Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scharlau, Rudolf / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard

4 Issues per year

IMPACT FACTOR 2017: 0.734

CiteScore 2017: 0.70

SCImago Journal Rank (SJR) 2017: 0.695
Source Normalized Impact per Paper (SNIP) 2017: 0.891

Mathematical Citation Quotient (MCQ) 2017: 0.62

See all formats and pricing
More options …
Ahead of print


On non-Kähler degrees of complex manifolds

Daniele Angella
  • Dipartimento di Matematica e Informatica “Ulisse Dini”, Università degli Studi di Firenze, viale Morgagni 67/a, 50134 Firenze, Italy
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Adriano Tomassini
  • Corresponding author
  • Dipartimento di Scienze Matematiche, Fisiche, ed Informatiche, Plesso Matematico e Informatico, Università di Parma, Parco Area delle Scienze 53/A, 43124, Parma, Italy.
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Misha Verbitsky
  • Instituto Nacional de Matemática Pura e Aplicada, Estrada Dona Castor ina, 110, Jardim Botânico, CEP 22460-320, Rio de Janeiro, RJ, Brasil
  • Laboratory of Algebraic Geometry, Faculty of Mathematics, National Research University HSE, 7 Vavilova Str. Moscow, Russia
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2018-10-25 | DOI: https://doi.org/10.1515/advgeom-2018-0026


We study cohomological properties of complex manifolds. In particular, under suitable metric conditions, we extend to higher dimensions a result by A. Teleman, which provides an upper bound for the Bott– Chern cohomology in terms of Betti numbers for compact complex surfaces according to the dichotomy b1 even or odd.

Keywords: Cohomology; complex surface; non-Kähler

MSC 2010: 32Q55; 32C35; 53C55


  • [1]

    A. Aeppli, On the cohomology structure of Stein manifolds. In: Proc. Conf. Complex Analysis (Minneapolis, Minn., 1964), 58–70, Springer 1965. MR0221536 Zbl 0166.33902Google Scholar

  • [2]

    D. Angella, The cohomologies of the Iwasawa manifold and of its small deformations. J. Geom. Anal. 23 (2013), 1355–1378. MR3078358 Zbl 1278.32013Google Scholar

  • [3]

    D. Angella, G. Dloussky, A. Tomassini, On Bott–Chern cohomology of compact complex surfaces. Ann. Mat. Pura Appl. (4) 195 (2016), 199–217. MR3453598 Zbl 1343.32012Google Scholar

  • [4]

    D. Angella, M. G. Franzini, F. A. Rossi, Degree of non-Kählerianity for 6-dimensional nilmanifolds. Manuscripta Math. 148 (2015), 177–211. MR3377754 Zbl 1326.57057Google Scholar

  • [5]

    D. Angella, H. Kasuya, Bott–Chern cohomology of solvmanifolds. Ann. Global Anal. Geom. 52 (2017), 363–411. MR3735904 Zbl 06825293Google Scholar

  • [6]

    D. Angella, H. Kasuya, Cohomologies of deformations of solvmanifolds and closedness of some properties. North-West. Eur. J. Math. 3 (2017), 75–105, i. MR3654464 Zbl 06828939Google Scholar

  • [7]

    D. Angella, N. Tardini, Quantitative and qualitative cohomological properties for non-Kähler manifolds. Proc. Amer. Math. Soc. 145 (2017), 273–285. MR3565379 Zbl 1355.32020Google Scholar

  • [8]

    D. Angella, A. Tomassini, On cohomological decomposition of almost-complex manifolds and deformations. J. Symplectic Geom. 9 (2011), 403–428. MR2817781 Zbl 1228.53035Google Scholar

  • [9]

    D. Angella, A. Tomassini, On the ¯-lemma and Bott–Chern cohomology. Invent. Math. 192 (2013), 71–81. MR3032326 Zbl 1271.32011Google Scholar

  • [10]

    D. Angella, A. Tomassini, Inequalities à la Frölicher and cohomological decompositions. J. Noncommut. Geom. 9 (2015), 505–542. MR3359019 Zbl 1325.32018Google Scholar

  • [11]

    R. Bott, S. S. Chern, Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections. Acta Math. 114 (1965), 71–112. MR0185607 Zbl 0148.31906Google Scholar

  • [12]

    N. Buchdahl, On compact Kähler surfaces. Ann. Inst. Fourier (Grenoble) 49 (1999), 287–302. MR1688136 Zbl 0926.32025Google Scholar

  • [13]

    K. Chan, Y.-H. Suen, A Frölicher-type inequality for generalized complex manifolds. Ann. Global Anal. Geom. 47 (2015), 135–145. MR3313137 Zbl 1309.32007Google Scholar

  • [14]

    P. Deligne, P. Griffiths, J. Morgan, D. Sullivan, Real homotopy theory of Kähler manifolds. Invent. Math. 29 (1975), 245–274. MR0382702 Zbl 0312.55011Google Scholar

  • [15]

    N. Enrietti, A. Fino, L. Vezzoni, Tamed symplectic forms and strong Kähler with torsion metrics. J. Symplectic Geom. 10 (2012), 203–223. MR2926995 Zbl 1248.53070Google Scholar

  • [16]

    P. Gauduchon, La constante fondamentale d’un fibré en droites au-dessus d’une variété hermitienne compacte. C. R. Acad. Sci. Paris Sér. A-B 281 (1975), Aii, A393–A396. MR0425191 Zbl 0312.53046Google Scholar

  • [17]

    A. Gray, L. M. Hervella, The sixteen classes of almost Hermitian manifolds and their linear invariants. Ann. Mat. Pura Appl. (4) 123 (1980), 35–58. MR581924 Zbl 0444.53032Google Scholar

  • [18]

    J. Jost, S.-T. Yau, A nonlinear elliptic system for maps from Hermitian to Riemannian manifolds and rigidity theorems in Hermitian geometry. Acta Math. 170 (1993), 221–254. MR1226528 Zbl 0806.53064Google Scholar

  • [19]

    K. Kodaira, On the structure of compact complex analytic surfaces. I. Amer. J. Math. 86 (1964), 751–798. MR0187255 Zbl 0137.17501Google Scholar

  • [20]

    A. Lamari, Courants kählériens et surfaces compactes. Ann. Inst. Fourier (Grenoble) 49 (1999), 263–285. MR1688140 Zbl 0926.32026Google Scholar

  • [21]

    A. Latorre, L. Ugarte, R. Villacampa, On the Bott–Chern cohomology and balanced Hermitian nilmanifolds. Internat. J. Math. 25 (2014), 1450057, 24 pages. MR3225581 Zbl 1302.32013Google Scholar

  • [22]

    M. Lübke, A. Teleman, The Kobayashi–Hitchin correspondence. World Scientific, River Edge, NJ 1995. MR1370660 Zbl 0849.32020Google Scholar

  • [23]

    A. McHugh, Narrowing cohomologies on complex S6. Eur. J. Pure Appl. Math. 10 (2017), 440–454. MR3647611 Zbl 1366.53053Google Scholar

  • [24]

    Y. Miyaoka, Kähler metrics on elliptic surfaces. Proc. Japan Acad. 50 (1974), 533–536. MR0460730 Zbl 0354.32011Google Scholar

  • [25]

    M. Schweitzer, Autour de la cohomologie de Bott–Chern. Prépublication de l’Institut Fourier no. 703 (2007). Preprint 2007, arXiv:0709.3528v1[math.AG]Google Scholar

  • [26]

    Y. T. Siu, Every K3 surface is Kähler. Invent. Math. 73 (1983), 139–150. MR707352 Zbl 0557.32004Google Scholar

  • [27]

    J. Streets, G. Tian, A parabolic flow of pluriclosed metrics. Int. Math. Res. Not. 2010, no. 16, 3101–3133. MR2673720 Zbl 1198.53077Google Scholar

  • [28]

    A. Teleman, The pseudo-effective cone of a non-Kählerian surface and applications. Math. Ann. 335 (2006), 965–989. MR2232025 Zbl 1096.32011Google Scholar

  • [29]

    L. Ugarte, Hodge numbers of a hypothetical complex structure on the six sphere. Geom. Dedicata 81 (2000), 173–179. MR1772200 Zbl 0996.53046Google Scholar

About the article

E-mail: daniele.angella@unifi.it

Received: 2016-12-13

Published Online: 2018-10-25

Citation Information: Advances in Geometry, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2018-0026.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in