Duke Mathematical Journal

Absolute and relative Gromov-Witten invariants of very ample hypersurfaces

Andreas Gathmann

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Abstract

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

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

Dates
First available: 26 May 2004

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1085598142

Mathematical Reviews number (MathSciNet)
MR1944571

Digital Object Identifier
doi:10.1215/S0012-7094-02-11521-X

Zentralblatt MATH identifier
1042.14032

Subjects
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)

Citation

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


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