Complex analysis/Differential geometry
Extension formulas and deformation invariance of Hodge numbers
Comptes Rendus. Mathématique, Volume 353 (2015) no. 11, pp. 979-984.

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.

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.

Published online:
DOI: 10.1016/j.crma.2015.09.004
Keywords: Deformations of complex structures, Deformations and infinitesimal methods, Formal methods, Deformations, Hermitian and Kählerian manifolds
Zhao, Quanting 1, 2; Rao, Sheng 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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     title = {Extension formulas and deformation invariance of {Hodge} numbers},
     journal = {Comptes Rendus. Math\'ematique},
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Zhao, Quanting; Rao, Sheng. 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.

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

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

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

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

[6] Griffiths, P. 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] Li, Yi On deformations of generalized complex structures the generalized Calabi–Yau case | arXiv

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

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

[10] Liu, K.; Sun, X.; Yau, S.-T. 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] Morrow, J.; Kodaira, K. Complex Manifolds, Holt, Rinehart and Winston, Inc., New York–Montreal, Quebec–London, 1971

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

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

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

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

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

[17] Sun, X.; Yau, S.-T. 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] Tian, G. 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] Todorov, A. The Weil–Petersson geometry of the moduli space of SU(n3) (Calabi–Yau) manifolds I, Commun. Math. Phys., Volume 126 (1989) no. 2, pp. 325-346

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

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

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