Advertisement

An algebraic proof of Bogomolov-Tian-Todorov theorem

  • Donatella Iacono
  • Marco Manetti

Abstract

We give a completely algebraic proof of the Bogomolov-Tian-Todorov theorem. More precisely, we shall prove that if X is a smooth projective variety with trivial canonical bundle defined over an algebraically closed field of characteristic 0, then the L -algebra governing infinitesimal deformations of X is quasi-isomorphic to an abelian differential graded Lie algebra.

Keywords

Open Cover Deformation Theory Ahler Manifold Canonical Bundle Smooth Projective Variety 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. S. Barannikov, M. Kontsevich: Frobenius manifolds and formality of Lie algebras of polyvector fields. Internat. Math. Res. Notices 4 (1998) 201–215.CrossRefMathSciNetGoogle Scholar
  2. F. Bogomolov: Hamiltonian Kählerian manifolds. Dokl. Akad. Nauk SSSR 243 (1978) 1101–1104. Soviet Math. Dokl. 19 (1979) 1462–1465.Google Scholar
  3. X.Z. Cheng, E. Getzler: Homotopy commutative algebraic structures. J. Pure Appl. Algebra 212 (2008) 2535–2542; arXiv:math.AT/0610912.MATHCrossRefMathSciNetGoogle Scholar
  4. H. Clemens: Geometry of formal Kuranishi theory. Adv. Math. 198 (2005) 311–365.MATHCrossRefMathSciNetGoogle Scholar
  5. P. Deligne, L. Illusie: Relévements modulo p 2 et décomposition du complexe de de Rham. Invent. Math. 89 (1987) 247–270.MATHCrossRefMathSciNetGoogle Scholar
  6. J.L. Dupont: Simplicial de Rham cohomology and characteristic classes of flat bundles. Topology 15 (1976) 233–245.MATHCrossRefMathSciNetGoogle Scholar
  7. J.L. Dupont: Curvature and characteristic classes. Lecture Notes in Mathematics 640, Springer-Verlag, New York Berlin, (1978).Google Scholar
  8. S. Eilenberg, J.A. Zilber: Semi-simplicial complexes and singular homology. Ann. of Math. (2) 51 (1950) 499–513.CrossRefMathSciNetGoogle Scholar
  9. G. Faltings: p-adic Hodge theory. J. Amer. Math. Soc. 1 (1988) 255–299.MATHCrossRefMathSciNetGoogle Scholar
  10. B. Fantechi, M. Manetti: On the T 1 -lifting theorem. J. Algebraic Geom. 8 (1999) 31–39.MATHMathSciNetGoogle Scholar
  11. Y. Félix, S. Halperin, J. Thomas: Rational homotopy theory. Graduate texts in mathematics 205, Springer-Verlag, New York Berlin, (2001).Google Scholar
  12. D. Fiorenza, M. Manetti: L algebras, Cartan homotopies and period maps. Preprint arXiv:math/0605297.Google Scholar
  13. D. Fiorenza, M. Manetti: L structures on mapping cones. Algebra Number Theory, 1, (2007), 301–330; arXiv:math.QA/0601312.MATHCrossRefMathSciNetGoogle Scholar
  14. D. Fiorenza, M. Manetti: A period map for generalized deformations. J. Noncommut. Geom. (to appear); arXiv:0808.0140v1.Google Scholar
  15. D. Fiorenza, M. Manetti, E. Martinengo: Semicosimplicial DGLAs in deformation theory. Preprint arXiv:0803.0399v1.Google Scholar
  16. K. Fukaya: Deformation theory, homological algebra and mirror symmetry. Geometry and physics of branes (Como, 2001), Ser. High Energy Phys. Cosmol. Gravit., IOP Bristol, (2003), 121–209. Electronic version available at http://www.math.kyoto-u.ac.jp/%7Efukaya/como.dvi
  17. E. Getzler: Lie theory for nilpotent L -algebras. Ann. of Math. 170 (1), (2009) 271–301; arXiv:math/0404003v4. Ann. of Math., 170 (1), (2009), 271–301. arXiv:math/0404003v4.Google Scholar
  18. W.M. Goldman, J.J. Millson: The deformation theory of representations of fundamental groups of compact kähler manifolds. Publ. Math. I.H.E.S. 67 (1988) 43–96.MATHMathSciNetGoogle Scholar
  19. W.M. Goldman, J.J. Millson: The homotopy invariance of the Kuranishi space. Illinois J. Math. 34 (1990) 337–367.MATHMathSciNetGoogle Scholar
  20. V. Hinich: Descent of Deligne groupoids. Internat. Math. Res. Notices 1997, no. 5, 223–239.Google Scholar
  21. V. Hinich, V. Schechtman: Deformation theory and Lie algebra homology. I. Algebra Colloq. 4 (1997), no. 2, 213–240.MATHMathSciNetGoogle Scholar
  22. V. Hinich, V. Schechtman: Deformation theory and Lie algebra homology. II. Algebra Colloq. 4 (1997), no. 3, 291–316.MATHMathSciNetGoogle Scholar
  23. F. Hirzebruch: Topological Methods in Algebraic Geometry. Classics in Mathematics, Springer-Verlag, New York Berlin, (1978).Google Scholar
  24. J. Huebschmann, T. Kadeishvili: Small models for chain algebras. Math. Z. 207 (1991) 245–280.MATHCrossRefMathSciNetGoogle Scholar
  25. D. Iacono: Differential Graded Lie Algebras and Deformations of Holomorphic Maps. Phd Thesis (2006) arXiv:math.AG/0701091.Google Scholar
  26. D. Iacono: A semiregularity map annihilating obstructions to deforming holomorphic maps. Canad. Math. Bull. (to appear); arXiv:0707.2454.Google Scholar
  27. D. Iacono: L -algebras and deformations of holomorphic maps. Int. Math. Res. Not. 8 (2008) 36 pp. arXiv:0705.4532.Google Scholar
  28. T. V. Kadeishvili: The algebraic structure in the cohomology of A(∞)-algebras. Soobshch. Akad. Nauk Gruzin. SSR 108 (1982), 249–252.MATHMathSciNetGoogle Scholar
  29. Y. Kawamata: Unobstructed deformations - a remark on a paper of Z. Ran. J. Algebraic Geom. 1 (1992), 183–190.MATHMathSciNetGoogle Scholar
  30. K. Kodaira: Complex manifold and deformation of complex structures. Springer-Verlag (1986).Google Scholar
  31. M. Kontsevich: Deformation quantization of Poisson manifolds, I. Letters in Mathematical Physics 66 (2003) 157–216; arXiv:q-alg/9709040.MATHCrossRefMathSciNetGoogle Scholar
  32. M. Kontsevich, Y. Soibelman: Deformations of algebras over operads and Deligne’s conjecture. In: G. Dito and D. Sternheimer (eds) Conférence Moshé Flato 1999, Vol. I (Dijon 1999), Kluwer Acad. Publ., Dordrecht (2000) 255–307; arXiv:math.QA/0001151.Google Scholar
  33. M. Kontsevich, Y. Soibelman: Homological mirror symmetry and torus fibrations. K. Fukaya, (ed.) et al., Symplectic geometry and mirror symmetry. Proceedings of the 4th KIAS annual international conference, Seoul, South Korea, August 14–18, 2000. Singapore: World Scientific. (2001) 203–263; arXiv:math.SG/0011041.Google Scholar
  34. T. Lada, M. Markl: Strongly homotopy Lie algebras. Comm. Algebra 23 (1995) 2147–2161; arXiv:hep-th/9406095.MATHCrossRefMathSciNetGoogle Scholar
  35. T. Lada, J. Stasheff: Introduction to sh Lie algebras for physicists. Int. J. Theor. Phys. 32 (1993) 1087–1104; arXiv:hep-th/9209099.MATHCrossRefMathSciNetGoogle Scholar
  36. M. Manetti: Deformation theory via differential graded Lie algebras. In Seminari di Geometria Algebrica 1998–1999 Scuola Normale Superiore (1999); arXiv:math.AG/0507284.Google Scholar
  37. M. Manetti: Extended deformation functors. Int. Math. Res. Not. 14 (2002) 719–756; arXiv:math.AG/9910071.CrossRefMathSciNetGoogle Scholar
  38. M. Manetti: Cohomological constraint to deformations of compact Kähler manifolds. Adv. Math. 186 (2004) 125–142; arXiv:math.AG/0105175.MATHCrossRefMathSciNetGoogle Scholar
  39. M. Manetti: Lectures on deformations of complex manifolds. Rend. Mat. Appl. (7) 24 (2004) 1–183; arXiv:math.AG/0507286.MathSciNetGoogle Scholar
  40. M. Manetti: Lie description of higher obstructions to deforming submanifolds. Ann. Sc. Norm. Super. Pisa Cl. Sci. 6 (2007) 631–659; arXiv:math.AG/0507287.MATHMathSciNetGoogle Scholar
  41. M. Manetti: Differential graded Lie algebras and formal deformation theory. In Algebraic Geometry: Seattle 2005. Proc. Sympos. Pure Math. 80 (2009) 785–810.MathSciNetGoogle Scholar
  42. S.A. Merkulov: Strong homotopy algebras of a Kähler manifold. Intern. Math. Res. Notices (1999) 153–164; arXiv:math.AG/9809172.Google Scholar
  43. V. Navarro Aznar: Sur la théorie de Hodge-Deligne. Invent. Math. 90 (1987) 11–76.MATHCrossRefMathSciNetGoogle Scholar
  44. J. P. Pridham: Deformations via Simplicial Deformation Complexes. Preprint arXiv:math/0311168v6.Google Scholar
  45. D. Quillen: Rational homotopy theory. Ann. of Math. 90 (1969) 205–295.CrossRefMathSciNetGoogle Scholar
  46. Z. Ran: Deformations of manifolds with torsion or negative canonical bundle. J. Algebraic Geom. 1 (1992), 279–291.MATHMathSciNetGoogle Scholar
  47. V. Schechtman: Local structure of moduli spaces. arXiv:alg-geom/9708008.Google Scholar
  48. M. Schlessinger, J. Stasheff: Deformation Theory and Rational Homotopy Type. Preprint (1979).Google Scholar
  49. E. Sernesi: Deformations of Algebraic Schemes. Grundlehren der mathematischen Wissenschaften, 334, Springer-Verlag, New York Berlin, (2006).Google Scholar
  50. G. Tian: Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric. Mathematical Aspects of String Theory (San Diego, 1986), Adv. Ser. Math. Phys. 1, World Sci. Publishing, Singapore, (1987), 629–646.Google Scholar
  51. A.N. Todorov: The Weil-Petersson geometry of the moduli space of SU (n ≥ 3) (Calabi-Yau) Manifolds I. Commun. Math. Phys., 126, (1989), 325–346.MATHCrossRefGoogle Scholar
  52. C.A. Weibel: An introduction to homological algebra. Cambridge Studies in Advanced Mathematics 38, Cambridge Univesity Press, Cambridge, (1994).Google Scholar
  53. H. Whitney: Geometric integration theory. Princeton University Press, Princeton, N. J., (1957).MATHGoogle Scholar

Copyright information

© Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden GmbH 2010

Authors and Affiliations

  • Donatella Iacono
  • Marco Manetti

There are no affiliations available

Personalised recommendations