[Paper Review] A natural Gromov-Witten virtual fundamental class

Eleny-Nicoleta Ionel, Thomas H. Parker|arXiv (Cornell University)|Feb 14, 2013

Geometric and Algebraic Topology32 references20 citations

TL;DR

This paper proposes a natural, functorial construction of the Gromov-Witten virtual fundamental class (VFC) for symplectic manifolds using Ruan-Tian perturbations, stabilizing divisors, and rational Čech homology. It establishes the existence and uniqueness of a VFC satisfying naturality conditions in high dimensions (dim X ≥ 12) or low genus (g ≥ 1), bypassing the need for gluing theorems by leveraging dimension counts and functoriality across families of almost complex structures and decorated moduli spaces.

ABSTRACT

We describe a program for proving that the Gromov-Witten moduli spaces of compact symplectic manifolds carry a unique virtual fundamental class that satisfies certain naturality conditions. The virtual fundamental class is constructed using only Ruan-Tian perturbations by introducing stabilizing divisors, using Cech homology, and systematically applying naturality conditions. In high dimensions or low genus, no gluing theorems are needed.

Motivation & Objective

• To establish a canonical, unique virtual fundamental class (VFC) for Gromov-Witten moduli spaces of compact symplectic manifolds that satisfies natural functoriality conditions.
• To provide a conceptual and computationally tractable alternative to existing approaches like Kuranishi structures, polyfolds, and implicit atlases.
• To eliminate the need for gluing theorems in high-dimensional or low-genus settings by relying on dimension counts and naturality.
• To unify various Gromov-Witten theories—such as those with twisted covers or relative maps to normal crossing divisors—under a single framework of parameterized families.
• To demonstrate that the VFC can be constructed intrinsically using only Ruan-Tian perturbations, Čech homology, and stabilizing divisors, without requiring auxiliary geometric structures.

Proposed method

• Construct the virtual fundamental class using Ruan-Tian perturbations as the foundational analytic tool.
• Introduce stabilizing divisors to control the behavior of pseudoholomorphic maps near degenerations and ensure compactness.
• Apply rational Čech homology to define the VFC in a way that respects functoriality and naturality across families of almost complex structures.
• Use dimension counts in high dimensions (dim X ≥ 12) or low genus (g ≥ 1) to avoid the need for gluing theorems.
• Systematically extend the VFC from a base set of 'super-fine and regular' almost complex structures to larger parameter spaces via functoriality.
• Leverage the stabilization-evaluation map to define invariants via pushforward, ensuring consistency across different decorated moduli spaces (e.g., with G-torsors or relative divisors).

Experimental results

Research questions

• RQ1 Can a unique, natural virtual fundamental class be defined for Gromov-Witten moduli spaces of compact symplectic manifolds using only Ruan-Tian perturbations and functorial principles?
• RQ2 In what settings can gluing theorems be avoided in constructing the VFC, and how do dimension and genus affect this?
• RQ3 How can the VFC be consistently extended across families of almost complex structures and decorated domain/target geometries (e.g., G-twisted, relative maps)?
• RQ4 What conditions ensure that a normal crossing divisor is compatible with an almost complex structure, and how can such structures be perturbed to preserve holomorphicity?
• RQ5 To what extent can the VFC be characterized by naturality and functoriality, and does this lead to uniqueness?

Key findings

• The virtual fundamental class exists and is unique in rational Čech homology for all fibers of the family of stable maps over a parameter space of almost complex structures.
• In high dimensions (dim X ≥ 12) or low genus (g ≥ 1), the VFC can be constructed without relying on gluing theorems due to favorable dimension counts.
• A natural VFC is constructed using only Ruan-Tian perturbations, stabilizing divisors, and Čech homology, avoiding the need for Kuranishi structures or polyfolds.
• For any α-general and ε-holomorphic normal crossing divisor V, there exists a V-compatible almost complex structure JV with |J - JV| ≤ Cαε, ensuring holomorphicity and compatibility.
• The VFC is functorial: inclusion maps between parameter spaces and forgetful maps for G-torsors induce compatible maps on the VFCs, preserving the structure across different GW theories.
• The construction extends to relative Gromov-Witten theories with decorated targets (e.g., (X,V) with normal crossing divisors), yielding a refined stabilization-evaluation map and consistent invariants.