Abstract
In this paper, we consider a proper modification \(f : \tilde{M} \rightarrow M\) between complex manifolds, and study when a generalized p-Kähler property goes back from M to \(\tilde{M}\). When f is the blow-up at a point, every generalized p-Kähler property is conserved, while when f is the blow-up along a submanifold, the same is true for \(p=1\). For \(p=n-1\), we prove that the class of compact generalized balanced manifolds is closed with respect to modifications, and we show that the fundamental forms can be chosen in the expected cohomology class. We also get some partial results in the non-compact case; finally, we end the paper with some examples of generalized p-Kähler manifolds.
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References
Alessandrini, L.: Classes of compact non-Kähler manifolds. C. R. Acad. Sci. Paris Ser. I 349, 1089–1092 (2011)
Alessandrini, L.: Holomorphic Submersions onto Kähler or Balanced Manifolds. arXiv:1410.2396 [math.DG], to appear in Tohoku Math. J. (2016)
Alessandrini, L., Bassanelli, G.: Compact \(p\)-Kähler manifolds. Geom. Dedicata 38, 199–210 (1991)
Alessandrini, L., Bassanelli, G.: Smooth proper modifications of compact Kähler manifolds. In: Diederich, K. (ed.) Complex Analysis. Aspects of Mathematics E, vol. 17, pp. 1–7. Vieweg, Braunschweig (1991)
Alessandrini, L., Bassanelli, G.: A balanced proper modification of \({\mathbb{P}}_3\). Comment. Math. Helvetici 66, 505–511 (1991)
Alessandrini, L., Bassanelli, G.: Positive \({\partial }\overline{\partial }\)-closed currents and non-Kähler geometry. J. Geom. Anal. 2, 291–316 (1992)
Alessandrini, L., Bassanelli, G.: Metric properties of manifolds bimeromorphic to compact Kähler spaces. J. Differ. Geom. 37, 95–121 (1993)
Alessandrini, L., Bassanelli, G.: Modifications of compact balanced manifolds. C. R. Acad. Sci. Paris Ser. I 320, 1517–1522 (1995)
Alessandrini, L., Bassanelli, G., Leoni, M.: 1-Convex manifolds are p-Kähler. Abh. Math. Sem. Univ. Hamburg 72, 255–268 (2002)
Bassanelli, G.: A cut-off theorem for plurisubharmonic currents. Forum Math. 6, 567–595 (1994)
Deligne, P., Griffiths, P., Morgan, J., Sullivan, D.: Real homotopy theory of Kähler manifolds. Invent. Math. 29, 245–274 (1975)
Demailly, J.-P.: Complex Analytic and Algebraic Geometry. http://www.fourier.ujf-grenoble.fr/demailly/books.html
Egidi, N.: Special metrics on compact complex manifolds. Differ. Geom. Appl. 14, 217–234 (2001)
Fino, A., Tomassini, A.: Blow-ups and resolutions of strong Kähler with torsion metrics. Adv. Math. 221, 914–935 (2009)
Fu, J., Xiao, J.: Relations between the Kähler cone and the balanced cone of a Kähler manifold. Adv. Math. 263, 230–252 (2014)
Fujiki, A.: Closedness of the Douady spaces of Compact Kähler spaces. Publ. RIMS Kyoto 14, 1–52 (1978)
Grauert, H., Remmert, G.: Coherent Analytic Sheaves. Springer, Berlin (1984)
Griffiths, PhA, Harris, J.: Principles of Algebraic Geometry, Reprint of the 1978 Original, Wiley Classics Library. Wiley, New York (1994)
Hartshorne, R.: Algebraic Geometry. Springer, New York (1977)
Harvey, R., Knapp, A.W.: Positive \((p,p)\)-forms, Wirtinger’s inequality and currents. In: Proceedings of the Tulane University. 1972–1973, pp. 43–62. Dekker, New York (1974)
Harvey, R., Lawson, H.B.: An intrinsic characterization of Kähler manifolds. Invent. Math. 74, 169–198 (1983)
Lelong, P.: Plurisubharmonic Functions and Positive Differential Forms. Gordon and Breach, Dunod, Paris (1968)
Michelsohn, M.L.: On the existence of special metrics in complex geometry. Acta Math. 149, 261–295 (1982)
Popovici, D.: Deformation limits of projective manifolds: Hodge numbers and strongly Gauduchon metrics. Invent. Math. 194, 515–534 (2013)
Popovici, D.: Stability of strongly Gauduchon manifolds under modifications. J. Geom. Anal. 23, 653–659 (2013)
Popovici, D., Ugarte, L.: The sGG Class of Compact Complex Manifolds. arXiv e-print arXiv:1407.5070v1 [math.DG]
Streets, J., Tian, G.: A parabolic flow of pluriclosed metrics. Int. Math. Res. Not. 16, 3101–3133 (2010)
Sullivan, D.: Cycles for the dynamical study of foliated manifolds and complex manifolds. Invent. Math. 36, 225–255 (1976)
Varouchas, J.: Sur l’image d’une variété Kählerienne compacte, Séminaire Norguet, années 1983–1984. Lecture Notes in Mathematics, vol. 1188, pp. 245–259. Springer, Berlin (1985)
Varouchas, J., Propriétés cohomologiques d’une classe de variétés analytiques complexes compactes, Séminaire d’analyse Lelong, P., Dolbeault, P., Skoda, H., années 1983–1984. Lecture Notes in Mathematics, vol. 1198, pp. 233–243. Springer, Berlin (1986)
Varouchas, J.: Kähler spaces and proper open morphisms. Math. Ann. 283, 13–52 (1989)
Xiao, J.: On Strongly Gauduchon Metrics of Compact Complex Manifolds. arXiv e-print arXiv:1306.0833v1 [math.DG]
Yachou, A.: Une classe de variétés semi-kählériennes. C. R. Acad. Sci. Paris Ser. I 321, 763–765 (1995)
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Alessandrini, L. Proper Modifications of Generalized p-Kähler Manifolds. J Geom Anal 27, 947–967 (2017). https://doi.org/10.1007/s12220-016-9705-z
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DOI: https://doi.org/10.1007/s12220-016-9705-z