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Positive \(\partial\bar\partial\)-closed currents and non-Kähler geometry. (English) Zbl 0735.32008
Some new results on positive \(\partial\bar\partial\)-closed currents (in particular a Support Theorem) are applied to modifications \(f:\tilde M\to M\). The main result in this topic is that if \(f:\tilde M\to M\) is a modification, \(M\) and \(\tilde M\) are compact complex manifolds and \(M\) is Kähler, then \(\tilde M\) is balanced. Moreover, under suitable hypotheses on the map, the Kähler degrees of \(\tilde M\) corresponds to homological properties of the exceptional set of the modification. More examples of \(p\)-Kähler manifolds are discussed in the last section of the paper.

32C30 Integration on analytic sets and spaces, currents
32Q15 Kähler manifolds
53C55 Global differential geometry of Hermitian and Kählerian manifolds
Full Text: DOI
[1] Alessandrini, L., and Andrentia, M. Closed transverse (p, p)-forms on compact complex manifolds. Compositio Math.61, 181–200 (1987). Erratum ibid.63, 143 (1987). · Zbl 0619.53019
[2] Alessandrini, L., and Bassanelli, G. Compactp-Kähler manifolds. Geometriac Dedicata38, 199–210 (1991). · Zbl 0724.53040
[3] Alessandrini, L., and Bassanelli, G. A balanced proper modification ofP 3. Comment. Math. Helvetici66, 505–511 (1991). · Zbl 0761.53034
[4] Barlet, D. Convexité de l’espace des cycles. Bull. Soc. Math. France106, 373–397 (1978). · Zbl 0395.32009
[5] Bigolin, B. Gruppi di Aeppli. Ann. Sc. Norm. Sup.23, 259–287 (1969).
[6] Bigolin, B. Osservazioni sulla coomologia del $$\(\backslash\)partial \(\backslash\)bar \(\backslash\)partial $$ . Ann. Sc. Norm. Sup.24, 571–583 (1970). · Zbl 0223.32015
[7] Fujiki, A. Closedness of the Douady spaces of compact Kähler spaces. Publ. RIMS Kyoto14, 1–52 (1978). · Zbl 0409.32016
[8] Gauduchon, P. Fibrés hermitiens a endomorphisme de Ricci non négatif. Bull. Soc. Math. France105, 113–140 (1977). · Zbl 0382.53045
[9] Grauert, H., and Remmert, R. Plurisubharmonische Funktionen in komplexen Räumen. Math. Z.65, 175–194 (1957). · Zbl 0070.30403
[10] Grauert, H., and Remmert, R. Coherent Analytic Sheaves. Berlin: Springer Verlag 1984.
[11] Grauert, H., and Riemenschneider, O. Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen. Inv. Math.11, 263–292 (1970). · Zbl 0202.07602
[12] Harvey, R. Removable singularities for positive currents. Am. J. Math.96, 67–78 (1974). · Zbl 0293.32015
[13] Harvey, R. Holomorphic chains and their boundaries. Proc. Symp. Pure Math., Vol. 30, Part 1, pp. 309–382. Providence, RI: American Mathematical Society 1977. · Zbl 0374.32002
[14] Harvey, R., and Knapp, A. W. Positive (p, p)-forms, Wirtinger’s inequality and currents. In: Proceedings Tulane Univ. Program on Value Distribution Theory in Complex Analysis and Related Topics in Differential Geometry 1972–73, pp. 43–62, New York: Marcel Dekker 1974. · Zbl 0287.53046
[15] Harvey, R., and Lawson, J. R. An intrinsec characterization of Kähler manifolds. Inv. Math.74, 169–198 (1983). · Zbl 0553.32008
[16] Hironaka H. Flattening theorems in complex analytic geometry. Am. J. Math.97, 503–547 (1975). · Zbl 0307.32011
[17] Hörmander, L. The Analysis of Linear Partial Differential Operators I. Grundlehren der mat. Wissenschaften 256. Berlin: Springer-Verlag 1983.
[18] King, J. R. The currents defined by analytic varieties. Acta Math.127, 185–220 (1971). · Zbl 0224.32008
[19] Lelong, P. Plurisubharmonic Functions and Positive Diffential Form. New York: Gordon and Breach 1969. · Zbl 0195.11604
[20] Michelson, M. L. On the existence of special metrics in complex geometry. Acta Math.143, 261–295 (1983).
[21] Miyaoka, Y. Extension theorems for Kähler metrics. In: Proc. Japan Acad.50, 407–410 (1974). · Zbl 0354.32010
[22] Sibony, N. Quelques problèmes de prolongement de courants en analyse complexe. Duke Math. J.52, 157–197 (1985). · Zbl 0578.32023
[23] Siu, Y. T. Analyticity of sets associated to Lelong numbers and the extension of closed positive currents. Inv. Math.27, 53–156 (1974). · Zbl 0289.32003
[24] Varouchas J. Propriétés cohomologiques d’une classe de variétés analitiques complexes compacies. Sem. d’Analyse Lelong-Dolbeault-Skoda 1983–84, Lecture Notes in Math. Vol. 1198, pp. 233–243, Berlin: Springer-Verlag 1985.
[25] Varouchas, J. Sur l’image d’une variété Kähleriénne compacie. Seminalre Norguet 1983–84, Lecture Notes in Math., Vol. 1188, pp. 245–259. Berlin: Springer-Verlag 1985.
[26] Varouchas, J. Kähler spaces and proper open morphisms. Math. Ann.283, 13–52 (1989). · Zbl 0632.53059
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