Symplectic Homology of complements of smooth divisors

Abstract

If $(X, \omega)$ is a symplectic manifold, and $\Sigma$ is a smooth symplectic submanifold Poincar\'e dual to a positive multiple of $\omega$, $X \setminus \Sigma$ admits a compactification as a Liouville domain, which we then complete to $(W, d\lambda)$. Under monotonicity assumptions on $X$ and on $\Sigma$, we construct a chain complex whose homology computes the Symplectic Homology of $W$. We show the differential is given in terms of Morse contributions, terms computed from Gromov-Witten invariants of $X$ relative to $\Sigma$ and terms computed from the Gromov-Witten invariants of $\Sigma$. We use a Morse-Bott model for symplectic homology. Our proof involves comparing Floer cylinders with punctures to pseudoholomorphic curves in in the symplectization of the unit normal bundle to $\Sigma$.