Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Mirror symmetry via logarithmic degeneration data, II


Authors: Mark Gross and Bernd Siebert
Journal: J. Algebraic Geom. 19 (2010), 679-780
DOI: https://doi.org/10.1090/S1056-3911-2010-00555-3
Published electronically: June 22, 2010
MathSciNet review: 2669728
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Abstract | References | Additional Information

Abstract: This paper continues the authors' program of studying mirror symmetry via log geometry and toric degenerations, relating affine manifolds with singularities, log Calabi-Yau spaces, and toric degenerations of Calabi-Yau spaces. The main focus of this paper is the calculation of the cohomology of a Calabi-Yau variety associated to a given affine manifold with singularities $ B$. We show that the Dolbeault cohomology groups of the Calabi-Yau associated to $ B$ are described in terms of certain cohomology groups of sheaves on $ B$, as expected. This is proved first by calculating the log de Rham and log Dolbeault cohomology groups on the log Calabi-Yau space associated to $ B$, and then proving a base-change theorem for cohomology in our logarithmic setting. As applications, this shows that our mirror symmetry construction via Legendre duality of affine manifolds results in the usual interchange of Hodge numbers expected in mirror symmetry and gives an explicit description of the monodromy of a smoothing.


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  • 1. Batyrev, V., ``Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties,'' J. Algebraic Geom. 3 (1994), 493-535. MR 1269718 (95c:14046)
  • 2. Batyrev, V., and Borisov, L., ``On Calabi-Yau complete intersections in toric varieties,'' in Higher-dimensional complex varieties (Trento, 1994), 39-65, de Gruyter, Berlin, 1996. MR 1463173 (98j:14052)
  • 3. Batyrev, V., and Dais, D., ``Strong McKay correspondence, string-theoretic Hodge numbers and mirror symmetry,'' Topology 35 (1996), 901-929. MR 1404917 (97e:14023)
  • 4. Bredon, G., Sheaf Theory, second edition, Springer-Verlag, 1997. MR 1481706 (98g:55005)
  • 5. Deligne, P., ``Théorème de Lefschetz et critères de dégénérescence de suites spectrales,'' Inst. Hautes Études Sci. Publ. Math. 35 (1968), 259-278. MR 0244265 (39:5582)
  • 6. Friedman, R., ``Global smoothings of varieties with normal crossings,'' Ann. of Math. (2) 118 (1983), 75-114. MR 707162 (85g:32029)
  • 7. Fukaya, K., ``Multivalued Morse theory, asymptotic analysis and mirror symmetry,'' in Graphs and patterns in mathematics and theoretical physics, 205-278, Proc. Sympos. Pure Math., vol. 73, Amer. Math. Soc., Providence, RI, 2005. MR 2131017 (2006a:53100)
  • 8. Goldman, W., and Hirsch, M., ``The radiance obstruction and parallel forms on affine manifolds,'' Trans. Amer. Math. Soc. 286 (1984), 629-649. MR 760977 (86f:57032)
  • 9. Gross, M., ``Special Lagrangian Fibrations I: Topology,'' in Integrable Systems and Algebraic Geometry, eds. M.-H. Saito, Y. Shimizu and K. Ueno, World Scientific, 1998, 156-193. MR 1672120 (2000e:14066)
  • 10. Gross, M., ``Toric degenerations and Batyrev-Borisov duality,'' Math. Ann. 333 (2005), 645-688. MR 2198802 (2007b:14086)
  • 11. Gross, M., and Siebert, B., ``Affine manifolds, log structures, and mirror symmetry,'' Turkish J. Math. 27 (2003), 33-60. MR 1975331 (2004g:14041)
  • 12. Gross, M., and Siebert, B., ``Mirror symmetry via logarithmic degeneration data. I,'' J. Differential Geom. 72 (2006), 169-338. MR 2213573 (2007b:14087)
  • 13. Gross, M., and Siebert, B., ``From affine geometry to complex geometry,'' preprint (2007).
  • 14. Gross, M., and Siebert, B., ``Torus fibrations and toric degenerations,'' in preparation.
  • 15. Gross, M., Pandharipande, R., and Siebert, B., ``The tropical vertex,'' Duke Math. J., Vol. 153, no. 2 (2010), 297-362.
  • 16. Grothendieck, A., Revêtements étales et groupe fondamental (SGA I), Lecture notes in Mathematics, 224, Springer-Verlag, 1963. MR 0354651 (50:7129)
  • 17. Haase, C., and Zharkov, I., ``Integral affine structures on spheres and torus fibrations of Calabi-Yau toric hypersurfaces I,'' preprint, 2002, math.AG/0205321.
  • 18. Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, No. 52., Springer-Verlag, New York-Heidelberg, 1977. MR 0463157 (57:3116)
  • 19. Hitchin, N., ``The Moduli Space of Special Lagrangian Submanifolds,'' Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 25 (1997), 503-515. MR 1655530 (2000c:32075)
  • 20. Kato, F., ``Log smooth deformation theory,'' Tohoku Math. J. 48 (1996), 317-354. MR 1404507 (99a:14012)
  • 21. Kato, F., ``Functors of log Artin rings,'' Manuscripta Math. 96 (1998), 97-112. MR 1624360 (99f:14012)
  • 22. Kato, K., ``Logarithmic structures of Fontaine-Illusie,'' in: Algebraic analysis, geometry, and number theory (J.-I. Igusa et al., eds.), 191-224, Johns Hopkins Univ. Press, Baltimore, 1989. MR 1463703 (99b:14020)
  • 23. Katz, K., and Oda, T., ``On the differentiation of de Rham cohomology classes with respect to a parameter,'' J. Math. Kyoto Univ. 8 (1968), 199-213. MR 0237510 (38:5792)
  • 24. Kawamata, K., and Namikawa, Y., ``Logarithmic deformations of normal crossing varieties and smooothing of degenerate Calabi-Yau varieties,'' Invent. Math. 118 (1994), 395-409. MR 1296351 (95j:32030)
  • 25. Kontsevich, M., and Soibelman, Y., ``Homological mirror symmetry and torus fibrations,'' in: Symplectic geometry and mirror symmetry (Seoul, 2000), 203-263, World Sci. Publishing, River Edge, NJ, 2001. MR 1882331 (2003c:32025)
  • 26. Kontsevich, M., and Soibelman, Y., ``Affine structures and non-Archimedean analytic spaces,'' in The unity of mathematics, 321-385, Progr. Math., 244, Birkhaüser Boston, Boston, MA, 2006. MR 2181810 (2006j:14054)
  • 27. Leung, N.C., ``Mirror symmetry without corrections,'' Comm. Anal. Geom. 13 (2005), 287-331. MR 2154821 (2006c:32028)
  • 28. Milne, J.S., Étale Cohomology, Princeton University Press, 1980. MR 559531 (81j:14002)
  • 29. Oda, T., Convex bodies and algebraic geometry. An introduction to the theory of toric varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 15. Springer-Verlag, Berlin, 1988. MR 922894 (88m:14038)
  • 30. Ogus, A., Lectures on logarithmic algebraic geometry, in progress.
  • 31. Ran, Z., ``Unobstructedness of Calabi-Yau orbi-Kleinfolds,'' J. Math. Phys. 39 (1998), 625-629. MR 1489640 (98m:14043)
  • 32. Ruddat, H., ``Log Hodge groups on a toric Calabi-Yau degeneration,'' preprint, 2009.
  • 33. Schlessinger, M., ``Functors of Artin rings,'' Trans. Amer. Math. Soc. 130 (1968), 208-222. MR 0217093 (36:184)
  • 34. Steenbrink, J., ``Limits of Hodge structures'', Invent. Math. 31 (1976), 229-257. MR 0429885 (55:2894)
  • 35. Strominger, A., Yau, S.-T., and Zaslow, E., ``Mirror symmetry is $ T$-duality,'' Nucl. Phys. B, 479 (1996), 243-259. MR 1429831 (97j:32022)
  • 36. Tian, G., ``Smoothness of the Universal Deformation Space of Compact Calabi-Yau Manifolds and its Petersson-Weil Metric,'' in Mathematical Aspects of String Theory, 629-646, ed. S.-T. Yau, World Scientific, Singapore, 1987. MR 915841
  • 37. Todorov, A., ``The Weil-Petersson Geometry of the Moduli Space of $ SU(n\ge 3)$ (Calabi-Yau) Manifolds I,'' Commun. Math. Phys 126 (1989), 325-346. MR 1027500 (91f:32022)


Additional Information

Mark Gross
Affiliation: UCSD Mathematics, 9500 Gilman Drive, La Jolla, California 92093-0112
Email: mgross@math.ucsd.edu

Bernd Siebert
Affiliation: Mathematisches Institut, Albert-Ludwigs-Universität, Eckerstraße 1, 79104 Freiburg, Germany
Address at time of publication: Department Mathematik, Bundesstraße 55, 20146 Hamburg, Germany
Email: bernd.siebert@math.uni-freiburg.de, bernd.siebert@math.uni-hamburg.de

DOI: https://doi.org/10.1090/S1056-3911-2010-00555-3
Received by editor(s): September 14, 2007
Received by editor(s) in revised form: December 8, 2009
Published electronically: June 22, 2010
Additional Notes: This work was partially supported by NSF grant 0505325 and DFG priority programs “Globale Methoden in der komplexen Geometrie” and “Globale Differentialgeometrie”.

American Mathematical Society