On the Structure of Certain Natural Cones over Moduli Spaces of Genus-One Holomorphic Maps

Details On the Structure of Certain Natural Cones over Moduli Spaces of Genus-One Holomorphic Maps On the Structure of Certain Natural Cones over Moduli Spaces of Genus-One Holomorphic Maps

2004-06-06

arxiv.org/abs/math/0406104v2

Mathematics

Symplectic Geometry

an error corrected; 45 pages, 3 figures

Scientific paper

We show that certain naturally arising cones over the main component of a moduli space of $J_0$-holomorphic maps into $P^n$ have a well-defined euler class. We also prove that this is the case if the standard complex structure $J_0$ on $P^n$ is replaced by a nearby almost complex structure $J$. The genus-zero analogue of the cone considered in this paper is always a vector bundle. The genus-zero Gromov-Witten invariant of a projective hypersurface is the euler class of such a vector bundle. As shown in a separate paper, this is also the case for the "genus-one part" of the genus-one GW-invariant. The remaining part is a multiple of the genus-zero GW-invariant.

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