THE $\partial\bar{\partial}$-LEMMA UNDER SURJECTIVE MAPS

doi:10.1007/s10473-022-0303-9

数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (3): 865-875.doi: 10.1007/s10473-022-0303-9

THE $\partial\bar{\partial}$-LEMMA UNDER SURJECTIVE MAPS

孟令旭

Department of Mathematics, North University of China, Taiyuan, 030051, China

收稿日期:2020-06-15 修回日期:2021-04-14 发布日期:2022-06-24
通讯作者: Lingxu MENG,E-mail:menglingxu@nuc.edu.cn E-mail:menglingxu@nuc.edu.cn
基金资助:
The author is supported by the National Natural Science Foundation of China (12001500, 12071444), the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (2020L0290) and the Natural Science Foundation of Shanxi Province of China (201901D111141).

THE $\partial\bar{\partial}$-LEMMA UNDER SURJECTIVE MAPS

Lingxu MENG

Department of Mathematics, North University of China, Taiyuan, 030051, China

Received:2020-06-15 Revised:2021-04-14 Published:2022-06-24
Contact: Lingxu MENG,E-mail:menglingxu@nuc.edu.cn E-mail:menglingxu@nuc.edu.cn
Supported by:
The author is supported by the National Natural Science Foundation of China (12001500, 12071444), the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (2020L0290) and the Natural Science Foundation of Shanxi Province of China (201901D111141).

摘要/Abstract

摘要： We consider the $\partial\bar{\partial}$-lemma for complex manifolds under surjective holomorphic maps. Furthermore, using Deligne-Griffiths-Morgan-Sullivan's theorem, we prove that a product compact complex manifold satisfies the $\partial\bar{\partial}$-lemma if and only if all of its components do as well.

关键词: $\partial\bar{\partial}$-lemma, surjective holomorphic map, product complex manifold, fiber bundle, E₁-isomorphism

Abstract: We consider the $\partial\bar{\partial}$-lemma for complex manifolds under surjective holomorphic maps. Furthermore, using Deligne-Griffiths-Morgan-Sullivan's theorem, we prove that a product compact complex manifold satisfies the $\partial\bar{\partial}$-lemma if and only if all of its components do as well.

Key words: $\partial\bar{\partial}$-lemma, surjective holomorphic map, product complex manifold, fiber bundle, E₁-isomorphism

中图分类号:

32Q99

孟令旭. THE $\partial\bar{\partial}$-LEMMA UNDER SURJECTIVE MAPS[J]. 数学物理学报(英文版), 2022, 42(3): 865-875.

Lingxu MENG. THE $\partial\bar{\partial}$-LEMMA UNDER SURJECTIVE MAPS[J]. Acta mathematica scientia,Series B, 2022, 42(3): 865-875.

参考文献

[1] Alessandrini L, Bassanelli G. Compact p-Kähler manifolds. Geom Dedicata, 1991, 38:199-210
[2] Angella D, Kasuya H. Bott-Chern cohomology of solvmanifolds. Ann Global Anal Geom, 2017, 52:363-411
[3] Angella D, Kasuya H. Cohomologies of deformations of solvmanifolds and closedness of some properties. North-West Eur J Math, 2017, 3:75-105
[4] Angella D, Suwa T, Tardini N, Tomassini A. Note on Dolbeault cohomology and Hodge structures up to bimeromorphisms. Complex Manifolds, 2020, 7(1):194-214
[5] Angella D, Tardini N. Quantitative and qualitative cohomological properties for non-Kähler manifolds. Proc Amer Math Soc, 2017, 145:273-285
[6] Angella D, Tomassini A. On the $\partial\bar{\partial}$-lemma and Bott-Chern cohomology. Invent Math, 2013, 192(3):71-81
[7] Deligne P, Griffiths P, Morgan J, Sullivan D. Real homotopy theory of Kähler manifolds. Invent Math, 1975, 29(3):245-274
[8] Friedman R. The $\partial\bar{\partial}$-lemma for general Clemens manifolds. Pure Appl Math Q, 2019, 15(4):1001-1028
[9] Fujiki A. Closedness of the Douady spaces of compact Kähler spaces. Publ Res Inst Math Sci, 1978, 14:1-52
[10] Kasuya H. Hodge symmetry and decomposition on non-Kähler solvmanifolds. J Geom Phys, 2014, 76:61-65
[11] Meng L X. Leray-Hirsch theorem and blow-up formula for Dolbeault cohomology. Ann Mat Pura Appl (4), 2020, 199(5):1997-2014
[12] Meng L X. The heredity and bimeromorphic invariance of the $\partial\bar{\partial}$-lemma property. C R Math Acad Sci Paris, 2021, 359:645-650
[13] Rao S, Yang S, Yang X D. Dolbeault cohomologies of blowing up complex manifolds. J Math Pures Appl (9), 2019, 130:68-92
[14] Stelzig J. Double complexes and Hodge stuctures as vector bundles[D]. Mänster:Westfälischen Wilhelms-Universität Münster, 2018. https://d-nb.info/1165650959/34
[15] Stelzig J. The double complex of a blow-up. Int Math Res Not IMRN, 2021(14):10731-10744
[16] Stelzig J. On the structure of double complexes. J Lond Math Soc (2), 2021, 104(2):956-988
[17] Stelzig J. Private communications. May, 2019
[18] Voisin C. Hodge Theory and Complex Algebraic Geometry. Vol I. Cambridge Stud Adv Math 76. Cambridge:Cambridge University Press, 2003
[19] Wu C C. On the geometry of superstrings with torsions[D]. Massachusetts:Harvard University, 2006
[20] Yang S, Yang X D. Bott-Chern blow-up formula and bimeromorphic invariance of the $\partial\bar{\partial}$-Lemma for threefolds. Trans Amer Math Soc, 2020, 373(12):8885-8909

THE $\partial\bar{\partial}$-LEMMA UNDER SURJECTIVE MAPS

THE $\partial\bar{\partial}$-LEMMA UNDER SURJECTIVE MAPS

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