Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori. (English) Zbl 0812.14035

The polynomial representation of cohomology classes in the middle cohomology group of hypersurfaces allows one to explicitly calculate the Picard-Fuchs equations, or in fact the Gauss-Manin connection for families of hypersurfaces. These differential equations seem to contain sufficient information to describe some superconformal theories. We remark that there exists a general philosophy that \(N = 2\) supergravity and superconformal models can be identified with special variations of abstract mixed Hodge structures. – Projective spaces and weighted projective spaces are special cases of so-called toric varieties. It is natural to ask whether the method of Jacobian rings for the calculation of the Hodge structure can be extended to the case of hypersurfaces in projective toric varieties.
There is one more strong reason why the calculation of the variation of the Hodge structure of hypersurfaces in toric varieties is especially important for physicists. The most interesting class of variations of Hodge structure in physics is connected with families of Calabi-Yau varieties. First, it is possible to construct many examples of families of Calabi-Yau hypersurfaces in toric varieties \(\mathbb{P}_ \Delta\) associated with reflexive polyhedra \(\Delta\). Second, for Calabi-Yau hypersurfaces in toric varieties we can use a precise mathematical definition of mirrors via the notion of dual polyhedron. The main idea can be also explained for families of Calabi-Yau 3-folds as follows:
There are two kinds of Yukawa couplings on the Hodge spaces \(H^{2,1} (W)\) and \(H^{1,1} (W)\) considered by physicists for Calabi-Yau 3-folds \(W\). It is known that the Yukawa coupling on \(H^{2,1} (W)\) is defined by the deformation of the complex structure, but the definition of the Yukawa coupling on \(H^{1,1} (W)\) is outside the classical algebraic geometry. Physicists have discovered a trick, the so-called mirror symmetry, which allows one to replace the calculation of the “mysterious Yukawa coupling” on \(H^{1,1} (W)\) by the calculation of the Yukawa coupling on the space \(H^{2,1} (W')\) corresponding to the deformation of the complex structure on another Calabi-Yau 3-fold \(W'\). The Calabi- Yau variety \(W'\) is called a mirror of \(W\). Since the mirror mapping for Calabi-Yau hypersurfaces in toric varieties admits a precise mathematical definition, we can compute both of the Yukawa couplings using only deformations of the classical Hodge structures. One should remark that in the general case the construction of mirrors for Calabi-Yau manifolds is a rather difficult problem, and Calabi-Yau hypersurfaces in toric varieties are now the only known cases in which this problem is solved by mathematical methods.
Denote by \(\mathbb{T}\) the \(n\)-dimensional algebraic torus \((\mathbb{C}^*)^ n\). The main purpose of this paper is to give the explicit description of the mixed Hodge structure of the affine \(\Delta\)-regular hypersurface \(Z_ f \subset \mathbb{T}\) defined by a Laurent polynomial \(f\) with the Newton polyhedron \(\Delta\). As an application of our calculation of the infinitesimal variations of mixed Hodge structures, we show under some restrictions the following property of the mirror mapping for Calabi-Yau varieties:
The topological cup product on the cohomology ring \(\bigoplus^{n- 1}_{i = 0} H^{i,i}(W)\) of smooth Calabi-Yau hypersurfaces in toric Fano \(n\)-folds can be obtained as a “limit” of the multiplicative structure on the chiral ring \(\bigoplus^{n-1}_{i = 0} H^{n-1-i,i} (W')\) of the mirrors \(W'\), where the “limit” corresponds to some degeneration of \(W'\) at a boundary point of the moduli space of complex structures on \(W'\).
This property of the mirror mapping agrees with predictions of physicists.


14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14D07 Variation of Hodge structures (algebro-geometric aspects)
14J30 \(3\)-folds
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
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[1] M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra , Addison-Wesley Ser. Math., Oxford, 1969. · Zbl 0175.03601
[2] P. S. Aspinwall, C. A. Lütken, and G. G. Ross, Construction and couplings of mirror manifolds , Phys. Lett. B 241 (1990), no. 3, 373-380.
[3] P. S. Aspinwall and C. A. Lütken, Geometry of mirror manifolds , Nuclear Phys. B 353 (1991), no. 2, 427-461.
[4] P. S. Aspinwall and C. A. Lütken, Quantum algebraic geometry of superstring compactifications , Nuclear Phys. B 355 (1991), no. 2, 482-510.
[5] H. Bass, On the ubiquity of Gorenstein rings , Math. Z. 82 (1963), 8-28. · Zbl 0112.26604
[6] V. V. Batyrev, On the classification of smooth projective toric varieties , Tohoku Math. J. (2) 43 (1991), no. 4, 569-585. · Zbl 0792.14026
[7] V. V. Batyrev, Dual polyhedra and the mirror symmetry for Calabi-Yau hypersurfaces in toric varieties , preprint, Univ.-GH-Essen, 1992. · Zbl 0829.14023
[8] V. V. Batyrev, Classification of toric Fano \(4\)-folds , preprint, Univ.-GH-Essen, 1992. · Zbl 0929.14024
[9] J.-M. Bismut and D. S. Freed, The analysis of elliptic families. I. Metrics and connections on determinant bundles , Comm. Math. Phys. 106 (1986), no. 1, 159-176. · Zbl 0657.58037
[10] P. Candelas, X. C. de la Ossa, P. S. Green, and L. Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory , Nuclear Phys. B 359 (1991), no. 1, 21-74. · Zbl 1098.32506
[11] J. Carlson and P. Griffiths, Infinitesimal variations of Hodge structure and the global Torelli problem , Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff and Noordhoff, Alphen aan den Rijn, 1980, pp. 51-76. · Zbl 0479.14007
[12] J. Carlson, M. Green, P. Griffiths, and J. Harris, Infinitesimal variations of Hodge structure I , Compositio Math. 50 (1983), no. 2-3, 109-205. · Zbl 0531.14006
[13] S. Cecotti, \(N=2\) Landau-Ginzburg vs. Calabi-Yau \(\sigma\)-models: non-perturbative aspects , Internat. J. Modern Phys. A 6 (1991), no. 10, 1749-1813. · Zbl 0743.57022
[14] C. H. Clemens, A Scrapbook of Complex Curve Theory , Univ. Ser. Math., Plenum Press, New York, 1980. · Zbl 0456.14016
[15] V. I. Danilov, The geometry of toric varieties , Russian Math. Surveys 33 (1978), 97-154. · Zbl 0425.14013
[16] V. I. Danilov and A. G. Khovanskiî, Newton polyhedra and an algorithm for computing Hodge-Deligne numbers , Math. USSR-Izv. 29 (1987), 279-298. · Zbl 0669.14012
[17] V. I. Danilov, de Rham complex on toroidal variety , Algebraic geometry (Chicago, IL, 1989) eds. I. Dolgachev and W. Fulton, Lecture Notes in Math., vol. 1479, Springer, Berlin, 1991, pp. 26-38. · Zbl 0773.14011
[18] 1 P. Deligne, Théorie de Hodge. II , Inst. Hautes Études Sci. Publ. Math. (1971), no. 40, 5-57. · Zbl 0219.14007
[19] 2 P. Deligne, Théorie de Hodge. III , Inst. Hautes Études Sci. Publ. Math. (1974), no. 44, 5-77. · Zbl 0237.14003
[20] I. Dolgachev, Weighted projective varieties , Group actions and vector fields (Vancouver, B.C., 1981) ed. J. B. Carrell, Lecture Notes in Math., vol. 956, Springer, Berlin, 1982, pp. 34-71. · Zbl 0516.14014
[21] B. Dwork, On the zeta function of a hypersurface , Inst. Hautes Études Sci. Publ. Math. (1962), no. 12, 5-68. · Zbl 0173.48601
[22] B. Dwork, On the zeta function of a hypersurface II , Ann. of Math. (2) 80 (1964), 227-299. JSTOR: · Zbl 0173.48601
[23] B. Dwork, Generalized Hypergeometric Functions , Oxford Math. Monographs, Clarendon Press, Oxford, 1990. · Zbl 0747.33001
[24] E. Ehrhart, Démonstration de la loi de réciprocité pour un polyèdre entier , C. R. Acad. Sci. Paris Sér. A-B 265 (1967), A5-A7. · Zbl 0147.30701
[25] A. Font, Periods and duality symmetries in Calabi-Yau compactifications , preprint, UCVFC/DF-1-92. · Zbl 1360.32009
[26] I. M. Gelfand, A. V. Zelevinsky, and M. M. Kapranov, Equations of hypergeometric type and toric varieties , Functional Anal. Appl. 28 (1989), no. 2, 12-26, English trans. 94-106.
[27] I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky, Generalized Euler integrals and \(A\)-hypergeometric functions , Adv. Math. 84 (1990), no. 2, 255-271. · Zbl 0741.33011
[28] I. M. Gelfand, A. V. Zelevinsky, and M. M. Kapranov, Discriminants of polynomials in several variables and triangulations of Newton polyhedra , Algebra i Analiz 2 (1990), no. 3, 1-62, English trans., Leningard Math. J. 2 (1991), 449-505. · Zbl 0741.14033
[29] D. Gepner, Exactly solvable string compactifications on manifolds of \(\mathrm SU(N)\) holonomy , Phys. Lett. B 199 (1987), no. 3, 380-388.
[30] B. R. Greene, S.-S. Roan, and S.-T. Yau, Geometric singularities and spectra of Landau-Ginzburg models , Comm. Math. Phys. 142 (1991), no. 2, 245-259. · Zbl 0735.53054
[31] B. Grossman, \(p\)-adic strings, the Weil conjectures and anomalies , Phys. Lett. B 197 (1987), no. 1-2, 101-104. · Zbl 0694.22006
[32] P. Griffiths, On the periods of certain rational integrals. I, II , Ann. of Math. (2) 90 (1969), 460-495; ibid. (2) 90 (1969), 496-541. JSTOR: · Zbl 0215.08103
[33] A. Grothendieck, On the de Rham cohomology of algebraic varieties , Inst. Hautes Études Sci. Publ. Math. (1966), no. 29, 95-103. · Zbl 0145.17602
[34] R. Hartshorne, Algebraic Geometry , Grad. Texts in Math., vol. 52, Springer-Verlag, Berlin, 1977. · Zbl 0367.14001
[35] T. Hibi, Ehrhart polynomials of convex polytopes, \(h\)-vectors of simplicial complexes, and nonsingular projective toric varieties , Discrete and Computational Geometry (New Brunswick, NJ, 1989/1990), DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 6, Amer. Math. Soc., Providence, 1991, pp. 165-177. · Zbl 0755.05098
[36] T. Hibi, Dual polytopes of rational convex polytopes , Combinatorica 12 (1992), no. 2, 237-240. · Zbl 0758.52009
[37] M. M. Kapranov, B. Sturmfels, and A. V. Zelevinsky, Quotients of toric varieties , Math. Ann. 290 (1991), no. 4, 643-655. · Zbl 0762.14023
[38] N. Katz, On the differential equation satisfied by a period matrix , Publ. Math. I.H.E.S. 39 (1968), 71-106. · Zbl 0159.22502
[39] N. Katz, Internal reconstruction of unit root \(F\)-crystals via expansion coefficients , Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 2, 245-285. · Zbl 0592.14021
[40] A. G. Hovanskiî, Newton polyhedra and the genus of full intersections , Funktsional. Anal. i Prilozhen. 12 (1978), no. 1, 51-61.
[41] A. G. Kouchnirenko, Polyèdres de Newton et nombres de Milnor , Invent. Math. 32 (1976), no. 1, 1-31. · Zbl 0328.32007
[42] W. Lerche, C. Vafa, and N. P. Warner, Chiral rings in \(N=2\) superconformal theories , Nuclear Phys. B 324 (1989), no. 2, 427-474.
[43] D. Mumford and J. Fogarty, Geometric Invariant Theory , 2nd ed., Ergeb. Math. Grenzgeb. (2), vol. 34, Springer-Verlag, Berlin, 1982. · Zbl 0504.14008
[44] S. Mori, Projective manifolds with ample tangent bundles , Ann. of Math. (2) 110 (1979), no. 3, 593-606. JSTOR: · Zbl 0423.14006
[45] D. Morrison, Mirror symmetry and rational curves on quintic \(3\)-folds: A guide for mathematicians , preprint, Duke University, DUK-M-90-01, July 1991. · Zbl 0932.14022
[46] D. Morrison, Picard-Fuchs equations and mirror maps for hyperspaces , preprint, Duke University, DUK-M-91-14, October 1991.
[47] A. Klemm and S. Theisen, Consideration of one-modulus Calabi-Yau compactifications: Picard-Fuchs equations, Kähler potentials and mirror maps , preprint, KA-THEP-03/92, TUM-TH-132-92, April 1992.
[48] T. Oda, Convex Bodies and Algebraic Geometry-An Introduction to the Theory of Toric Varieties , Ergeb. Math. Grenzgeb. (3), vol. 15, Springer-Verlag, Berlin, 1988. · Zbl 0628.52002
[49] T. Oda and H. S. Park, Linear Gale transforms and Gel fand-Kapranov-Zelevinskij decompositions , Tohoku Math. J. (2) 43 (1991), no. 3, 375-399. · Zbl 0782.52006
[50] R. Stanley, Decompositions of rational convex polytopes , Ann. Discrete Math. 6 (1980), 333-342. · Zbl 0812.52012
[51] R. Stanley, Enumerative Combinatorics, Vol. I , Wadsworth & Brooks/Cole Math. Ser., Wadsworth & Brooks/Cole, Monterey, Calif., 1986. · Zbl 0608.05001
[52] R. Stanley, On the Hilbert function of a graded Cohen-Macaulay domain , J. Pure Appl. Algebra 73 (1991), no. 3, 307-314. · Zbl 0735.13010
[53] J. Steenbrink, Intersection form for quasi-homogeneous singularities , Compositio Math. 34 (1977), no. 2, 211-223. · Zbl 0347.14001
[54] J. Stienstra, Formal group laws arising from algebraic varieties , Amer. J. Math. 109 (1987), no. 5, 907-925. JSTOR: · Zbl 0633.14022
[55] J. Stienstra, Marius van der Put, and Bert van der Marel, On \(p\)-adic monodromy , Math. Z. 208 (1991), no. 2, 309-325. · Zbl 0748.14006
[56] J. Stienstra, A variation of mixed Hodge structure for a special case of Appell’s \(F4\) , Proc. Tanaguchi Workshop “Special Differential Equations”, Kyoto, 1991, pp. 109-116.
[57] A. N. Varchenko, Zeta-function of monodromy and Newton’s diagram , Invent. Math. 37 (1976), no. 3, 253-262. · Zbl 0333.14007
[58] E. Witten, Mirror manifolds and topological field theory , preprint, IASSNS-HEP-91, December 1991.
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