Comptes Rendus
no. 11
Complex analysis/Differential geometry
Extension formulas and deformation invariance of Hodge numbers
[Formules d'extension et invariance par déformation des nombres de Hodge]

Présenté par : the Editorial Board

Quanting Zhao1, 2 ; Sheng Rao3, 4
1 School of Mathematics and statistics, Central China Normal University, Wuhan 430079, China
2 Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China
3 School of Mathematics and statistics, Wuhan University, Wuhan 430072, China
4 Department of Mathematics, University of California at Los Angeles, CA 90095-1555, USA
Comptes Rendus. Mathématique, Volume 353 (2015) no. 11, pp. 979-984.

Nous introduisons un isomorphisme canonique entre l'espace des formes différentielles complexes de type pur sur une variété complexe, compacte, et celui de ses déformations infinitésimales, et nous l'utilisons pour généraliser la formule d'extension récemment obtenue par K. Liu, X. Yang et le second auteur. Comme corollaire direct des formules d'extension, nous établissons plusieurs théorèmes d'invariance par déformation des nombres de Hodge des variétés complexes, sans avoir recours à l'inégalité de Frölicher ou à l'invariance topologique des nombres de Betti.

We introduce a canonical isomorphism from the space of pure-type complex differential forms on a compact complex manifold to the one on its infinitesimal deformations. By use of this map, we generalize an extension formula in a recent work of K. Liu, X. Yang and the second author. As a direct corollary of the extension formulas, we prove several deformation invariance theorems for Hodge numbers on some certain classes of complex manifolds, without using the Frölicher inequality or the topological invariance of the Betti numbers.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.09.004
Mots clés : Deformations of complex structures, Deformations and infinitesimal methods, Formal methods, Deformations, Hermitian and Kählerian manifolds
Quanting Zhao 1, 2 ; Sheng Rao 3, 4

1 School of Mathematics and statistics, Central China Normal University, Wuhan 430079, China
2 Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China
3 School of Mathematics and statistics, Wuhan University, Wuhan 430072, China
4 Department of Mathematics, University of California at Los Angeles, CA 90095-1555, USA 
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Quanting Zhao; Sheng Rao. Extension formulas and deformation invariance of Hodge numbers. Comptes Rendus. Mathématique, Volume 353 (2015) no. 11, pp. 979-984. doi : 10.1016/j.crma.2015.09.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.09.004/

[1] S. Barannikov; M. Kontsevich Frobenius manifolds and formality of Lie algebras of polyvector fields, Int. Math. Res. Not., Volume 4 (1998), pp. 201-215

[2] W. Barth; K. Hulek; C. Peters; A. Van de Ven Compact Complex Surfaces, Ergeb. Math. Ihrer Grenzgeb. 3. Folge, Ser. Mod. Surv. Math., vol. 4, Springer-Verlag, Berlin, 2004

[3] H. Clemens Geometry of formal Kuranishi theory, Adv. Math., Volume 198 (2005), pp. 311-365

[4] R. Friedman On threefolds with trivial canonical bundle, Sundance, UT, 1989 (Proc. Symp. Pure Math.), Volume vol. 53, Amer. Math. Soc., Providence, RI (1991), pp. 103-134

[5] H. Grauert Ein Theorem der analytischen Garbentheorie und die Modulräume komplexer Strukturen, Publ. Math. IHÉS, Volume 5 (1960), p. 64 (German)

[6] P. Griffiths The extension problem for compact submanifolds of complex manifolds. I. The case of a trivial normal bundle, Minneapolis, 1964, Springer, Berlin (1964), pp. 113-142

[7] Yi Li On deformations of generalized complex structures the generalized Calabi–Yau case | arXiv

[8] K. Liu; S. Rao Remarks on the Cartan formula and its applications, Asian J. Math., Volume 16 (March 2012) no. 1, pp. 157-170

[9] K. Liu; S. Rao; X. Yang Quasi-isometry and deformations of Calabi–Yau manifolds, Invent. Math., Volume 199 (2015) no. 2, pp. 423-453

[10] K. Liu; X. Sun; S.-T. Yau Recent development on the geometry of the Teichmüller and moduli spaces of Riemann surfaces, Geometry of Riemann Surfaces and Their Moduli Spaces, Surv. Diff. Geom., vol. XIV, 2009, pp. 221-259

[11] J. Morrow; K. Kodaira Complex Manifolds, Holt, Rinehart and Winston, Inc., New York–Montreal, Quebec–London, 1971

[12] I. Nakamura Complex parallelisable manifolds and their small deformations, J. Differ. Geom., Volume 10 (1975), pp. 85-112

[13] A. Newlander; L. Nirenberg Complex analytic coordinates in almost complex manifolds, Ann. of Math. (2), Volume 65 (1957), pp. 391-404

[14] D. Popovici Holomorphic deformations of balanced Calabi–Yau ∂∂¯-manifolds | arXiv

[15] S. Rao, Q. Zhao, Several special complex structures and their deformation properties, preprint, 2015.

[16] X. Sun Deformation of canonical metrics I, Asian J. Math., Volume 16 (2012) no. 1, pp. 141-155

[17] X. Sun; S.-T. Yau Deformation of Kähler–Einstein metrics, Surveys in Geometric Analysis and Relativity, Adv. Lect. Math. (ALM), vol. 20, Int. Press, Somerville, MA, 2011, pp. 467-489

[18] G. Tian Smoothness of the universal deformation space of compact Calabi–Yau manifolds and its Petersson–Weil metric, San Diego, Calif., 1986 (Adv. Ser. Math. Phys.), Volume vol. 1, World Sci. Publishing, Singapore (1987), pp. 629-646

[19] A. Todorov The Weil–Petersson geometry of the moduli space of SU(n≥3) (Calabi–Yau) manifolds I, Commun. Math. Phys., Volume 126 (1989) no. 2, pp. 325-346

[20] C. Voisin Hodge Theory and Complex Algebraic Geometry. I, Cambridge Stud. Adv. Math., vol. 76, Cambridge University Press, Cambridge, 2002 (Translated from the French original by Leila Schneps)

[21] X. Ye The jumping phenomenon of Hodge numbers, Pac. J. Math., Volume 235 (2008) no. 2, pp. 379-398

[22] Q. Zhao; S. Rao Applications of deformation formula of holomorphic one-forms, Pac. J. Math., Volume 266 (2013) no. 1, pp. 221-255

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