Virtual neighborhood technique for pseudo-holomorphic spheres
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, )-action on spaces of W1, p-maps from the Riemann sphere to a symplectic manifold (X, ω, J) with a non-zero homology class A. In particular, we establish the slice and tubular neighbourhood theorems for PSL(2, )-action along smooth maps, and construct a PSL(2, )-obstruction bundle along PSL(2, )-orbit of a pseudo-holomorphic map representing a point in the moduli space 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 0, 0(X, A) of pseudo-holomorphic spheres in (X, ω, J) is compact, applying the virtual neighborhood technique developed in Section 3, we obtain a virtual system for the moduli space 0, 0(X, A) of pseudo-holomorphic spheres in (X, ω, J) and show that the genus zero Gromov-Witten invariant is well-defined.