## Relative Gromov-Witten invariants.(English)Zbl 1039.53101

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 [L. Caporaso and 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 A. Gathmann in [Duke Math. J. 115, 171–203 (2002; Zbl 1042.14032)].

### MSC:

 53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds 57R17 Symplectic and contact topology in high or arbitrary dimension 57R57 Applications of global analysis to structures on manifolds

### Keywords:

Gromov-Witten invariants; symplectic manifolds

### Citations:

Zbl 0943.53046; Zbl 0934.14040; Zbl 1042.14032
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