Meng, Lingxu The heredity and bimeromorphic invariance of the \(\partial\bar{\partial}\)-lemma property. (English) Zbl 1477.32050 C. R., Math., Acad. Sci. Paris 359, No. 6, 645-650 (2021). Summary: We give a simple proof of a result on the \(\partial\bar{\partial}\)-lemma property under a blow-up transformation by Deligne-Griffiths-Morgan-Sullivan’s criterion. Here, we use an explicit blow-up formula for Dolbeault cohomology given in our previous work, which can be induced by a morphism expressed on the level of spaces of forms and currents. At last, we discuss the heredity and bimeromorphic invariance of the \(\partial\bar{\partial}\)-lemma property. Cited in 2 Documents MSC: 32Q99 Complex manifolds 32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators Keywords:\(\partial\bar{\partial}\)-lemma; Dolbeault cohomology PDFBibTeX XMLCite \textit{L. Meng}, C. R., Math., Acad. Sci. Paris 359, No. 6, 645--650 (2021; Zbl 1477.32050) Full Text: DOI arXiv References: [1] Abramovich, Dan; Karu, Kalle; Matsuki, Kenji; Wlodarczyk, Jaroslaw, Torification and factorization of birational maps, J. Am. Math. Soc., 15, 3, 531-572 (2002) · Zbl 1032.14003 [2] Alessandrini, Lucia, Proper modifications of generalized \(p\)-Kähler manifolds, J. Geom. Anal., 27, 2, 947-967 (2017) · Zbl 1369.53051 [3] Angella, Daniele; Suwa, Tatsuo; Tardini, Nicoletta; Tomassini, Adriano, Note on Dolbeault cohomology and Hodge structures up to bimeromorphisms, Complex Manifolds, 7, 1, 194-214 (2002) · Zbl 1462.32021 [4] Angella, Daniele; Tomassini, Adriano, On the \(\partial \bar{\partial } \)-lemma and Bott-Chern cohomology, Invent. Math., 192, 3, 71-81 (2013) · Zbl 1271.32011 [5] Buchdahl, Nicholas, On compact Kähler surfaces, Ann. Inst. Fourier, 49, 1, 287-302 (1999) · Zbl 0926.32025 [6] Deligne, Pierre; Griffiths, Phillip; Morgan, John; Sullivan, Dennis, Real homotopy theory of Kähler manifolds, Invent. Math., 29, 3, 245-274 (1975) · Zbl 0312.55011 [7] Friedman, Robert, The \(\partial \bar{\partial } \)-lemma for general Clemens manifolds, Pure Appl. Math. Q., 15, 4, 1001-1028 (2019) · Zbl 1439.32037 [8] Lamari, Ahcène, Courants kählériens et surfaces compactes, Ann. Inst. Fourier, 49, 1, 263-285 (1999) · Zbl 0926.32026 [9] Meng, Lingxu, Mayer-Vietoris systems and their applications (2019) [10] Meng, Lingxu, Leray-Hirsch theorem and blow-up formula for Dolbeault cohomology, Ann. Mat. Pura Appl., 199, 5, 1997-2014 (2020) · Zbl 1476.32008 [11] Parshin, Alexei N., On a certain generalization of Jacobian manifold, Izv. Akad. Nauk SSSR, Ser. Mat., 30, 1, 175-182 (1966) · Zbl 0163.15303 [12] Rao, Sheng; Yang, Song; Yang, Xiangdong, Dolbeault cohomologies of blowing up complex manifolds, J. Math. Pures Appl., 130, 68-92 (2019) · Zbl 1425.32028 [13] Rao, Sheng; Yang, Song; Yang, Xiangdong, Dolbeault cohomologies of blowing up complex manifolds II: bundle-valued case, J. Math. Pures Appl., 133, 1-38 (2020) · Zbl 1430.32010 [14] Rao, Sheng; Zou, Yongpan, \( \partial \bar{\partial } \)-lemma, double complex and \({L}^2\) cohomology (2020) [15] Stelzig, Jonas, Double complexes and Hodge stuctures as vector bundles (2018) · Zbl 1397.14002 [16] Stelzig, Jonas, The double complex of a blow-up, Int. Math. Res. Not., 2019 (2019) [17] Yang, Song; Yang, Xiangdong, Bott-Chern blow-up formula and bimeromorphic invariance of the \(\partial \bar{\partial } \)-lemma for threefolds, Trans. Am. Math. Soc., 373, 12, 8885-8909 (2020) · Zbl 1471.32039 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.