  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.
