Hostname: page-component-76c49bb84f-s7zp8 Total loading time: 0 Render date: 2025-07-10T23:03:12.702Z Has data issue: false hasContentIssue false
  • English
  • Français

On local stabilities of $p$-Kähler structures

Part of: Global differential geometry Deformations of analytic structures Families, fibrations Commutative algebra: Homological methods

Published online by Cambridge University Press:  07 March 2019

Sheng Rao ,
Xueyuan Wan  and
Quanting Zhao
Sheng Rao
Affiliation:
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China email likeanyone@whu.edu.cn, raoshengmath@gmail.com
Xueyuan Wan
Affiliation:
Mathematical Sciences, Chalmers University of Technology, University of Gothenburg, 412 96 Gothenburg, Sweden email xwan@chalmers.se
Quanting Zhao
Affiliation:
School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, China email zhaoquanting@126.com, zhaoquanting@mail.ccnu.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

By use of a natural extension map and a power series method, we obtain a local stability theorem for $p$-Kähler structures with the $(p,p+1)$th mild $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$-lemma under small differentiable deformations.

Keywords

deformations of complex structuresdeformations and infinitesimal methodsformal methodsdeformationsHermitian and Kählerian manifolds

MSC classification

Primary: 32G05: Deformations of complex structures
Secondary: 13D10: Deformations and infinitesimal methods 14D15: Formal methods; deformations 53C55: Hermitian and Kählerian manifolds

Information

Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 3 , March 2019 , pp. 455 - 483
Copyright
© The Authors 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

Rao is partially supported by NSFC (Grant No. 11671305, 11771339). Zhao is partially supported by China Postdoctoral Science Foundation and NSFC (Grant No. 2016M592356 and 11801205).

References

Alessandrini, L., Proper modifications of generalized p-Kähler manifolds , J. Geom. Anal. 27 (2017), 947–967.Google Scholar
Alessandrini, L. and Andreatta, M., Closed transverse (p, p)-forms on compact complex manifolds , Compositio Math. 61 (1987), 181–200; Erratum 63 (1987), 143.Google Scholar
Alessandrini, L. and Bassanelli, G., Small deformations of a class of compact non-Kähler manifolds , Proc. Amer. Math. Soc. 109 (1990), 1059–1062.Google Scholar
Alessandrini, L. and Bassanelli, G., Compact p-Kähler manifolds , Geom. Dedicata 38 (1991), 199–210.Google Scholar
Alessandrini, L. and Bassanelli, G., Positive ∂ ∂ -closed currents and non-Kähler geometry , J. Geom. Anal. 2 (1992), 291–361.Google Scholar
Angella, D., The cohomologies of the Iwasawa manifold and its small deformations , J. Geom. Anal. 23 (2013), 1355–1378.Google Scholar
Angella, D. and Kasuya, H., Hodge theory for twisted differentials , Complex Manifolds 1 (2014), 64–85.Google Scholar
Angella, D. and Kasuya, H., Bott–Chern cohomology of solvmanifolds , Ann. Global Anal. Geom. 52 (2017), 363–411.Google Scholar
Angella, D., Suwa, T., Tardini, N. and Tomassini, A., Note on Dolbeault cohomology and Hodge structures up to bimeromorphisms, Preprint (2017), arXiv:1712.08889.Google Scholar
Angella, D. and Tardini, N., Quantitative and qualitative cohomological properties for non-Kähler manifolds , Proc. Amer. Math. Soc. 145 (2017), 273–285.Google Scholar
Angella, D. and Tomassini, A., On the ∂ ∂ -lemma and Bott–Chern cohomology , Invent. Math. 192 (2013), 71–81.Google Scholar
Angella, D. and Ugarte, L., Locally conformal Hermitian metrics on complex non-Kähler manifolds , Mediterr. J. Math. 13 (2016), 2105–2145.Google Scholar
Angella, D. and Ugarte, L., On small deformations of balanced manifolds , Differential Geom. Appl. 54 (2017), 464–474.Google Scholar
Barannikov, S. and Kontsevich, M., Frobenius manifolds and formality of Lie algebras of polyvector fields , Int. Math. Res. Not. IMRN 1998 (1998), 201–215.Google Scholar
Clemens, H., Geometry of formal Kuranishi theory , Adv. Math. 198 (2005), 311–365.Google Scholar
Console, F. and Fino, A., Dolbeault cohomology of compact nilmanifolds , Transform. Groups 6 (2001), 111–124.Google Scholar
Console, F., Fino, A. and Poon, Y.-S., Stability of abelian complex structures , Int. J. Math. 17 (2006), 401–416.Google Scholar
Deligne, P., Griffiths, P., Morgan, J. and Sullivan, D., Real homotopy theory of Kähler manifolds , Invent. Math. 29 (1975), 245–274.Google Scholar
Demailly, J.-P., Complex analytic and differential geometry (2012),http://www-fourier.ujf-grenoble.fr/demailly/books.html.Google Scholar
Douglis, A. and Nirenberg, L., Interior estimates for elliptic systems of partial differential equations , Comm. Pure Appl. Math. 8 (1955), 503–538.Google Scholar
Friedman, R., On threefolds with trivial canonical bundle , in Complex geometry and Lie theory (Sundance, UT, 1989), Proceedings of Symposia in Pure Mathematics, vol. 53 (American Mathematical Society, Providence, RI, 1991), 103–134.Google Scholar
Fu, J. and Yau, S.-T., A note on small deformations of balanced manifolds , C. R. Math. Acad. Sci. Paris 349 (2011), 793–796.Google Scholar
Griffiths, P. and Harris, J., Principles of algebraic geometry, Pure and Applied Mathematics (Wiley-Interscience, New York, 1978).Google Scholar
Harvey, R., Holomorphic chains and their boundaries , in Several complex variables, Proceedings of Symposia in Pure Mathematics, vol. XXX, Part 1, Williams College, Williamstown, MA, 1975 (American Mathematics Society, Providence, RI, 1977), 309–382.Google Scholar
Harvey, R. and Knapp, A., Positive (p, p)-forms, Wirtinger’s inequality and currents , in Proc. Tulane Univ. program on value distribution theory in complex analysis and related topics in differential geometry 1972–73 (Dekker, New York, 1974), 43–62.Google Scholar
Kasuya, H., Techniques of computations of Dolbeault cohomology of solvmanifolds , Math. Z. 273 (2013), 437–447.Google Scholar
Kodaira, K., Complex manifolds and deformations of complex structures, Grundlehren der mathematischen Wissenschaften, vol. 283 (Springer, 1986).Google Scholar
Kodaira, K., Nirenberg, L. and Spencer, D. C., On the existence of deformations of complex analytic structures , Ann. of Math. (2) 68 (1958), 450–459.Google Scholar
Kodaira, K. and Spencer, D., On deformations of complex analytic structures. III. Stability theorems for complex structures , Ann. of Math. (2) 71 (1960), 43–76.Google Scholar
Kuranishi, M., New proof for the existence of locally complete families of complex structures , in Proc. conf. complex analysis (Minneapolis, 1964) (Springer, Berlin, 1965), 142–154.Google Scholar
Li, Y., On deformations of generalized complex structures the generalized Calabi–Yau case, Preprint (2005), arXiv:0508030v2.Google Scholar
Liu, K. and Rao, S., Remarks on the Cartan formula and its applications , Asian J. Math. 16 (2012), 157–170.Google Scholar
Liu, K., Rao, S. and Wan, X., Geometry of logarithmic forms and deformations of complex structures. J. Algebraic Geom., to appear. Preprint (2017), arXiv:1708.00097v2.Google Scholar
Liu, K., Rao, S. and Yang, X., Quasi-isometry and deformations of Calabi–Yau manifolds , Invent. Math. 199 (2015), 423–453.Google Scholar
Liu, K., Sun, X. and Yau, S.-T., Recent development on the geometry of the Teichmüller and moduli spaces of Riemann surfaces , in Geometry of Riemann surfaces and their moduli spaces, Surveys in Differential Geometry, vol. XIV (International Press, Somerville, MA, 2009), 221–259.Google Scholar
Liu, K. and Zhu, S., Solving equations with Hodge theory, Preprint (2018), arXiv:1803.01272v1.Google Scholar
Morrow, J. and Kodaira, K., Complex manifolds (Holt, Rinehart and Winston, Inc., New York, Montreal, London, 1971).Google Scholar
Parshin, A., A generalization of the Jacobian variety (Russian) , Isvestia 30 (1966), 175–182.Google Scholar
Popovici, D., Aeppli cohomology classes associated with Gauduchon metrics on compact complex manifolds , Bull. Soc. Math. France 143 (2015), 763–800.Google Scholar
Rao, S., Wan, X. and Zhao, Q., Power series proofs for local stabilities of Kähler and balanced structures with mild $\unicode[STIX]{x2202}\bar{\unicode[STIX]{x2202}}$ -lemma, Preprint (2016), arXiv:1609.05637v1.Google Scholar
Rao, S., Yang, S. and Yang, X.-D., Dolbeault cohomologies of blowing up complex manifolds, J. Math. Pures Appl., to appear. Preprint (2017), arXiv:1712.06749v1; (2018),arXiv:1712.06749v4.Google Scholar
Rao, S. and Zhao, Q., Several special complex structures and their deformation properties , J. Geom. Anal. 28 (2018), 2984–3047.Google Scholar
Rollenske, S., Lie algebra Dolbeault cohomology and small deformations of nilmanifolds , J. Lond. Math. Soc. 79 (2009), 346–362.Google Scholar
Sullivan, D., Cycles for the dynamical study of foliated manifolds and complex manifolds , Invent. Math. 36 (1976), 225–255.Google Scholar
Sun, X., Deformation of canonical metrics I , Asian J. Math. 16 (2012), 141–155.Google Scholar
Sun, X. and Yau, S.-T., Deformation of Kähler–Einstein metrics , in Surveys in geometric analysis and relativity, Advanced Lectures in Mathematics (ALM), vol. 20 (International Press, Somerville, MA, 2011), 467–489.Google Scholar
Tian, G., Smoothness of the universal deformation space of compact Calabi–Yau manifolds and its Petersson–Weil metric , in Mathematical aspects of string theory (San Diego, Calif., 1986), Advanced Series in Mathematical Physics, vol. 1 (World Scientific Publishing, Singapore, 1987), 629–646.Google Scholar
Todorov, A., The Weil–Petersson geometry of the moduli space of SU(n 3) (Calabi–Yau) manifolds I , Comm. Math. Phys. 126 (1989), 325–346.Google Scholar
Ueno, K., Classification theory of algebraic varieties and compact complex spaces , Lecture Notes in Mathematics, vol. 439 (Springer, Berlin, Heidelberg, New York, 1975).Google Scholar
Ugarte, L. and Villacampa, R., Balanced Hermitian geometry on 6-dimensional nilmanifolds , Forum Math. 27 (2015), 1025–1070.Google Scholar
Voisin, C., Hodge theory and complex algebraic geometry. I, Cambridge Studies in Advanced Mathematics, vol. 76 (Cambridge University Press, Cambridge, 2002), Translated from the French original by Leila Schneps.Google Scholar
Wells, R. O., Comparison of de Rham and Dolbeault cohomology for proper surjective mappings , Pacific J. Math. 53 (1974), 281–300.Google Scholar
Wu, C.-C., On the geometry of superstrings with torsion, PhD thesis, Harvard University (2006).Google Scholar
Yang, S. and Yang, X.-D., Bott–Chern cohomology of blowing up manifolds, Preprint (2017),arXiv:1712.08901.Google Scholar
Zhao, Q. and Rao, S., Applications of deformation formula of holomorphic one-forms , Pacific J. Math. 266 (2013), 221–255.Google Scholar
Zhao, Q. and Rao, S., Extension formulas and deformation invariance of Hodge numbers , C. R. Math. Acad. Sci. Paris 353 (2015), 979–984.Google Scholar