Relative Gromov-Witten invariants
type: Item
instance of: scholarly article
title: Relative Gromov-Witten invariants
zbMATH Open document ID: 1039.53101
author: Eleny-Nicoleta Ionel
author: Thomas H. Parker
published in: Annals of Mathematics. Second Series
publication date: 2003-01-01T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/9907155
review text: The authors define Gromov-Witten invariants for a symplectic manifold \((X,\omega)\) relative to a codimension two symplectic submanifold \(V\). These relative invariants are designed for use in formulas describing how GW invariants behave under symplectic connected sums along \(V\) [the authors, Math. Res. Lett. 5, 563--576 (1998; Zbl 0943.53046)]. In very rough terms, the GW invariants of \(X\) count the number of stable curves in \(X\); so their relative version has to keep track of how these curves intersect \(V\). A stable map \(f\colon C\to X\) is called \(V\)-regular if no component of \(C\) is mapped entirely into \(V\). Any such map intersects \(V\) in a finite number of points, with multiplicity. The authors prove that the space \({\mathcal M}^V_{g,n,\vec{s}}(X,A)\) of \(V\)-regular stable maps from a genus \(g\) \(n\)-pointed curve to \(X\) with \(f_*[C]=A\) which intersect \(V\) in \(\ell\) points, with multiplicities \(\vec{s}= (s_1, \dots,s_\ell)\), is an orbifold. Moreover, since the interest is in symplectic connected sums, the relative invariants have to record the homology class of a curve in \(X\setminus V\). This leads to the construction of a space \({\mathcal H}^V_X\), which is a covering of \(H_2(X)\times V^\ell\). The space of \(V\)-regular maps has an orbifold compactification equipped with a natural map \(\overline{\mathcal M}^V_{g,n,\vec{s}} (X,A)\to \overline{\mathcal M}_{g,n+\ell} \times X^n\times {\mathcal H}^V_X\). The relative GW invariant \(GW_X^V\) is then defined as the homology class of the image of this map; it depends only on the symplectic deformation class of \((X,V,\omega)\). By evaluating this homology class on dual cohomology classes, one expresses \(GW_X^V\) as a collection of numbers, which count the number of stable maps with prescribed intersection with \(V\). It should be remarked that the entire construction of relative GW invariants carries through when \(V\) is the empty set: in this case one finds the absolute GW i…
reviewed by: Domenico Fiorenza
Mathematics Subject Classification ID: 53D45
Mathematics Subject Classification ID: 57R17
Mathematics Subject Classification ID: 57R57
zbMATH DE Number: 1933155
zbMATH Keywords: Gromov-Witten invariants
zbMATH Keywords: symplectic manifolds
MaRDI profile type: Publication
OpenAlex ID: W2154417895
arXiv ID: math/9907155
DOI: 10.4007/ANNALS.2003.157.45
description: scientific article