Algebraic reduced genus one Gromov-Witten invariants for complete intersections in projective spaces

Abstract

A. Zinger defined reduced Gromov-Witten (GW) invariants and proved a comparison theorem of standard and reduced genus one GW invariants for every symplectic manifold (with all dimension). H. -L. Chang and J. Li provided a proof of the comparison theorem for quintic Calabi-Yau 3-folds in algebraic geometry by taking a definition of reduced invariants as an Euler number of certain vector bundle. T. Coates and C. Manolache have defined reduced GW invariants in algebraic geometry following the idea by Vakil and Zinger and proved the comparison theorem for every Calabi-Yau threefold. In this paper, we prove the comparison theorem for every (not necessarily Calabi-Yau) complete intersection of dimension 2 or 3 in projective spaces by taking a definition of reduced GW invariants in the paper of Coates and Manolache.