Duke Mathematical Journal

Absolute and relative Gromov-Witten invariants of very ample hypersurfaces

Andreas Gathmann

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For any smooth complex projective variety $X$ and any smooth very ample hypersurface $Y\subset X$, we develop the technique of genus zero relative Gromov-Witten invariants of $Y$ in $X$ in algebro-geometric terms. We prove an equality of cycles in the Chow groups of the moduli spaces of relative stable maps which relates these relative invariants to the Gromov-Witten invariants of $X$ and $Y$. Given the Gromov-Witten invariants of $X$, we show that these relations are sufficient to compute all relative invariants, as well as all genus zero Gromov-Witten invariants of $Y$ whose homology and cohomology classes are induced by $X$.

Article information

Duke Math. J., Volume 115, Number 2 (2002), 171-203.

First available in Project Euclid: 26 May 2004

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Zentralblatt MATH identifier

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]
Secondary: 14H10: Families, moduli (algebraic) 14J70: Hypersurfaces 14N10: Enumerative problems (combinatorial problems)


Gathmann, Andreas. Absolute and relative Gromov-Witten invariants of very ample hypersurfaces. Duke Math. J. 115 (2002), no. 2, 171--203. doi:10.1215/S0012-7094-02-11521-X. https://projecteuclid.org/euclid.dmj/1085598142

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