Singularities and Semistable Degenerations for Symplectic Topology

TL;DR

This paper explores the introduction and smoothing of normal crossings singularities in symplectic topology, answering Gromov's question and establishing conditions for their smoothing, with connections to broader mathematical areas.

Contribution

It provides a framework for defining and smoothing normal crossings symplectic varieties, advancing understanding of singularities in symplectic topology.

Findings

Established necessary and sufficient conditions for smoothing normal crossings symplectic varieties

Answered Gromov's question on singular subvarieties in symplectic topology

Connected symplectic singularities with other mathematical fields

Abstract

We overview our recent work defining and studying normal crossings varieties and subvarieties in symplectic topology. This work answers a question of Gromov on the feasibility of introducing singular (sub)varieties into symplectic topology in the case of normal crossings singularities. It also provides a necessary and sufficient condition for smoothing normal crossings symplectic varieties. In addition, we explain some connections with other areas of mathematics and discuss a few directions for further research.