Normal Crossings Degenerations of Symplectic Manifolds

TL;DR

This paper introduces a method to degenerate symplectic manifolds into normal crossings varieties using local Hamiltonian torus actions, extending symplectic cut techniques for complex degenerations.

Contribution

It presents a novel construction that generalizes symplectic cuts to produce normal crossings degenerations, inspired by Gross-Siebert and Parker's programs.

Findings

Constructs a smooth one-parameter family degenerating to a normal crossings variety.

Generalizes symplectic cut to a multifold version for complex degenerations.

Enables splitting of symplectic manifolds into components with shared divisors.

Abstract

We use local Hamiltonian torus actions to degenerate a symplectic manifold to a normal crossings symplectic variety in a smooth one-parameter family. This construction, motivated in part by the Gross-Siebert and B. Parker's programs, contains a multifold version of the usual (two-fold) symplectic cut construction and in particular splits a symplectic manifold into several symplectic manifolds containing normal crossings symplectic divisors with shared irreducible components in one step.