[Paper Review] Holomorphic curves in Exploded Torus Fibrations: Compactness

Brett Parker|ArXiv.org|Jun 26, 2007

Geometric and Algebraic Topology8 references3 citations

TL;DR

This paper establishes a compactness theorem for pseudoholomorphic curves in exploded torus fibrations—a generalized category of geometric objects that unify smooth manifolds and tropical geometry. By extending Gromov’s compactness to this framework, the author provides a foundational tool for defining Gromov–Witten invariants in degenerate or tropicalized settings, particularly in adiabatic limits common in mirror symmetry and symplectic topology.

ABSTRACT

The category of exploded torus fibrations is an extension of the category of smooth manifolds in which some adiabatic limits look smooth. (For example, the limits considered in tropical geometry appear smooth, also degenerations corresponding to an algebraic family with normal crossing singularities are smooth.) In this paper we prove a compactness theorem for (pseudo)-holomorphic curves in exploded torus fibrations. In the case of smooth manifolds, this is just a version of Gromov's compactness theorem in a topology strong enough for gluing analysis.

Motivation & Objective

• To establish a compactness theorem for pseudoholomorphic curves in the category of exploded torus fibrations, extending Gromov’s original result.
• To address the limitations of the smooth category in handling moduli spaces with bubbling and topology change.
• To provide a geometric framework where degenerations (e.g., symplectic sums, normal crossing divisors) appear as smooth families.
• To support the construction of Kuranishi-type structures for defining holomorphic curve invariants in non-smooth or tropicalized settings.
• To unify the study of holomorphic curves and tropical curves via a single geometric category that captures both smooth and combinatorial structures.

Proposed method

• The paper introduces exploded torus fibrations as an extension of smooth manifolds, incorporating a tropical (or integral affine) stratification to model degenerations.
• It defines a $C^{k, u}$-regularity condition on maps and almost complex structures to ensure sufficient differentiability for analysis.
• The construction uses refined local charts and gluing maps involving vector fields $\phi^{\pm}$ to define families of curves across different components.
• A key technical tool is the $G_{(\phi^+ among the $\phi^{i\pm}_n$ converge in $C^{\infty,\delta}$ to $\phi^{\pm}_n$, ensuring $C^{\infty,\delta}$ convergence of the maps $\hat{f}^i$ to $\hat{f}$.
• The proof relies on estimates from Lemma 3.13 showing $|\phi^{i\pm}_n(z^\pm)| \leq c|z^\pm|^\delta$ for $\delta < 1$, which controls Hölder-type regularity.
• The convergence of the maps $\hat{f}^i$ to $\hat{f}$ in $C^{\infty,\delta}$ is established by combining convergence of the vector fields and the exponential map $e^{G_{(\phi^{i+},\phi^{i-})}}$.

Experimental results

Research questions

• RQ1 Can a compactness theorem for pseudoholomorphic curves be established in the category of exploded torus fibrations, analogous to Gromov’s theorem in the smooth category?
• RQ2 How can degenerations of symplectic manifolds—such as symplectic sums or normal crossing divisors—be treated as smooth families in a generalized geometric category?
• RQ3 To what extent do tropical curves in the stratified tropical part $\lfloor\mathfrak{B}\rfloor$ capture information about holomorphic curve invariants in the total space?
• RQ4 Can the moduli space of holomorphic curves in exploded fibrations admit a Kuranishi structure, given the compactness result?
• RQ5 How does the $C^{\infty,\delta}$-topology on maps ensure convergence of families of curves under degeneration?

Key findings

• The paper proves that a sequence of pseudoholomorphic curves in an exploded torus fibration converges in the $C^{\infty,\delta}$ topology to a limit curve, establishing compactness in a strong topology suitable for gluing analysis.
• The convergence of the maps $\hat{f}^i$ to $\hat{f}$ is established via uniform $C^{\infty,\delta}$ bounds on the vector fields $\phi^{i\pm}_n$, which decay as $|z^\pm|^\delta$ for $\delta < 1$.
• The gluing construction using the exponential of the vector field flow $G_{(\phi^{i+},\phi^{i-})}$ ensures smoothness and convergence of the family of maps across different components.
• The maps $F^i_n$ and $F_n$ converge in $C^\infty$, and the exponential factor $e^{G_{(\phi^{i+},\phi^{i-})}}$ converges in $C^{\infty,\delta}$, leading to the overall $C^{\infty,\delta}$ convergence of $\hat{f}^i$ to $\hat{f}$.
• The result confirms that exploded torus fibrations support a well-behaved moduli theory for pseudoholomorphic curves, with compactness in a topology strong enough for analysis and gluing.
• The framework allows degenerations such as symplectic sums and normal crossing central fibers to be treated as smooth families, with holomorphic curves in the total space projecting to both smooth and tropical components.