Abstract
Given a holomorphic family of pairs \(\{(X_t,E_t)\}\) where each \(E_t\) is a holomorphic vector bundle over a compact complex manifold \(X_t\), we get a correspondence between the Dolbeault complex of \(E_t\)-valued (p, q)-forms on \(X_t\) and the one of \(E_0\)-valued (p, q)-forms on \(X_0\) for small enough t.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chan, K.K., Suen, Y.H.: On the jumping phenomenon of \(\dim _CH^q({\cal{X}}_t,{\cal{E}}_t)\). Asian J. Math. (2016). arXiv:1601.06472(preprint)
Clemens, H.: Geometry of formal Kuranishi theory. Adv. Math. 198(1), 311–365 (2005)
Green, M., Lazarsfeld, R.K.: A deformation theory for cohomology of analytic vector bundles on Kähler manifolds, with applications. In: Yau, S.T. (ed.) Mathematical Aspects of String Theory, pp. 416–440. World Scientific, World Scientific (1987)
Green, M., Lazarsfeld, R.K.: Higher obstructions to deforming cohomology groups of line bundles. J. Am. Math. Soc. 4(1), 87–103 (1991)
Greub, W., Halperin, S., Vanstone, R.: Connections, curvature, and cohomology, vol. II: lie groups, principal bundles, and characteristic classes, vol. 47-II. Pure Appl. Math. Academic Press (A subsidiary of Harcourt Brace Jovanovich, Publishers), New York, London. xxi + 541 pp (1973)
Huang, L.: On joint moduli spaces. Math. Ann. 302(1), 61–79 (1995)
Liu, K.F., Rao, S., Yang, X.K.: Quasi-isometry and deformations of Calabi–Yau manifolds. Invent. Math. 199(2), 423–453 (2015)
Liu, K.F., Tu, J.: The deformation of pairs \((X, E)\) lifting from base family. Asian J. Math. 22(5), 841–861 (2018)
Manetti, M.: Lectures on deformations of complex manifolds (deformations from differential graded viewpoint. Rend. Mat. Appl. (7) 24(1), 1–183 (2004)
Rao, S., Zhao, Q.T.: Several special complex structures and their deformation properties. J. Geom. Anal. 28(4), 2984–3047 (2018)
Rao, S., Wan, X.Y., Zhao, Q.T.: Power series proofs for local stabilities of Kähler and balanced structures with mild \(\partial \bar{\partial }\)-lemma (2016). arXiv:1609.05637(preprint)
Rao, S., Wan, X.Y., Zhao, Q.T.: On local stabilities of p-Kähler structures. Compos. Math. 155(3), 455–483 (2019)
Vosin, C.: Hodge theory and complex algebraic geometry I, Translated from the French by Leila Schneps. Reprint of the 2002 English edition. Cambridge Studies in Advanced Mathematics, 76. Cambridge University Press, Cambridge (2007). x + 322 pp
Zhao, Q., Rao, S.: Extension formulas and deformation invariance of Hodge numbers. C. R. Math. Acad. Sci. Paris 353(11), 979–984 (2015)
Acknowledgements
The author would like to express her gratitude to Prof. Kefeng Liu for his constant support and Prof. Sheng Rao for many useful comments and discussions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Tu, J. The correspondence formula of Dolbeault complex on pair deformation. Geom Dedicata 212, 365–378 (2021). https://doi.org/10.1007/s10711-020-00562-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-020-00562-2