zbMATH — the first resource for mathematics

String cohomology of Calabi-Yau hypersurfaces via mirror symmetry. Appendix: \(G\)-polynomials. (English) Zbl 1055.14044
The paper under review proposes a construction of string cohomology spaces for Calabi-Yau hypersurfaces that arise in the Batyrev mirror symmetry construction [cf. V. Batyrev, J. Algebr. Geom. 3, No. 3, 493–535 (1994; Zbl 0829.14023)], with spaces defined explicitly in terms of the corresponding reflexive polyhedra. The construction starts with the description of the cohomology of semiample hypersurfaces in toric varieties and then uses mirror symmetry to provide a conjectural string cohomology of Calabi-Yau hypersurfaces. This construction gives the correct (bigraded) dimension of the space, and further, a finite-dimensional family of string cohomology spaces rather than just a single-string cohomology space. That is, string cohomology space depends not only on the complex structure (the defining polynomial \(f\)), but also on some extra parameter \(\omega\). For special values of this parameter of an orbifold Calabi-Yau hypersurface, the construction gives the orbifold Dolbeault cohomology, recovering the result of W. Chen and Y. Ruan [Commun. Math. Phys. 248, 1–31 (2004; Zbl 1063.53091)]. However, for non-orbifold Calabi-Yau hypersurfaces, there is no natural choice for \(\omega\), and this implies the dependence of the general definition of string cohomology space on some parameter. In case of Calabi-Yau hypersurfaces, this particular parameter \(\omega\) corresponds to the defining polynomial of the mirror Calabi-Yau hypersurface. In general, this parameter should be related to the “string complexified Kähler class” which is yet to be defined. An attempt is made to extend the definition of string cohomology space beyond the Calabi-Yau hypersurfaces. A conjectural definition of string cohomology vector spaces is presented for stratified varieties with \({\mathbb Q}\)-Gorenstein toroidal singularities that satisfy certain restrictions on the types of singular strata. This definition would involve intersection cohomology of the closures of strata as well as perverse sheaves. This conjectural definition gives the correct bigraded dimension and also reproduces orbifold cohomology of a \({\mathbb Q}\)-Gorenstein toric variety as a special case.

14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
32Q25 Calabi-Yau theory (complex-analytic aspects)
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
Full Text: DOI arXiv
[1] Abramovich, D.; Karu, K.; Matsuki, K.; Włodarsczyk, J., Torification and factorization of birational maps, J. amer. math. soc., 15, 3, 531-572, (2002) · Zbl 1032.14003
[2] G. Barthel, J.-P. Brasselet, K.-H. Fieseler, L. Kaup, Combinatorial intersection cohomology for fans, preprint math.AG/0002181. · Zbl 1055.14024
[3] Batyrev, V.V., Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori, Duke math. J., 69, 349-409, (1993) · Zbl 0812.14035
[4] Batyrev, V.V., Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. algebraic geom., 6, 493-535, (1994) · Zbl 0829.14023
[5] V.V. Batyrev, Stringy Hodge numbers of varieties with Gorenstein canonical singularities, in: Integrable Systems and Algebraic Geometry (Kobe/Kyoto, 1997), World Scientific Publishing, River Edge, NJ, 1998, pp. 1-32. · Zbl 0963.14015
[6] Batyrev, V.V., Non-Archimedean integrals and stringy Euler numbers of log-terminal pairs, J. eur. math. soc. (JEMS), 1, 1, 5-33, (1999) · Zbl 0943.14004
[7] Batyrev, V.V.; Borisov, L.A., Mirror duality and string-theoretic Hodge numbers, Invent. math., 126, 183-203, (1996) · Zbl 0872.14035
[8] Batyrev, V.V.; Cox, D.A., On the Hodge structure of projective hypersurfaces in toric varieties, Duke math. J., 75, 293-338, (1994) · Zbl 0851.14021
[9] Batyrev, V.V.; Dais, D., Strong mckay correspondence, string-theoretic Hodge numbers and mirror symmetry, Topology, 35, 901-929, (1996) · Zbl 0864.14022
[10] Borisov, L.A., String cohomology of a toroidal singularity, J. algebraic geom., 9, 2, 289-300, (2000) · Zbl 0949.14029
[11] Borisov, L.A., Vertex algebras and mirror symmetry, Comm. math. phys., 215, 3, 517-557, (2001) · Zbl 0990.17023
[12] P. Bressler, V. Lunts, Intersection cohomology on nonrational polytopes, preprint math.AG/0002006. · Zbl 1024.52005
[13] Cox, D.A., The homogeneous coordinate ring of a toric variety, J. algebraic geom., 4, 17-50, (1995) · Zbl 0846.14032
[14] D.A. Cox, Recent developments in toric geometry, in: Algebraic Geometry (Santa Cruz, 1995), Proceedings of Symposia in Pure Mathematics, bf-62, Part 2, American Mathematical Society, Providence, RI, 1997, pp. 389-436. · Zbl 0899.14025
[15] D.A. Cox, S. Katz, Algebraic Geometry and Mirror Symmetry, Math, Surveys Monographs, Vol. 68, American Mathematical Society, Providence, RI, 1999. · Zbl 0951.14026
[16] W. Chen, Y. Ruan, A new cohomology theory for orbifold, preprint math.AG/0004129.
[17] Danilov, V.I., The geometry of toric varieties, Russian math. surveys, 33, 97-154, (1978) · Zbl 0425.14013
[18] L. Dixon, J. Harvey, C. Vafa, E. Witten, Strings on orbifolds I, II, Nucl. Phys. B 261, B274 (1985) (1986).
[19] Danilov, V.; Khovanskii, A., Newton polyhedra and an algorithm for computing Hodge-Deligne numbers, Math. USSR-izv., 29, 279-298, (1987) · Zbl 0669.14012
[20] D. Eisenbud, Commutative Algebra with a view Toward Algebraic Geometry, in: Graduate Texts in Mathematics, Vol. 150, Springer, New York, 1995. · Zbl 0819.13001
[21] B.R. Greene, String theory on Calabi-Yau manifolds, Fields, strings and duality (Boulder, CO, 1996), World Sci. Publishing, River Edge, NJ, 1997, pp. 543-726.
[22] J. Kollár, S. Mori, Birational geometry of algebraic varieties, With the collaboration of C.H. Clemens and A. Corti, Cambridge Tracts in Mathematics, Vol. 134, Cambridge University Press, Cambridge, 1998. · Zbl 0926.14003
[23] F. Malikov, V. Schechtman, Deformations of chiral algebras and quantum cohomology of toric varieties, preprint math.AG/0003170. · Zbl 1033.58022
[24] Matsumura, Commutative ring theory, Translated from Japanese by M. Reid, 2nd Edition, Cambridge Studies in Advanced Mathematics, Vol. 8, Cambridge University Press, Cambridge, 1989. · Zbl 0666.13002
[25] Mavlyutov, A.R., Semiample hypersurfaces in toric varieties, Duke math. J., 101, 85-116, (2000) · Zbl 1023.14027
[26] A.R. Mavlyutov, On the chiral ring of Calabi-Yau hypersurfaces in toric varieties, preprint math. AG/0010318.
[27] Mavlyutov, A.R., The Hodge structure of semiample hypersurfaces and a generalization of the monomial-divisor mirror map, (), 199-227 · Zbl 1052.14045
[28] M. Poddar, Orbifold Hodge numbers of Calabi-Yau hypersurfaces, preprint math.AG/0107152. · Zbl 1052.32018
[29] Stanley, R., Generalized H-vectors, intersection cohomology of toric varieties, and related results, Adv. stud. in pure math., 11, 187-213, (1987)
[30] Stanley, R., Subdivisions and local h-vectors, Jams, 5, 805-851, (1992) · Zbl 0768.05100
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.