  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.
