Rational Gromov-witten Invariants of Higher Codimensional Subvarieties

Abstract

Let X be a smooth projective variety. It was shown by A. Gathmann that in case X is a very ample hypersurface of some other smooth projective variety Y , the genus-0 (so-called “restricted” and “unrestricted” with a sufficiently low number of marked points) Gromov-Witten invariants of X can be computed in terms of genus-0 Gromov-Witten invariants of Y. The purpose of this article is to generalize this result. More precisely, we will try to answer the following questions: “When can we compute the rational invariants of X from those of the projective space that contains X?” or, if this is not possible, “When can we compute the invariants of X from those of a bigger variety Z that contains X?”. In the first section we prove our main theorem that allows the computation of Gromov-Witten invariants of nef hypersurfaces. We will then try to compute the invariants of an s-codimensional subvariety X of a given variety Z, in case we can find a sequence of varieties Y1, . . . , Ys−1, such that X ↪→ Y1 ↪→ Y2 · · · ↪→ Ys := Z and each two consequent varieties in the above row respect the hypothesis of our main theorem. In particular, we develop an algorithm for the computation of the Gromov-Witten invariants of complete intersections. One example is the number of lines and conics on a degree-9 three-fold in P that is a complete intersection of two cubic hypersurfaces, numbers that were first predicted by A. Libgober and J. Teitelbaum ([12]) in 1993.

Cite this paper

@inproceedings{Manolache2008RationalGI, title={Rational Gromov-witten Invariants of Higher Codimensional Subvarieties}, author={Cristina Manolache}, year={2008} }