Relative Gromov-witten Invariants and the Mirror Formula

Let X be a smooth complex projective variety, and let Y ⊂ X be a smooth very ample hypersurface such that −KY is nef. Using the technique of relative Gromov-Witten invariants, we give a new short and geometric proof of (a version of) the “mirror formula”, i.e. we show that the generating function of the genus zero 1-point Gromov-Witten invariants of Y can be obtained from that of X by a certain change of variables (the so-called “mirror transformation”). Moreover, we use the same techniques to give a similar expression for the (virtual) numbers of degree-d plane rational curves meeting a smooth cubic at one point with multiplicity 3d, which play a role in local mirror symmetry. For a smooth very ample hypersurface Y of a smooth complex projective variety X , the theory of relative Gromov-Witten invariants gives rise to an algorithm that allows one to compute the genus zero Gromov-Witten invariants of Y from those of X [Ga]. The goal of this paper is to show that in the case when −KY is nef, this algorithm can be “solved” explicitly to obtain a formula that expresses the generating function of the 1-point Gromov-Witten invariants of Y in terms of that ofX . This so-called “mirror formula” (also denoted “quantum Lefschetz hyperplane theorem” by some authors) has already been known for some time ([Gi], [LLY], [K], [B], [L]). Our approach however is entirely different and essentially “elementary” in the sense that it does not use any of the special techniques that have been used in the previous proofs, like e.g. torus actions, equivariant cohomology, or moduli spaces other than the usual spaces of stable maps to X and their subspaces. This does not only make our proof much simpler than the previous ones, but also hopefully easier to generalize, e.g. to more general hypersurfaces, or to higher genus of the curves. Let us briefly recall the ideas and results from [Ga]. For n ≥ 0 and a homology class β ∈ H2(X)/torsion we denote by M̄n(X, β) the moduli space of n-pointed genus zero stable maps to X of class β. For any m ≥ 0 there are closed subspaces M̄(m)(X, β) of M̄1(X, β) that can be thought of as parametrizing 1-pointed rational curves in X having multiplicity (at least) m to Y at the marked point. (For simplicity, we suppress in the notation the dependence of these spaces on Y .) These moduli spaces have expected codimension m in M̄1(X, β). In fact, they come equipped with natural virtual fundamental classes [M̄(m)(X, β)] virt of this expected dimension. If X is a projective space and Y a hyperplane, then these moduli spaces do have the expected dimension, and their virtual fundamental classes are equal to the usual ones. The idea is now to raise the multiplicity m of the curves from 0 up to Y · β + 1 by one at a time. Curves with multiplicity (at least) 0 are just unrestricted curves 1991 Mathematics Subject Classification. 14N35,14N10,14J70. Funded by the DFG scholarships Ga 636/1–1 and Ga 636/1–2. 1