Algebraic Geometry, Differential Geometry
On the Morse–Novikov Cohomology of blowing up complex manifolds
Zou, Yongpan1
1 School of Mathematics and Statistics, Wuhan University, Wuhan 430072, P. R. China
Comptes Rendus. Mathématique, Volume 358 (2020) no. 1, pp. 67-77.

Inspired by the recent works of S. Rao–S. Yang–X.-D. Yang and L. Meng on the blow-up formulae for de Rham and Morse–Novikov cohomology groups, we give a new simple proof of the blow-up formula for Morse–Novikov cohomology by introducing the relative Morse–Novikov cohomology group via sheaf cohomology theory and presenting the explicit isomorphism therein.

Inspiré par les récents travaux de S. Rao, S. Yang, X.-D. Yang et L. Meng sur les formules donnant le comportement des groupes de cohomologie de de Rham et Morse-Novikov dans les éclatements, nous donnons une nouvelle preuve simple de la formule pour la cohomologie de Morse-Novikov en introduisant le groupe de cohomologie de Morse-Novikov relatif via la cohomologie des faisceaux et en explicitant l’isomorphisme de la formule.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.12
Zou, Yongpan 1

1 School of Mathematics and Statistics, Wuhan University, Wuhan 430072, P. R. China 
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Zou, Yongpan. On the Morse–Novikov Cohomology of blowing up complex manifolds. Comptes Rendus. Mathématique, Volume 358 (2020) no. 1, pp. 67-77. doi : 10.5802/crmath.12. http://archive.numdam.org/articles/10.5802/crmath.12/

[1] Angella, Daniele; Kasuya, Hisashi Hodge theory for twisted differentials, Complex Manifolds, Volume 1 (2014), pp. 64-85 | MR | Zbl

[2] Belgun, Florin A. On the metric structure of non-Kähler complex surfaces, Math. Ann., Volume 317 (2000) no. 1, pp. 1-40 | DOI | MR | Zbl

[3] Bredon, Glen E. Sheaf theory, Graduate Texts in Mathematics, 170, Springer, 1997 | MR | Zbl

[4] Guedira, Fouzia; Lichnerowicz, André Géométrie des algèbres de Lie locales de Kirillov, J. Math. Pures Appl., IX. Sér., Volume 63 (1984) no. 4, pp. 407-484 | Zbl

[5] Iversen, Birger Cohomology of sheaves, Universitext, Springer, 1986 | Zbl

[6] Lee, John M. Introduction to smooth manifolds, Graduate Texts in Mathematics, 218, Springer, 2013 | MR | Zbl

[7] Lichnerowicz, André Les variétés de Poisson et leurs algébres de Lie associées, J. Differ. Geom., Volume 12 (1977), pp. 253-300 | DOI | Zbl

[8] Meng, Lingxu Morse-Novikov cohomology for blow-ups of complex manifolds, (2018) (https://arxiv.org/abs/1806.06622v3)

[9] Meng, Lingxu Mayer–Vietoris systems and their applications (2019) (https://arxiv.org/abs/1811.10500v3)

[10] Novikov, S. P. The Hamiltonian formalism and a multivalued analogue of Morse theory (Russian), Usp. Mat. Nauk, Volume 37 (1982) no. 5, pp. 3-43 | Zbl

[11] Ornea, Liviu; Vuletescu, Victor; Verbitsky, Misha Classification of non-Kähler surfaces and locally conformally Kähler geometry (2018) (https://arxiv.org/abs/1810.05768v2)

[12] Rao, Sheng; Yang, Song; Yang, Xiangdong Dolbeault cohomologies of blowing up complex manifolds, J. Math. Pures Appl., Volume 130 (2019), pp. 68-92 | MR | Zbl

[13] Rao, Sheng; Yang, Song; Yang, Xiangdong Dolbeault cohomologies of blowing up complex manifolds II: bundle-valued case, J. Math. Pures Appl., Volume 130 (2020), pp. 1-38 | DOI | MR | Zbl

[14] Rao, Sheng; Yang, Song; Yang, Xiangdong; Yu, Xun Hodge cohomology on blow-ups along subvarieties (2019) (https://arxiv.org/abs/1907.13281)

[15] Voisin, Claire Hodge theory and complex algebraic geometry. I, Cambridge Studies in Advanced Mathematics, 76, Cambridge University Press, 2002 (translated from the French original by Leila Schneps) | MR | Zbl

[16] Wells, R. O. jun. Comparison of de Rham and Dolbeault cohomology for proper surjective mappings, Pac. J. Math., Volume 53 (1974), pp. 281-300 | DOI | MR | Zbl

[17] Yang, Xiangdong; Zhao, Guosong A note on the Morse-Novikov cohomology of blow-ups of locally conformal Kähler manifolds, Bull. Aust. Math. Soc., Volume 91 (2015) no. 1, pp. 155-166 | DOI | Zbl

[18] Zou, Yongpan Morse–Novikov cohomology of blowing up complex manifolds and deformation of CR structure, Masters thesis, Wuhan University (2019)

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