Abstract.
Let X be a smooth complex projective variety, and let \( Y \subset X \) be a smooth very ample hypersurface such that \( -K_Y \) 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.
Similar content being viewed by others
Author information
Authors and Affiliations
Additional information
Funded by the DFG scholarships Ga 636/1–1 and Ga 636/1–2.
Rights and permissions
About this article
Cite this article
Gathmann, A. Relative Gromov-Witten invariants and the mirror formula. Math. Ann. 325, 393–412 (2003). https://doi.org/10.1007/s00208-002-0345-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-002-0345-1