Symplectic Homology of complements of smooth divisors

TL;DR

This paper develops a new method to compute symplectic homology of complements of smooth divisors in symplectic manifolds, linking Morse theory and Gromov-Witten invariants under monotonicity assumptions.

Contribution

It introduces a chain complex for symplectic homology of divisor complements, incorporating Morse and Gromov-Witten invariants, and employs a Morse-Bott model with novel curve comparisons.

Findings

Constructed a chain complex computing symplectic homology.

Expressed the differential in terms of Morse and Gromov-Witten invariants.

Established a comparison between Floer cylinders and pseudoholomorphic curves.

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 $\Sigma$.