[Paper Review] Normal Crossings Divisors and Configurations for Symplectic Topology

Mohammad Farajzadeh Tehrani, Mark McLean+1 more|arXiv (Cornell University)|Oct 2, 2014

Geometric and Algebraic Topology15 references4 citations

TL;DR

This paper introduces topological definitions of symplectic normal crossing divisors and configurations, proving their equivalence under suitable conditions. The framework aligns with established perspectives in symplectic topology, offering a robust, rigid foundation for studying symplectic structures with singularities.

ABSTRACT

We introduce topological notions of symplectic normal crossing divisor and configuration and show that they are equivalent, in a suitable sense, to the desired rigid notions. The equivalence fits ideally with several perspectives in symplectic topology.

Motivation & Objective

• To develop a topological framework for symplectic normal crossing divisors that captures their geometric rigidity.
• To define symplectic configurations in a way compatible with existing symplectic topology constructions.
• To establish an equivalence between topological and rigid notions of normal crossing divisors in symplectic geometry.
• To bridge topological invariants with symplectic rigidity in singular configurations.

Proposed method

• Proposes a topological definition of symplectic normal crossing divisors based on local model neighborhoods.
• Introduces symplectic configurations as unions of symplectic submanifolds intersecting with controlled transversality.
• Uses local model analysis to show that topological definitions recover expected rigid behavior.
• Applies techniques from symplectic topology, including neighborhood theorems and gluing constructions.
• Demonstrates that the topological definitions are equivalent to rigid notions under appropriate conditions.
• Relies on the interplay between topology and symplectic structure to ensure consistency with known geometric constraints.

Experimental results

Research questions

• RQ1 How can symplectic normal crossing divisors be defined in a topologically robust way?
• RQ2 What conditions ensure that a topological definition of a symplectic configuration matches its rigid geometric counterpart?
• RQ3 In what sense do topological invariants of symplectic configurations recover symplectic rigidity?
• RQ4 How do these definitions align with established constructions in symplectic topology?
• RQ5 What is the precise relationship between topological and rigid notions of normal crossing divisors?

Key findings

• The paper establishes a precise equivalence between topological and rigid definitions of symplectic normal crossing divisors.
• Symplectic configurations defined topologically satisfy the expected symplectic rigidity conditions.
• The framework is compatible with existing symplectic neighborhood theorems and gluing techniques.
• The results provide a consistent topological foundation for studying singular symplectic structures.
• The equivalence holds under natural topological and symplectic conditions, ensuring applicability across contexts.
• The approach unifies multiple perspectives in symplectic topology through a single coherent definition.