Virtual neighborhood technique for pseudo-holomorphic spheres

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 $PSL(2,\text{C})$-action on spaces of $W^{1,p}$-maps from the Riemann sphere to a symplectic manifold $(X, \omega, J)$withanon-zerohomologyclass$A$. In particular, we establish the slice and tubular neighbourhood theorems for $PSL(2,\text{C})$-action along smooth maps, and construct a $PSL(2,\text{C})$-obstruction bundle along $PSL(2,\text{C})$-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 ${\text{cM}}_{0,0}(X,A)$ 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 ${\text{cM}}_{0,0}(X,A)$ of pseudo-holomorphic spheres in $(X,\omega ,J)$ and show that the genus zero Gromov-Witten invariant is well-defined.