Relative Gromov-Witten invariants
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 invariants of \(X\). Special cases of this construction include the Hurwitz numbers, which are GW invariants of \({\mathbb P}^1\) relative to several points in \({\mathbb P}^1\), and the Caporaso-Harris enumerative invariants [\textit{L. Caporaso} and \textit{J. Harris}, Invent. Math. 131, 345--392 (1998; Zbl 0934.14040)], which are GW invariants of \({\mathbb P}^2\) relative to a line. Similar results for the genus zero Gromov-Witten invariants in the algebro-geometric setting, have been obtained by \textit{A.~Gathmann} in [Duke Math. J. 115, 171--203 (2002; Zbl 1042.14032)].
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