[Paper Review] Holomorphic cuves in Exploded Torus Fibrations: Regularity

Brett Parker|arXiv (Cornell University)|Jan 31, 2009

Geometric and Algebraic Topology14 references3 citations

TL;DR

This paper establishes regularity results for families of holomorphic curves in exploded torus fibrations—a category extending smooth geometry to include adiabatic limits in tropical geometry. It provides a local model for the moduli space of holomorphic curves, showing that transversality of the ∂̄ equation ensures appropriate regularity, including for bubbling phenomena, and sketches a method for constructing a virtual fundamental class using these results.

ABSTRACT

The category of exploded torus fibrations is an extension of the smooth category related to tropical geometry in which some adiabatic limits appear as smooth families. This paper contains regularity results for families of holomorphic curves in this category. The main result is a local model for the moduli space of holomorphic curves, which in the case of transversality of the ¯ @ equation implies that the moduli space of holomorphic curves has the appropriate regularity. (This includes regularity of families of holomorphic curves in the smooth category which exhibit bubbling behavior.) A sketch of one method for constructing a ‘virtual class’ for the moduli stack of holomorphic curves using these local regularity results is included.

Motivation & Objective

• To extend regularity theory for holomorphic curves to the category of exploded torus fibrations, which incorporates adiabatic limits relevant to tropical geometry.
• To provide a local model for the moduli space of holomorphic curves in this extended category.
• To establish conditions under which the moduli space exhibits desired regularity, particularly when the ∂̄ equation is transverse.
• To lay the foundation for constructing a virtual fundamental class for the moduli stack of holomorphic curves in this setting.

Proposed method

• Utilizes the category of exploded torus fibrations as a geometric framework that unifies smooth families and tropical limits.
• Analyzes families of holomorphic curves via the ∂̄-equation in this category, focusing on local behavior near degenerations.
• Establishes a local model for the moduli space by studying the kernel and cokernel of the ∂̄-operator under transversality assumptions.
• Applies techniques from differential geometry and complex analysis to control the regularity of the moduli space in the presence of bubbling.
• Demonstrates that transversality of the ∂̄-equation implies smoothness of the moduli space in this category.
• Sketches a construction of a virtual class using the local regularity results, enabling enumeration of curves in non-transverse settings.

Experimental results

Research questions

• RQ1 Under what conditions does the moduli space of holomorphic curves in exploded torus fibrations exhibit regularity?
• RQ2 How can a local model for the moduli space of holomorphic curves be constructed in this extended geometric category?
• RQ3 In what way does transversality of the ∂̄-equation ensure regularity of the moduli space, even in the presence of bubbling?
• RQ4 Can the local regularity results be leveraged to define a virtual fundamental class for the moduli stack of holomorphic curves?

Key findings

• The moduli space of holomorphic curves in exploded torus fibrations admits a local model that captures its regularity structure under transversality of the ∂̄-equation.
• Transversality of the ∂̄-equation ensures that the moduli space is smooth and of the expected dimension, even when bubbling occurs.
• The framework includes families of holomorphic curves in the classical smooth category that exhibit bubbling behavior, extending regularity results to such cases.
• The local regularity results provide a foundation for constructing a virtual fundamental class for the moduli stack of holomorphic curves.
• The method outlined offers a viable path toward virtual cycle constructions in enumerative geometry using exploded manifolds.
• The results unify aspects of tropical geometry and complex geometry by embedding adiabatic limits into a smooth family framework.