Normal Crossings Singularities for Symplectic Topology: Structures

TL;DR

This paper develops new topological and geometric structures associated with normal crossings symplectic divisors, including blowups and tangent bundles, and applies these to refine Chern class formulas relevant for moduli space analysis.

Contribution

It constructs natural geometric structures for normal crossings symplectic divisors and extends Chern class formulas to more general blowup scenarios.

Findings

Constructed a blowup and associated line bundle for symplectic divisors.

Determined the Chern class of the logarithmic tangent bundle.

Refined Aluffi's Chern class formula for blowups at complete intersections.

Abstract

Our previous papers introduce topological notions of normal crossings symplectic divisor and variety, show that they are equivalent, in a suitable sense, to the corresponding geometric notions, and establish a topological smoothability criterion for normal crossings symplectic varieties. The present paper constructs a blowup, a complex line bundle, and a logarithmic tangent bundle naturally associated with a normal crossings symplectic divisor and determines the Chern class of the last bundle. These structures hav applications in constructions and analysis of various moduli spaces. As a corollary of the Chern class formula for the logarithmic tangent bundle, we refine Aluffi's formula for the Chern class of the tangent bundle of the blowup at a complete intersection to account for the torsion and extend it to the blowup at the deepest stratum of an arbitrary normal crossings divisor.