Algebraic Reduced Genus One Gromov-Witten Invariants for Complete Intersections in Projective Spaces

In [17, 18], 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). In [3], Chang and Li provided a proof of the comparison theorem for quintic Calabi-Yau three-folds in algebraic geometry by taking a definition of reduced invariants as an Euler number of certain vector bundle. In [5], Coates and Manolache have defined reduced GW invariants in algebraic geometry following the idea by Vakil and Zinger in [14] 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 [5].