Virtual moduli cycles and Gromov-Witten invariants of noncompact symplectic manifolds

This paper constructs and studies the Gromov-Witten invariants and their properties for noncompact geometrically bounded symplectic manifolds. Two localization formulas for GW-invariants are also proposed and proved. As applications we get solutions of the generalized string equation and dilation equation and their variants. The more solutions of WDVV equation and quantum products on cohomology groups are also obtained for the symplectic manifolds with finitely dimensional cohomology groups. To realize these purposes we further develop the language introduced by Liu-Tian to describe the virtual moduli cycle (defined by Liu-Tian, Fukaya-Ono, Li-Tian, Ruan and Siebert).