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Relative Gromov-Witten invariants and the mirror formula. (English) Zbl 1043.14016
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. 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 A. Givental [Int. Math. Res. Not. 1996, 613–663 (1996; Zbl 0881.55006)] and B. Lian, K. Liu, and 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 N. Takahashi [Commun. Math. Phys. 220, 293–299 (2001; Zbl 1066.14048)].

##### MSC:
 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) 14N10 Enumerative problems (combinatorial problems) in algebraic geometry 14J70 Hypersurfaces and algebraic geometry 14J32 Calabi-Yau manifolds (algebro-geometric aspects)
##### Keywords:
Gromov-Witten invariants; mirror symmetry; hypersurface
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