Floer theory for negative line bundles via Gromov-Witten invariants

We prove that the GW theory of negative line bundles M = Tot(L→B) determines the symplectic cohomology: indeed SH *(M) is the quotient of QH *(M) by the kernel of a power of quantum cup product by c1(L). We prove this also for negative vector bundles and the top Chern class. We calculate SH * and QH * for O(-n)→CPm. For example: for O(-1), M is the blow-up of Cm+1 at the origin and SH *(M) has rank m. We prove Kodaira vanishing: for very negative L, SH * = 0; and Serre vanishing: if we twist a complex vector bundle by a large power of L, SH * = 0.Observe SH *(M) = 0 iff c1(L) is nilpotent in QH *(M). This implies Oancea's result: ωB(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)) to invertibles in SH *(X, ω). This is similar to Seidel's representation for closed X, except now they are not invertibles in QH *(X, ω). © 2014 Elsevier Inc