## 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)

### Citations:

Zbl 0826.32016; Zbl 0904.32019; Zbl 0829.14023; Zbl 0843.14016
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### References:

 [1] P. Candelas, X.C. de la Ossa, P.S. Green, L. Parkes.A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory. Nucl. Phys.,B359 (1991), 21. · Zbl 1098.32506 [2] V.V. Batyrev.Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties. Preprint, Essen Univ., 1992. · Zbl 0829.14023 [3] V.V. Batyrev, D. van Straten.Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections. Preprint, Essen Univ., 1993. · Zbl 0843.14016 [4] M. Atiyah, R. Bott.The moment map and equivariant cohomology. Topology23 (1984), 1–28. · Zbl 0521.58025 [5] M. Audin.The topology of torus actions on symplectic manifolds. Birkhäuser, Basel, 1991. · Zbl 0726.57029 [6] W.Y. Hsiang.Cohomology theory of topological transformation groups. Springer, N.-Y., 1975. · Zbl 0429.57011 [7] A. Floer.Symplectic fixed points and holomorphic spheres. Commun. Math. Phys.,120 (1989), 575–611. · Zbl 0755.58022 [8] A. Givental.A symplectic fixed point theorem for toric manifolds. Preprint, UC Berkeley, 1992 (to appear inThe Floer Memorial, Prog. in Math., Birkhäuser, 1995.). [9] Y. Ruan, G. Tian.A mathematical theory of quantum cohomology. Preprint, 1994. · Zbl 0860.58006 [10] A. Givental, B. Kim.Quantum cohomology of flag manifolds and Toda lattices. Preprint, hep-th/9312096 (to appear inCommun. Math. Phys.) [11] P. Aspinwall, D. Morrison.Topological field theory and rational curves. Preprint, Oxford, 1991 (to appear inCommun. Math. Phys.) · Zbl 0776.53043 [12] E. Witten.Phases of N=2 theories in two dimensions. Preprint, IASSNSHEP-93/3. · Zbl 0910.14020 [13] M. Kontsevich.Enumeration of rational curves via torus actions. Preprint, 1994. · Zbl 0885.14028 [14] N. Berline, M. Vergne.Fourier transform of orbits of the coadjoint representation. InProceedings of the conference on representation theory of reductive groups, (Park City, Utah, 1982, Prog. in Math., v. 40, Birkhäuser, Boston, 1983, p. 53–75. [15] A. Givental.Homological geometry and mirror symmetry. to appear inProceedings of the ICM Zürich 1994, Birkhäuser, 1995.
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