Floer theory for negative line bundles via Gromov-Witten invariants

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 π_1(Ham(X,ω)) on the symplectic cohomology of symplectic manifolds X conical at infinity.

53 pages; version 3: improved discussion of maximum principle for negative vector bundles. The final version is published in Advances in Mathematics

References in corpus (3)

• Functors and Computations in Floer homology with Applications Part II
• Topological quantum field theory structure on symplectic cohomology
• The monotone wrapped Fukaya category and the open-closed string map

Cited by in corpus (29)

• The monotone wrapped Fukaya category and the open-closed string map
• Floer theory and reduced cohomology on open manifolds
• Symplectic Homology of complements of smooth divisors
• Rabinowitz Floer homology and mirror symmetry
• Circle-actions, quantum cohomology, and the Fukaya category of Fano toric varieties
• Vanishing of Rabinowitz Floer homology on negative line bundles
• Symplectic fillings of asymptotically dynamically convex manifolds I
• Hamiltonian Floer homology for compact convex symplectic manifolds
• Homotopy type of the unitary group of the uniform Roe algebra on ${\mathbb{Z}}^n$
• $({\mathbb{RP}}^{2n-1},{\xi }_{std})$ is not exactly fillable for $n\ne 2^k$
• Quantum Steenrod squares and the equivariant pair-of-pants in symplectic cohomology
• Invariance of symplectic cohomology and twisted cotangent bundles over surfaces
• On filings of $\partial (V\times \mathbb{D})$
• Embedded contact homology of prequantization bundles
• Rabinowitz Floer homology of negative line bundles and Floer Gysin sequence
• Reeb orbits and the minimal discrepancy of an isolated singularity
• Holomorphic disks and the disk potential for a fibered Lagrangian
• Projective twists and the Hopf correspondence
• Disjoinable Lagrangian tori and semisimple symplectic cohomology
• Flat endomorphisms for mod $p$ equivariant quantum connections from quantum Steenrod operations
• An intertwining relation for equivariant Seidel maps
• Exact orbifold fillings of contact manifolds
• Shift operators and connections on equivariant symplectic cohomology
• Quantum Steenrod operations of symplectic resolutions
• Periodic leaf-wise intersection points from Lagrangians
• A computation of the ring structure in wrapped Floer homology
• Categorical non-properness in wrapped Floer theory
• A landscape of contact manifolds via rational SFT
• On the Filtered Symplectic Homology of Prequantization Bundles