A degeneration formula of Gromov-Witten invariants with respect to a curve class for degenerations from blow-ups

Chien-Hao Liu Shing-Tung Yau

Algebraic Geometry mathscidoc:1912.43652

arXiv preprint math/0408147, 2004.8
In two very detailed, technical, and fundamental works, Jun Li constructed a theory of Gromov-Witten invariants for a singular scheme of the gluing form Y_1\cup_D Y_2 that arises from a degeneration Y_1\cup_D Y_2 and a theory of relative Gromov-Witten invariants for a codimension-1 relative pair Y_1\cup_D Y_2 . As a summit, he derived a degeneration formula that relates a finite summation of the usual Gromov-Witten invariants of a general smooth fiber Y_1\cup_D Y_2 of Y_1\cup_D Y_2 to the Gromov-Witten invariants of the singular fiber Y_1\cup_D Y_2 via gluing the relative pairs Y_1\cup_D Y_2 and Y_1\cup_D Y_2 . The finite sum mentioned above depends on a relative ample line bundle Y_1\cup_D Y_2 on Y_1\cup_D Y_2 . His theory has already applications to string theory and mathematics alike. For other new applications of Jun Li's theory, one needs a refined degeneration formula that depends on a curve class Y_1\cup_D Y_2 in Y_1\cup_D Y_2 or Y_1\cup_D Y_2 , rather than on the line bundle Y_1\cup_D Y_2 . Some monodromy effect has to be taken care of to deal with this. For the simple but useful case of a degeneration Y_1\cup_D Y_2 that arises from blowing up a trivial family Y_1\cup_D Y_2 , we explain how the details of Jun Li's work can be employed to reach such a desired degeneration formula. The related set Y_1\cup_D Y_2 of admissible triples adapted to Y_1\cup_D Y_2 that appears in the formula can be obtained via an analysis on the intersection numbers of relevant cycles and a study of Mori cones that appear in the problem. This set is intrinsically determined by Y_1\cup_D Y_2 and the normal bundle Y_1\cup_D Y_2 of the smooth subscheme Y_1\cup_D Y_2 in Y_1\cup_D Y_2 to be blown up.
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@inproceedings{chien-hao2004a,
  title={A degeneration formula of Gromov-Witten invariants with respect to a curve class for degenerations from blow-ups},
  author={Chien-Hao Liu, and Shing-Tung Yau},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224204744589094216},
  booktitle={arXiv preprint math/0408147},
  year={2004},
}
Chien-Hao Liu, and Shing-Tung Yau. A degeneration formula of Gromov-Witten invariants with respect to a curve class for degenerations from blow-ups. 2004. In arXiv preprint math/0408147. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224204744589094216.
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