calculations and vanishing theorems.

Contribution

It proves an isomorphism between quotient quantum cohomology and symplectic cohomology for negative line bundles, extending to vector bundles and including explicit computations.

Findings

Symplectic cohomology vanishes iff the first Chern class is nilpotent in quantum cohomology.

Explicit calculations of symplectic and quantum cohomologies for O(-n) over P^m.

Established vanishing theorems and constructed a representation of a1_1(Ham) on symplectic cohomology.

Abstract

Let M be the total space of a negative line bundle over a closed symplectic manifold. We prove that the quotient of quantum cohomology by the kernel of a power of quantum cup product by the first Chern class of the line bundle is isomorphic to symplectic cohomology. We also prove this for negative vector bundles and the top Chern class. We explicitly calculate the symplectic and quantum cohomologies of O(-n) over P^m. For n=1, M is the blow-up of C^{m+1} at the origin and symplectic cohomology has rank m. The symplectic cohomology vanishes if and only if the first Chern class of the line bundle is nilpotent in quantum cohomology. We prove a Kodaira vanishing theorem and a Serre vanishing theorem for symplectic cohomology. In general, we construct a representation of \pi_1(Ham(X,\omega)) on the symplectic cohomology of symplectic manifolds X conical at infinity.