Relative Gromov-Witten invariants and the mirror formula
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 \(Y\) out of that of \(X\) by a formal change of variables (the so-called ``mirror transformation''). The same techniques are used to give a similar expression for the (virtual) numbers of degree \(d\) plane rational curves meeting a smooth cubic at one point with multiplicity \(3d\), which play a role in local mirror symmetry, as shown by \textit{N. Takahashi} [Commun. Math. Phys. 220, 293--299 (2001; Zbl 1066.14048)].
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