Virtual neighborhood technique for pseudo-holomorphic spheres

TL;DR

This paper develops a virtual neighborhood technique for pseudo-holomorphic spheres, addressing analytic challenges in Gromov-Witten theory, and establishes a framework for defining genus zero Gromov-Witten invariants.

Contribution

It introduces a novel virtual neighborhood method for moduli spaces of pseudo-holomorphic spheres, resolving differentiability issues of the $PSL(2, ext{C})$-action.

Findings

Established slice and tubular neighborhood theorems for $PSL(2, ext{C})$-action.

Constructed a $PSL(2, ext{C})$-obstruction bundle.

Defined a virtual system for the moduli space and proved the well-definedness of Gromov-Witten invariants.

Abstract

This is the first part of a trilogy where we apply the theory of virtual manifold/orbifolds developed by the first named author and Tian to study the Gromov-Witten moduli spaces. In this paper, we resolve the main analytic issue arising from the lack of differentiability of $PSL(2, \C)$-action on spaces of $W^{1, p}$-maps from the Riemann sphere to a symplectic manifold $(X, \omega, J)$ with a non-zero homology class $A$. In particular, we establish the slice and tubular neighbourhood theorems for $PSL(2, \C)$-action along smooth maps, and construct a $PSL(2, \C)$-obstruction bundle along $PSL(2, \C)$-orbit of a pseudo-holomorphic map representing a point in the moduli space $\cM_{0, 0}(X, A)$. In Sections 2 and 3 of this paper, we explain an integration theory on virtual orbifolds using proper \'etale groupoids and establish the virtual neighborhood technique for a general orbifold Fredholm system. When the moduli space $\cM_{0, 0}(X, A)$ of pseudo-holomorphic spheres in $(X, \omega, J)$ is compact, applying the virtual neighborhood technique developed in Section 3, we obtain a virtual system for the moduli space $\cM_{0, 0}(X, A)$ of pseudo-holomorphic spheres in $(X, \omega, J)$ and show that the genus zero Gromov-Witten invariant is well-defined.