Normal Crossings Singularities for Symplectic Topology, II

TL;DR

This paper extends the topological and geometric notions of normal crossings singularities in symplectic topology, establishing their equivalence and providing a smoothability criterion applicable to broader classes of singularities.

Contribution

It generalizes the concepts of normal crossings singularities to arbitrary cases and proves their equivalence to geometric notions, advancing the understanding of symplectic singularities.

Findings

Extended notions to arbitrary normal crossings singularities.

Proved equivalence between topological and geometric notions.

Established a smoothability criterion for these singularities.

Abstract

In recent work, we introduced topological notions of simple normal crossings symplectic divisor and variety, showed that they are equivalent, in a suitable sense, to the corresponding geometric notions, and established a topological smoothability criterion for them. The present paper extends these notions to arbitrary normal crossings singularities, from both local and global perspectives, and shows that they are also equivalent to the corresponding geometric notions. In subsequent papers, we extend our smoothability criterion to arbitrary normal crossings symplectic varieties and construct a variety of geometric structures associated with normal crossings singularities in algebraic geometry.