[Paper Review] Exploded Manifolds

Brett Parker|arXiv (Cornell University)|Oct 21, 2009

Polynomial and algebraic computation14 references4 citations

TL;DR

This paper introduces exploded manifolds, an extension of smooth manifolds that supports a robust holomorphic curve theory and enables computation of Gromov-Witten invariants relative to normal crossing divisors via tropical curve counts. The key contribution is a framework where the tropical part of an exploded manifold—a union of convex polytopes glued along faces—encodes geometric and enumerative information for invariants.

ABSTRACT

This paper provides an introduction to exploded manifolds. The category of exploded manifolds is an extension of the category of smooth manifolds with an excellent holomorphic curve theory. Each exploded manifold has a tropical part which is a union of convex polytopes glued along faces. Exploded manifolds are useful for defining and computing Gromov-Witten invariants relative to normal crossing divisors, and using tropical curve counts to compute Gromov-Witten invariants.

Motivation & Objective

• To extend the category of smooth manifolds to include a richer structure that supports holomorphic curve theory in degenerate or singular settings.
• To provide a geometric framework for defining and computing Gromov-Witten invariants relative to normal crossing divisors.
• To establish a connection between tropical curve counts and Gromov-Witten invariants through the tropical part of exploded manifolds.
• To formalize a category of exploded manifolds where the tropical part is a union of convex polytopes glued along faces, enabling combinatorial and geometric analysis.

Proposed method

• Introduce exploded manifolds as a category extending smooth manifolds, equipped with a sheaf of exploded structures that generalize smooth functions.
• Define the tropical part of an exploded manifold as a union of convex polytopes glued along faces, capturing the combinatorial data of degenerations.
• Utilize the tropical part to model and classify holomorphic curves in exploded manifolds, particularly those asymptotic to normal crossing divisors.
• Apply tropical curve counting techniques to compute Gromov-Witten invariants, leveraging the polyhedral structure of the tropical part.
• Establish a correspondence between holomorphic curves in exploded manifolds and combinatorial tropical curves in the tropical part.
• Use the exploded manifold framework to define relative Gromov-Witten invariants in settings where standard smooth manifold techniques fail.

Experimental results

Research questions

• RQ1 How can Gromov-Witten invariants be defined and computed in the presence of normal crossing divisors using a generalized geometric category?
• RQ2 What is the role of the tropical part of an exploded manifold in encoding enumerative invariants of holomorphic curves?
• RQ3 In what way does the polyhedral structure of the tropical part facilitate the counting of holomorphic curves?
• RQ4 How does the exploded manifold framework extend the classical holomorphic curve theory beyond smooth manifolds?
• RQ5 Can tropical curve counts in the tropical part of an exploded manifold recover or compute Gromov-Witten invariants?

Key findings

• Exploded manifolds provide a category that extends smooth manifolds and supports a well-behaved holomorphic curve theory in degenerate or singular settings.
• The tropical part of an exploded manifold is a union of convex polytopes glued along faces, offering a combinatorial model for curve counting.
• Gromov-Witten invariants relative to normal crossing divisors can be defined and computed using the exploded manifold framework.
• Tropical curve counts in the tropical part of an exploded manifold yield precise computations of Gromov-Witten invariants.
• The framework establishes a direct correspondence between holomorphic curves in exploded manifolds and tropical curves in their tropical parts.
• The structure of exploded manifolds allows for a systematic and geometric approach to relative invariants that is not available in the classical smooth category.