Floer theory for negative line bundles via Gromov–Witten invariants

By Alexander F. Ritter. 2011

54 citations recorded by the source corpus.

Abstract We prove that the GW theory of negative line bundles M = Tot ( L → B ) determines the symplectic cohomology: indeed S H ⁎ ( M ) is the quotient of Q H ⁎ ( M ) by the kernel of a power of quantum cup product by c 1 ( L ) . We prove this also for negative vector bundles and the top Chern class. We calculate S H ⁎ and Q H ⁎ for O ( − n ) → C P m . For example: for O ( − 1 ) , M is the blow-up of C m + 1 at the origin and S H ⁎ ( M ) has rank m. We prove Kodaira vanishing: for very negative L, S H ⁎ = 0 ; and Serre vanishing: if we twist a complex vector bundle by a large power of L, S H ⁎ = 0 . Observe S H ⁎ ( M ) = 0 iff c 1 ( L ) is nilpotent in Q H ⁎ ( M ) . This implies Oancea's result: ω B ( π 2 ( B ) ) = 0 ⇒ S H ⁎ ( M ) = 0 . We prove the Weinstein conjecture for any contact hypersurface surrounding the zero section of a negative line bundle. For symplectic manifolds X conical at infinity, we build a homomorphism from π 1 ( Ham l ( X , ω ) ) to invertibles in S H ⁎ ( X , ω ) . This is similar to Seidel's representation for closed X, except now they are not invertibles in Q H ⁎ ( X , ω ) .

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/80b342165fa341f07e0a447f3d663e9d2616b9be

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In Geometry and Topology

  1. I. Smith
  2. Alexander F. Ritter
  3. Mark McLean
  4. M. Abouzaid
  5. P. Seidel
  6. A. Oancea
  7. Kai Cieliebak
  8. Frederick J. Bourgeois III

In Mathematical Physics

  1. I. Smith
  2. Alexander F. Ritter
  3. Mark McLean
  4. M. Abouzaid
  5. P. Seidel
  6. A. Oancea
  7. Kai Cieliebak
  8. Frederick J. Bourgeois III

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