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On the Genus-One Gromov-Witten Invariants of Complete Intersections

Jun Li·Aleksey Zinger
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Abstract

As shown in a previous paper, certain naturally arising cones of holomorphic vector bundle sections over the main component $\ov\M_{1,k}^0(¶,d)$ of the moduli space of stable genus-one holomorphic maps into $¶$ have a well-defined euler class. In this paper, we extend this result to moduli spaces of perturbed, in a restricted way, $J$-holomorphic maps. We show that euler classes of such cones relate the reduced genus-one Gromov-Witten invariants of complete intersections to the corresponding GW-invariants of the ambient projective space. As a consequence, the standard genus-one GW-invariants of complete intersections can be expressed in terms of the genus-zero and genus-one GW-invariants of projective spaces. We state such a relationship explicitly for complete-intersection threefolds. A relationship for higher-genus invariants is conjectured as well.

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Jun Li, Aleksey Zinger. 2005-07-05. On the Genus-One Gromov-Witten Invariants of Complete Intersections. https://arxiv.org/abs/math/0507104

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