Relative Gromov-Witten invariants and the mirror formula
type: Item
instance of: scholarly article
title: Relative Gromov-Witten invariants and the mirror formula
zbMATH Open document ID: 1043.14016
author: Andreas Gathmann
published in: Mathematische Annalen
publication date: 2003-06-02T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0009190
review text: Let \(X\) be a smooth complex projective variety and let \(Y\subset X\) be a smooth very ample hypersurface such that \(-K_Y\) is nef. \textit{A. Gathmann} [Duke Math. J. 115, 171--203 (2002; Zbl 1042.14032)] defined relative Gromov-Witten invariants by considering the moduli spaces of relative stable maps \(\overline{M}_{(\alpha_1,\ldots,\alpha_n)}^Y(X,\beta)\) consisting of all stable genus zero curves \(f:C\rightarrow X\) representing \(\beta\in H_2(X)\) such that \(f\) cuts \(Y\) at the point \(x_i\) with multiplicity \(\alpha_i\). In the case of genus zero \(1\)-point Gromov-Witten relative invariants, the main formula obtained by A. Gathmann [op. cit.] is the following \[ (m \psi + ev^* [Y]) \cdot [\overline{M}_{(m)}^Y (X, \beta)]^{\text{virt}}= [\overline{M}_{(m+1)}^Y (X,\beta) ]^{\text{virt}} + [D_{(m)}(X,\beta)]^{\text{virt}}, \] where \(D_{(m)}(X, \beta)\) parametrizes reducible stable curves with the component containing the point \(x\) included in \(Y\); and \(\psi\) is the cotangent class (the Euler class of the bundle which puts over each stable \((C,f)\) the cotangent space of \(C\) at \(x\)). Now this formula is used recursively starting with the \(1\)-point Gromov-Witten invariants of \(X\) (which are associated to \(\overline{M}_{(0)}^Y(X,\beta)\)) and ending up with the \(1\)-point Gromov-Witten invariants of \(Y\) (which appear for \(\overline{M}_{(Y\cdot \beta +1)}^Y(X,\beta)= \overline{M}(Y,\beta)\)). The nef condition on \(-K_Y\) guarantees that there are few contributions from the correction terms \([D_{(m)}(X,\beta)]^{\text{virt}}\) to the Gromov-Witten invariants. This is used to give a new short and geometric proof of the ``mirror formula'' given by \textit{A. Givental} [Int. Math. Res. Not. 1996, 613--663 (1996; Zbl 0881.55006)] and \textit{B. Lian, K. Liu}, and \textit{S. Yau} [Asian J. Math. 1, 729--763 (1997; Zbl 0953.14026)]. Such a formula gives the generating function of the genus zero \(1\)-point Gromov-Witten invariants of \(Y\) o…
reviewed by: Vicente Muñoz
Mathematics Subject Classification ID: 14N35
Mathematics Subject Classification ID: 14N10
Mathematics Subject Classification ID: 14J70
Mathematics Subject Classification ID: 14J32
zbMATH DE Number: 1922390
zbMATH Keywords: Gromov-Witten invariants
zbMATH Keywords: mirror symmetry
zbMATH Keywords: hypersurface
MaRDI profile type: Publication
OpenAlex ID: W2014053041
arXiv ID: math/0009190
DOI: 10.1007/S00208-002-0345-1
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description: scientific article