Abstract
In our previous works on deformation limits of projective and Moishezon manifolds, we introduced and made crucial use of the notion of strongly Gauduchon metrics as a reinforcement of the earlier notion of Gauduchon metrics. Using direct and inverse images of closed positive currents of type (1,1) and regularization, we now show that compact complex manifolds carrying strongly Gauduchon metrics are stable under modifications. This stability property, known to fail for compact Kähler manifolds, mirrors the modification stability of balanced manifolds proved by Alessandrini and Bassanelli.
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Alessandrini, L., Bassanelli, G.: Smooth proper modifications of compact Kähler manifolds. In: Proc. Internat. Workshop on Complex Analysis, Wuppertal, 1990. Complex Analysis, Aspects of Mathematics, vol. E17, pp. 1–7. Vieweg, Braunschweig (1991)
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, t, Ser. I 320, 1517–1522 (1995)
Blanchard, A.: Les variétés analytiques complexes. Ann. Sci. Éc. Norm. Super. 73, 157–202 (1958)
Demailly, J.-P.: Regularization of closed positive currents and intersection theory. J. Algebr. Geom. 1, 361–409 (1992)
Demailly, J.-P.: Complex analytic and algebraic geometry. http://www-fourier.ujf-grenoble.fr/~demailly/books.html
Gauduchon, P.: Le théorème de l’excentricité nulle. C. R. Acad. Sci. Paris, Sér A, t. 285, 387–390 (1977)
Grauert, H., Riemenschneider, O.: Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen. Invent. Math. 11, 263–292 (1970)
Hörmander, L.: Notions of Convexity. Progress in Mathematics, vol. 127. Birkhäuser, Boston (1994)
Meo, M.: Image inverse d’un courant positif fermé par une application analytique surjective. C. R. Acad. Sci. Paris, t, Sér. I 322, 1141–1144 (1996)
Michelsohn, M.L.: On the existence of special metrics in complex geometry. Acta Math. 143, 261–295 (1983)
Popovici, D.: Limits of projective manifolds under holomorphic deformations. arXiv:0910.2032v1
Popovici, D.: Limits of Moishezon manifolds under holomorphic deformations. arXiv:1003.3605v1
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Communicated by Jiaping Wang.
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Popovici, D. Stability of Strongly Gauduchon Manifolds Under Modifications. J Geom Anal 23, 653–659 (2013). https://doi.org/10.1007/s12220-011-9257-1
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DOI: https://doi.org/10.1007/s12220-011-9257-1
Keywords
- Closed positive (1,1)-current
- Compact complex Hermitian manifold
- Direct and inverse image of a current
- Proper holomorphic bimeromorphic map
- Strongly Gauduchon manifold