Virtual Class of Zero Loci and Mirror Theorems

Let Y be the zero loci of a regular section of a convex vector bundle E over X . We provide a proof of a conjecture of Cox, Katz and Lee for the virtual class of the genus zero moduli of stable maps to Y . This in turn yields the expected relationship between Gromov-Witten theories of Y and X which together with Mirror Theorems allows for the calculation of enumerative invariants of Y inside of X . 0. Introduction Let X be a smooth, projective variety over C. A vector bundle E → X is called convex if H(f (E)) = 0 for any nonconstant morphism f : P → X. Let Y = Z(s) ⊂ X be the zero locus of a regular section s of a convex vector bundle E and let i denote the embedding of Y in X. It is the relationship between the Gromov-Witten theories of Y and X that we study here. 0.1. Virtual class of the zero loci. Let M 0,n(X, d) be the Q-scheme that represents coarsely genus zero, n-pointed stable maps (C, x1, x2, ..., xn, f : C → X) of class d ∈ H2(X,Z). Since E is convex the vector spaces H(f (E)) fit into a Q-vector bundle Ed onM 0,n(X, d). The section s of E induces a section s̃ of Ed over M 0,0(X, d) via s̃((C, f)) = s ◦ f . If i∗(β) = d the map i : Y →֒ X yields an inclusion iβ : M0,0(Y, β) → M0,0(X, d). Clearly Z(s̃) = ∐ i∗(β)=d M 0,0(Y, β). The map i∗ : H2Y → H2X is not injective in general, hence the zero locus Z(s̃) may have more then one connected component. An example is the quadric surface in P. 1991 Mathematics Subject Classification. Primary 14N35. Secondary 14C17. 1 The Q-normal bundle of Z(s̃) in M0,0(X, d) is Ed|Z(s̃). In Section 2.1 of this paper we prove the following theorem [5] Theorem 0.1.1. For any d ∈ H2(X,Z),