Uniruled symplectic divisors

TL;DR

This paper explores the relationship between uniruled symplectic divisors and the ambient symplectic manifold, demonstrating that the existence of such divisors often implies the manifold itself is uniruled, and provides new construction methods.

Contribution

It establishes a link between uniruled divisors and the uniruledness of the ambient manifold, extending previous results and offering a general construction approach.

Findings

Positive uniruled divisors imply the ambient manifold is uniruled.

Confirms part of the uniruled symplectic divisors dichotomy.

Provides a new general construction of uniruled symplectic manifolds.

Abstract

This is a paper devoted to the symplectic birational geometry program where many basic notions are defined in terms of genus 0 GW invariants. We show that the existence of a positive uniruled symplectic divisor often implies that the ambient manifold has a nonzero uniruled genus 0 GW invariant, hence is uniruled as well. This confirms a part of the dichotomy on uniruled symplectic divisors. In addition, it gives a rather general construction of uniruled symplectic manifolds, generalizing some beautiful results of McDuff.