Superheavy Skeleta for non-Normal Crossings Divisors

TL;DR

This paper constructs superheavy Lagrangian skeleta in the complements of certain divisors in projective varieties, extending beyond normal crossings, and uses these to analyze Reeb dynamics and Lagrangian Floer theory.

Contribution

It introduces a method to build superheavy skeleta for non-normal crossing divisors, broadening the scope of symplectic topology techniques.

Findings

Constructed smoothing families of contact hypersurfaces with controlled Reeb dynamics.

Identified superheavy subsets in divisor complements.

Provided examples where Lagrangian skeleta are superheavy, bypassing Floer theory complexities.

Abstract

Given an anticanonical divisor in a projective variety, one naturally obtains a monotone K\"ahler manifold. In this paper, for divisors in a certain class (larger than normal crossings), we construct smoothing families of contact hypersurfaces with controlled Reeb dynamics. We use these to obtain subsets of the divisor complement which are superheavy. In particular, we will show that several examples of Lagrangian skeleta of such divisor complements are superheavy, in cases where applying Lagrangian Floer theory may be intractable.