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Subcritical polarisations of symplectic manifolds have degree one

Hansjörg Geiges, Kevin Sporbeck, Kai Zehmisch

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.

Subcritical polarisations of symplectic manifolds have degree one

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.

Paper Structure

This paper contains 4 sections, 3 theorems, 4 equations.

Table of Contents

  1. Donaldson hypersurfaces and symplectic polarisations
  2. Subcritical polarisations
  3. The Kulkarni--Wood homomorphism
  4. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1

Let $(M,\omega)$ be a closed, integral symplectic manifold, and $\Sigma\subset M$ a compact symplectic submanifold of codimension $2$, Poincaré dual to the integral cohomology class $d[\omega]$ for some (positive) integer $d$. If $(M,\Sigma)$ is homologically subcritical, then $d[\omega]/\mathrm{tor

Theorems & Definitions (6)

  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Remark 4