Subcritical polarisations of symplectic manifolds have degree one

TL;DR

This paper proves that certain symplectic hypersurfaces must have degree one if their complement resembles a subcritical Stein manifold, confirming a conjecture in symplectic topology.

Contribution

It establishes a new link between the topology of symplectic hypersurfaces and subcritical Stein manifolds, confirming a conjecture by Biran and Cieliebak.

Findings

Hypersurfaces with subcritical Stein complements have degree one.

Confirms Biran and Cieliebak's conjecture on subcritical polarisations.

Uses a homological argument based on Kulkarni-Wood ideas.

Abstract

We show that if the complement of a Donaldson hypersurface in a closed, integral symplectic manifold has the homology of a subcritical Stein manifold, then the hypersurface is of degree one. In particular, this demonstrates a conjecture by Biran and Cieliebak on subcritical polarisations of symplectic manifolds. Our proof is based on a simple homological argument using ideas of Kulkarni-Wood.