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Homological geometry. I: Projective hypersurfaces. (English) Zbl 0920.14028

From the introduction: Consider a generic quintic hypersurface \(X\) in \(\mathbb{C}\mathbb{P}^4\). It is an example of a Calabi-Yau 3-fold. It follows from the Riemann-Roch formula that rational curves on a generic Calabi-Yau 3-fold should be situated in a discrete fashion. Therefore a natural question of enumerative algebraic geometry arises: Find the number \(n_d\) of rational curves in \(X\) of degree \(d\) for each \(d=1,2,3, \dots\). In 1991, P. Candelas, X. de la Ossa, P. S. Green and L. Parkes in: Mirror symmetry. I, Stud. Adv. Math. 9, 31-95 (1998; Zbl 0826.32016) “predicted” all the numbers \(n_d\) simultaneously by conjecturing that the generating function \[ K(Q)=5+\sum^\infty_{d=1}{n_dd^3Q^d\over 1-Q^d} \] can be found by studying the 4-th order linear differential operator annihilating some hypergeometric series, namely \(\sum^\infty_{d=0} {(5d)!q^d \over(d!)^5}\). The conjecture was motivated by some ideas of conformal topological field theory (CTFT): There is a 1-parametric family \(Y_q\) of Calabi-Yau 3-folds which are mirrors of the quintics \(X\) in the sense that their Hodge numbers satisfy \(h^{r,s}(Y)=h^{3-r,s}(X)\). This indicates that a model of CTFT dealing with rational curves in \(X\) might be equivalent to a model based on periods of holomorphic 3-forms in \(Y_q\). Therefore problems about rational curves in \(X\) can be transformed into those about the Picard-Fuchs equation for periods of holomorphic forms in \(Y\). The hypergeometric series in question is in fact one of such periods.
The mirror conjecture about equivalence of algebraic geometry of rational curves in Calabi-Yau manifolds and variations of Hodge structures on their mirrors, as well as an explicit construction of such mirror pairs, has been generalized to a broad class of Calabi-Yau complete intersections in toric varieties [see V. V. Batyrev, J. Algebr. Geom. 3, No. 3, 493-535 (1994; Zbl 0829.14023)] and V. V. Batyrev and D. van Straten, Commun. Math. Phys. 168, No. 3, 493-533 (1995; Zbl 0843.14016)] and supported by numerous verified corollaries and “experimental” data (see papers by D. Morrison and references therein). However the mirror phenomenon itself, and especially the relation between rational curves in \(X\) and the Picard-Fuchs differential equation for \(Y\) remain mysterious. In this paper, we take a step toward an explanation of the mirror conjecture. Namely we describe how hypergeometric differential equations arise in connection with algebraic geometry of rational curves in Kähler manifolds. In particular, the differential equations which one used to observe in the theory of mirror manifolds as Picard-Fuchs equations for periods of holomorphic forms “on \(Y\)” will be obtained naturally in terms of the geometry of rational curves in \(X\).

MSC:

14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
32G20 Period matrices, variation of Hodge structure; degenerations
14J70 Hypersurfaces and algebraic geometry
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
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