[Paper Review] Lie groupoid, deformation of unstable curve, and construction of equivariant Kuranishi charts

Kenji Fukaya|arXiv (Cornell University)|Jan 11, 2017

Geometric and Algebraic Topology30 references5 citations

TL;DR

This paper constructs $G$-equivariant Kuranishi charts for the moduli space of $J$-holomorphic curves in a symplectic manifold with a compact Lie group $G$-action, using deformation theory of unstable marked curves and Lie groupoid techniques. The key contribution is a systematic, transparent construction of $G$-equivariant local models that generalize standard Kuranishi structures to the equivariant setting, enabling the definition of equivariant Gromov-Witten invariants.

ABSTRACT

In this paper we give detailed construction of $G$-equivariant Kuranishi chart of moduli spaces of pseudo-holomorphic curves to a symplectic manifold with $G$-action, for an arbitrary compact Lie group $G$. The proof is based on the deformation theory of {\it unstable} marked curves using the language of Lie groupoid (which is {\it not} necessary etale) and the Riemannnian center of mass technique. This proof is actually similar to [FOn,Sections 13 and 15] except the usage of the language of Lie groupoid makes the argument more transparent. This version correct some errors of the previous version especially those pointed out by the referee.

Motivation & Objective

• To provide a rigorous, transparent construction of $G$-equivariant Kuranishi charts for the moduli space of $J$-holomorphic curves in a $G$-symplectic manifold.
• To overcome the limitations of ad-hoc constructions in prior works by using the language of Lie groupoids and Riemannian center of mass techniques.
• To establish a foundation for defining equivariant virtual fundamental classes and hence equivariant Gromov-Witten invariants in the symplectic setting.
• To generalize the Kuranishi structure framework to the equivariant case in a way that is compatible with gluing and global structure construction.

Proposed method

• Utilizes the language of Lie groupoids (not necessarily étale) to parametrize families of complex structures on unstable marked curves.
• Applies deformation theory of unstable curves to construct universal deformations, ensuring uniqueness and existence via groupoid-theoretic methods.
• Employs the Riemannian center of mass technique to construct $G$-equivariant sections of obstruction bundles in a controlled, $G$-invariant manner.
• Introduces $ε$-closeness conditions to control the geometry of the deformation space and ensure transversality.
• Uses exponential decay estimates for the obstruction bundle to ensure convergence and smoothness of the local model.
• Establishes independence of the local smooth structure from auxiliary choices via a detailed analysis of stabilization and trivialization data.

Experimental results

Research questions

• RQ1 How can one systematically construct $G$-equivariant Kuranishi charts for the moduli space of $J$-holomorphic curves when $G$ acts on the target symplectic manifold?
• RQ2 What role does the Lie groupoid formalism play in simplifying and clarifying the equivariant deformation theory of unstable curves?
• RQ3 How can the Riemannian center of mass technique be adapted to ensure $G$-equivariance in the construction of local models?
• RQ4 In what sense is the constructed $G$-equivariant Kuranishi chart canonical or independent of auxiliary choices?
• RQ5 How does the construction generalize prior ad-hoc methods used in toric or abelian group actions?

Key findings

• The paper proves the existence of a $G$-equivariant Kuranishi chart $(V_{\mathfrak{p}}, \mathcal{E}_{\mathfrak{p}}, s_{\mathfrak{p}}, \psi_{\mathfrak{p}})$ at each point $\mathfrak{p}$ in the moduli space $\mathcal{M}_{g,\ell}((X,\omega),\alpha)$, satisfying all required $G$-equivariance and smoothness conditions.
• The construction is based on a universal deformation of unstable marked curves parametrized by a Lie groupoid, ensuring uniqueness and compatibility with group actions.
• The use of Lie groupoids (non-étale) and the Riemannian center of mass technique renders the argument more transparent and systematic than previous ad-hoc approaches.
• Exponential decay estimates for the obstruction bundle are established, ensuring the convergence and smoothness of the local model.
• The local smooth structure is shown to be independent of auxiliary choices such as trivialization and stabilization data, via detailed analysis in Section 7.5.
• The method provides a foundational step toward constructing a global $G$-equivariant Kuranishi structure, with the core local construction being the novel and essential component.