Normal Crossings Singularities for Symplectic Topology

TL;DR

This paper develops topological and geometric frameworks for normal crossings symplectic divisors and varieties, establishing their equivalence and laying groundwork for future advances in symplectic topology and singular symplectic structures.

Contribution

It introduces topological notions of normal crossings symplectic divisors and varieties, proving their equivalence to geometric notions, and addresses foundational questions in symplectic singularities.

Findings

Topological and geometric notions of normal crossings symplectic divisors are equivalent.

Provides a foundation for defining and smoothing singular symplectic varieties.

Lays groundwork for symplectic analogues of logarithmic structures.

Abstract

We introduce topological notions of normal crossings symplectic divisor and variety and establish that they are equivalent, in a suitable sense, to the desired geometric notions. Our proposed concept of equivalence of associated topological and geometric notions fits ideally with important constructions in symplectic topology. This partially answers Gromov's question on the feasibility of defining singular symplectic (sub)varieties and lays foundation for rich developments in the future. In subsequent papers, we establish a smoothability criterion for symplectic normal crossings varieties, in the process providing the multifold symplectic sum envisioned by Gromov, and introduce symplectic analogues of logarithmic structures in the context of normal crossings symplectic divisors.