Floer theory for negative line bundles via Gromov–Witten invariants

We prove that the GW theory of negative line bundles M=Tot(L→B)M=Tot(L→B) determines the symplectic cohomology: indeed SH⁎(M)SH⁎(M) is the quotient of QH⁎(M)QH⁎(M) by the kernel of a power of quantum cup product by c1(L)c1(L). We prove this also for negative vector bundles and the top Chern class.We calculate SH⁎SH⁎ and QH⁎QH⁎ for O(−n)→CPmO(−n)→CPm. For example: for O(−1)O(−1), M   is the blow-up of Cm+1Cm+1 at the origin and SH⁎(M)SH⁎(M) has rank m.We prove Kodaira vanishing: for very negative L  , SH⁎=0SH⁎=0; and Serre vanishing: if we twist a complex vector bundle by a large power of L  , SH⁎=0SH⁎=0.Observe SH⁎(M)=0SH⁎(M)=0 iff c1(L)c1(L) is nilpotent in QH⁎(M)QH⁎(M). This implies Oancea's result: ωB(π2(B))=0⇒SH⁎(M)=0ωB(π2(B))=0⇒SH⁎(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 π1(Hamℓ(X,ω))π1(Hamℓ(X,ω)) to invertibles in SH⁎(X,ω)SH⁎(X,ω). This is similar to Seidel's representation for closed X  , except now they are not invertibles in QH⁎(X,ω)QH⁎(X,ω).