Abstract
A homogeneous polynomial equation in five variables determines a quintic 3-fp;d in ℂP4. Hodge numbers of a nonsingular quintic are know to be: hp, p = 1, p = 0, 1, 2, 3 (Kähler form and its powers), h3, 0 = h0,3 = 1 (a quintic happens to bear a holomorphic volume form), h2,1 = h1, 2 = 101 = 126 - 25 (it is the dimension of the space of all quintics modulo projective transformations, and h2,1 is responsible here for infinitesimal variations of the complex structure) and all the other hp,q = 0.
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References
V. I. Arnold, Critical points of smooth functions, in Proceedings of ICM 74, vol. 1, Vancouver, BC, 1974, 19–40.
P. Aspinwall and D. Morrison, Topological field theory and rational curves, preprint, Oxford, 1991.
M. Audin, The Topology of Torus Actions on Symplectic Manifolds, Birkhäuser, Basel and Boston, 1991.
V. V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, preprint, Essen Univ., 1992.
V. Batyrev and D. van Straten, Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties, preprint, Essen Univ., 1993.
P. Candelas, X.C. de la Ossa, P.S. Green, and L. Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal field theory, Nuclear Phys. B, 359 (1991), 21–74.
B. Dubrovin, Integrable systems and classification of 2-dimensional topological field theories, preprint, hepth 9209040.
A. Floer, Symplectic fixed points and holomorphic spheres, Comm. Math. Phys. 120 (1989), 575–611.
A. Givental, A symplectic fixed point theorem for toric manifolds, preprint, Berkeley, 1992 (to appear in Progr. Math., Floer’s memorial volume).
A. Givental, Homological geometry, I: Projective hypersurfaces, preprint, 1994. II: Integral representations, in preparation.
A. Givental and B. Kim, Quantum cohomology of flag manifolds and Toda lattices, preprint, hep-th 9312096 (to appear in Comm. Math. Phys.).
M. Kontsevich, Enumeration of rational curves via torus actions, preprint, 1994.
M. Kontsevich and Yu. Manin, Gromov-Witten invariants, quantum cohomology, and enumerative geometry, preprint, Max-Plank-Institut, 1994.
D. McDuff and D. Salamon, J-holomorphic Curves and Quantum Cohomology, Amer. Math. Soc., Providence, RI, 1994.
Y. Ruan and G. Tian, A mathematical theory of quantum cohomology, preprint, 1994.
K. Saito, On the periods of primitive integrals I, RIMS (1982), 1–235.
A. N. Varchenko and A. B. Givental, Period mappings and intersection forms, Functional Anal. Appl. 16:2 (1982), 11–25.
E. Witten, Two-dimensional gravity and intersection theory on moduli space, Surveys Differential Geom. 1 (1991), 243–310.
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© 1995 Birkhäser Verlag, Basel, Switzerland
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Givental, A.B. (1995). Homological Geometry and Mirror Symmetry. In: Chatterji, S.D. (eds) Proceedings of the International Congress of Mathematicians. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-9078-6_40
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DOI: https://doi.org/10.1007/978-3-0348-9078-6_40
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