[Paper Review] Smoothness of Kuranishi atlases on Gromov-Witten moduli spaces

Robert Castellano|arXiv (Cornell University)|Nov 13, 2015

Geometric and Algebraic Topology12 references5 citations

TL;DR

This paper establishes that Gromov-Witten moduli spaces of genus zero $J$-holomorphic curves admit a $C^1$ stratified smooth (C^1 SS) Kuranishi atlas, enabling the construction of a virtual fundamental class for any virtual dimension. By proving a stronger $C^1$ gluing theorem—showing the gluing map is $C^1$ in the gluing parameter—the authors resolve a key analytic obstruction in McDuff and Wehrheim’s Kuranishi atlas framework, extending its applicability beyond virtual dimension zero.

ABSTRACT

Kuranishi atlases were introduced by McDuff and Wehrheim to build a virtual fundamental class on moduli spaces of J-holomorphic curves and resolve some of the challenges in this field. This paper considers Gromov-Witten moduli spaces and shows they admit a smooth enough Kuranishi atlas to be able to define a Gromov-Witten virtual fundamental class in any virtual dimension. The key step for this result is the proof of a stronger gluing theorem.

Motivation & Objective

• To resolve the lack of smoothness in Kuranishi atlases for Gromov-Witten moduli spaces, which previously only allowed virtual fundamental classes in virtual dimension zero.
• To extend the Kuranishi atlas framework to allow construction of virtual fundamental classes in arbitrary virtual dimensions.
• To establish that genus zero Deligne-Mumford space $\overline{\mathcal{M}}_{0,k}$ admits a $C^1$ stratified smooth structure compatible with the atlas.
• To prove a stronger $C^1$ gluing theorem for the gluing map in the context of $J$-holomorphic curve moduli spaces.
• To enable the definition of genus zero Gromov-Witten invariants with homological constraints satisfying the Kontsevich-Manin axioms.

Proposed method

• Introduces a reparametrization of gluing variables using $\widetilde{\lambda} = \lambda |\lambda|^\varepsilon$ to achieve $C^1$ differentiability at $\lambda = 0$, where $\lambda$ is the complex gluing parameter.
• Replaces the standard gluing parameter $s = R^{-2/p}$ with $s^{1+\varepsilon}$ to ensure bounded derivatives and $C^1$ regularity at the origin.
• Applies the implicit function theorem to the gluing map in a Banach space setting, with careful estimates on the derivative and norm bounds in $L^p$ and $W^{1,p}$ spaces.
• Uses cross-ratio coordinates on $\overline{\mathcal{M}}_{0,k}$ to define a $C^1$ smooth structure on the Deligne-Mumford space, compatible with the Kuranishi atlas.
• Bounds the $W^{1,p}$ norm of the derivative of the gluing map by $C_9 \frac{1}{R_0} \frac{1}{(\delta R_0)^{2/p}}$, showing uniform decay as $R_0 \to \infty$, which ensures $C^1$ regularity.
• Constructs a $C^1$ SS Kuranishi atlas on $\overline{\mathcal{M}}_{0,k}(A,J)$ by ensuring all structural maps, including the forgetful map to $\overline{\mathcal{M}}_{0,k}^{\text{new}}$, are $C^1$ stratified smooth.

Experimental results

Research questions

• RQ1 Can a $C^1$ stratified smooth Kuranishi atlas be constructed on the genus zero Gromov-Witten moduli space $\overline{\mathcal{M}}_{0,k}(A,J)$ for arbitrary virtual dimension?
• RQ2 Is the gluing map in the Gromov-Witten setting $C^1$ smooth with respect to the gluing parameter, despite singular behavior at $\lambda = 0$?
• RQ3 Does the Deligne-Mumford space $\overline{\mathcal{M}}_{0,k}$ admit a $C^1$ stratified smooth structure compatible with the Kuranishi atlas?
• RQ4 Can the standard gluing theorem from [MS12] be strengthened to ensure $C^1$ differentiability of the gluing map at the origin in the gluing parameter?
• RQ5 Can the Kuranishi atlas framework be extended to support virtual fundamental classes in non-zero virtual dimensions?

Key findings

• The genus zero Gromov-Witten moduli space $\overline{\mathcal{M}}_{0,k}(A,J)$ admits an oriented, $d$-dimensional, weak $C^1$ SS Kuranishi atlas with $d = 2n + 2c_1(A) + 2k - 6$, valid for any virtual dimension.
• The Deligne-Mumford space $\overline{\mathcal{M}}_{0,k}$ admits a $C^1$ stratified smooth structure $\overline{\mathcal{M}}_{0,k}^{\text{new}}$ compatible with the atlas, ensuring the forgetful map $U \to \overline{\mathcal{M}}_{0,k}^{\text{new}}$ is $C^1$ SS.
• A new $C^1$ gluing theorem is proven: the gluing map is $C^1$ in the gluing parameter $\lambda$, with derivative bounded uniformly as $R_0 \to \infty$, resolving a key analytic obstruction.
• The $W^{1,p}$ norm of the derivative of the gluing map is bounded by $C_9 \frac{1}{R_0} \frac{1}{(\delta R_0)^{2/p}}$, which decays as $R_0 \to \infty$, confirming $C^1$ regularity.
• Reparametrizing the gluing parameter via $\widetilde{\lambda} = \lambda |\lambda|^\varepsilon$ with $\varepsilon > 0$ ensures $C^1$ differentiability at $\lambda = 0$, even though the original map is only smooth for $\lambda \neq 0$.
• The construction enables the definition of genus zero Gromov-Witten invariants with homological constraints satisfying the Kontsevich-Manin axioms, as shown in the follow-up work [Cas].