Absolute and Relative Gromov-witten Invariants of Very Ample Hypersurfaces

For any smooth complex projective variety X and smooth very ample hypersurface Y ⊂ X, we develop the technique of genus zero relative Gromov-Witten invariants of Y in X in algebro-geometric terms. We prove an equality of cycles in the Chow groups of the moduli spaces of relative stable maps that relates these relative invariants to the Gromov-Witten invariants of X and Y . Given the Gromov-Witten invariants of X, we show that these relations are sufficient to compute all relative invariants, as well as all genus zero Gromov-Witten invariants of Y whose homology and cohomology classes are induced by X. Much work has been done recently on Gromov-Witten invariants related to hypersurfaces. There are essentially two different problems that have been studied. The first one is the question: how can one compute the Gromov-Witten invariants of a hypersurface from those of the ambient variety [Be],[G],[K],[LLY]? The second problem, mainly studied from the point of view of symplectic geometry, is the theory of relative Gromov-Witten invariants of a hypersurface [IP1],[IP2],[LR],[R],[V]. The goal of this paper is to show that these two problems that have been studied completely independently so far are in fact very closely related. Let X be a smooth complex projective variety and Y ⊂ X a smooth very ample hypersurface. We start by giving a very short description of our method to compute the genus zero Gromov-Witten invariants of Y in terms of those of X , skipping all technical details. Fix n ≥ 1 and β ∈ H2(X). For m ≥ 0, we let M̄(m) (the official notation will be M̄(m,0,...,0)(X, β)) be a suitable compactification of the moduli space of all irreducible stable maps (P, x1, . . . , xn, f) to X such that f has multiplicity at least m to Y at the point x1. Obviously, M̄(0) should be just the ordinary moduli space of stable maps to X . On the other hand, M̄(Y ·β+1) should correspond to the moduli space of stable maps to Y , as all irreducible curves in X having multiplicity Y · β+1 to Y must actually lie inside Y . Moreover, M̄(m+1) is a subspace of M̄(m) of (expected) codimension one. The strategy is now obvious: if we can describe the (virtual) divisor M̄(m+1) in M̄(m) intersection-theoretically in terms of known classes (and our main theorem 2.6 does precisely that), then we can compute intersection products on M̄(m+1) if we can compute them on M̄(m). Iterating this procedure for m from 0 to Y ·β, this means that we can compute the Gromov-Witten invariants of Y if we can compute the Gromov-Witten invariants of X . In fact, we will show in a forthcoming paper that this method reproves and generalizes the well-known “mirror symmetry” type formulas for Gromov-Witten invariants of certain hypersurfaces [Be],[G],[LLY]. Funded by the DFG scholarship Ga 636/1–1. 1