Symplectic surgeries from singularities

TL;DR

This paper introduces symplectic surgeries derived from singularities, enabling the replacement of Lagrangian spheres with symplectic submanifolds, and explores their implications for the existence of certain symplectic manifolds.

Contribution

It presents new symplectic surgeries based on local Kahler degenerations and resolutions, expanding the toolkit for modifying symplectic manifolds.

Findings

Constructed symplectic surgeries from singularities.

Applied surgeries to examples in 4 and 6 dimensions.

Connected surgeries to existence questions for manifolds with different first Chern classes.

Abstract

We describe a variety of symplectic surgeries (not a priori compatible with Kahler structures) which are obtained by combining local Kahler degenerations and resolutions of singularities. The effect of the surgeries is to replace configurations of Lagrangian spheres with symplectic submanifolds. We discuss several examples in detail, relating them to existence questions for symplectic manifolds with $c_1>0, c_1=0, c_1<0$ in four and six dimensions.