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On deformations of Kähler spaces. I. (English) Zbl 0584.32042

Kähler spaces, i.e. Kähler manifolds with singularities were introduced by Grauert, and their relative situations are defined by Fujiki as Kähler maps. Contrary to the case of Kähler manifolds, local deformations of Kähler spaces are in general not Kähler. In this paper the following theorem is proved: Let \(f: X\to S\) be a proper holomorphic map and \(X_ s\) a fiber over \(s\in S\). If all infinitesimal neighborhoods of \(X_ s\) are Kähler spaces, then f is Kähler in a neighborhood of s.
Furthermore, the author introduces the notion of weakly Kähler metrics and weakly Kähler maps and prove a theorem similar to the above one for weakly Kähler maps. If f is flat, then the obstructions for lifting a real analytic weakly Kähler metric on \(X_ s\) to its infinitesimal neighborhoods are in the cokernel of the natural map \(H^ 2(X_ s,{\mathfrak R})\to H^ 2(X_ s,{\mathcal O}_{X_ s}).\) Therefore one concludes that if \(X_ s\) is Kähler and if the above natural map is surjective, then f is weakly Kähler in s.
Certain sufficient condition for a weakly Kähler map to be a Kähler map is also proved.
Reviewer: E.Horikawa

MSC:

32G07 Deformations of special (e.g., CR) structures
32H35 Proper holomorphic mappings, finiteness theorems
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
58H15 Deformations of general structures on manifolds
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References:

[1] Banica, C.: Sur les fibres infinitesimales d’un morphisme propre d’espaces complexes. In: S?m. F. Norguet, Functions de Plusieurs Variables Complexes IV. Lecture Notes in Mathematics807. Berlin-Heidelberg-New York: Springer 1980 · Zbl 0445.32019
[2] Bingener, J.: Endlichkeit der differenzierbaren de Rham-Kohomologie. Math. Z.161, 101-117 (1978) · Zbl 0378.32008 · doi:10.1007/BF01214923
[3] Bingener, J.: Offenheit der Versalit?t in der analytischen Geometrie. Math. Z.173, 241-281 (1980) · Zbl 0493.32020 · doi:10.1007/BF01159663
[4] Bingener, J.: On deformations of K?hler spaces II. Preprint · Zbl 0584.32043
[5] Bingener, J., Flenner, H.: Variation of the divisor class group. Preprint · Zbl 0542.14003
[6] Blanchard, A.: Sur les vari?t?s analytiques complexes. Ann. Sci. ?cole Norm. Sup.73, 157-202 (1956) · Zbl 0073.37503
[7] Deligne, P.: Th?or?me de Lefschetz et crit?res de d?g?n?rescence de suites spectrales. Inst. Hautes Etudes Sci. Publ. Math.35, 107-126 (1968) · Zbl 0159.22501 · doi:10.1007/BF02698925
[8] Deligne, P.: Th?orie de Hodge, II. Inst. Hautes Etudes Sci. Publ. Math.40, 5-58 (1971) · Zbl 0219.14007 · doi:10.1007/BF02684692
[9] Forster, O., Knorr, K.: Relativ-analytische R?ume und die Koh?renz von Bildgarben. Invent. Math.16, 113-160 (1972) · Zbl 0242.32020 · doi:10.1007/BF01391214
[10] Fujiki, A.: Closedness of the Douady spaces of Compact K?hler spaces. Publ. Res. Inst. Math. Sci.14, 1-52 (1978) · Zbl 0409.32016 · doi:10.2977/prims/1195189279
[11] Fujiki, A.: On Automorphism Groups of Compact K?hler Manifolds. Invent. Math.44, 225-258 (1978) · Zbl 0367.32004 · doi:10.1007/BF01403162
[12] Grauert, H.: ?ber Modifikationen und exzeptionelle analytische Mengen. Math. Ann.146, 331-368 (1962) · Zbl 0173.33004 · doi:10.1007/BF01441136
[13] Grothendieck, A., Dieudonn?, J.: ?l?ments de g?om?trie alg?brique. Inst. Hautes Etudes Sci. Publ. Math.4, 8, 11, 17, 20, 24, 28, 32 (1960-1967)
[14] Houzel, C.: Espaces analytiques relatifs et th?or?me de finitude. Math. Ann.205, 13-54 (1973) · Zbl 0264.32012 · doi:10.1007/BF01432513
[15] Katz, N. M.: Algebraic Solutions of Differential Equations (p-Curvature and the Hodge Filtration). Invent. Math.18, 1-118 (1972) · Zbl 0278.14004 · doi:10.1007/BF01389714
[16] Kiehl, R.: Relativ analytische R?ume. Invent. Math.16, 40-112 (1972) · Zbl 0242.32019 · doi:10.1007/BF01391213
[17] Kodaira, K., Morrow, J.: Complex Manifolds. New York: Holt, Rinehart and Winston 1971 · Zbl 0325.32001
[18] Kodaira, K., Spencer, D. C.: Deformations of complex analytic structures, III. Stability theorems for complex structures. Ann. of Math.71, 43-76 (1960) · Zbl 0128.16902 · doi:10.2307/1969879
[19] Kuhlmann, N.: ?ber holomorphe Abbildungen mit projektiven Fasern. Math. Z.135, 43-54 (1973) · Zbl 0268.32013 · doi:10.1007/BF01214304
[20] Moishezon, B. G.: Singular K?hlerian Spaces. In: Proceedings of the International Conference on Manifolds and Related Topics in Topology (Tokyo 1973), p. 343-351. Tokyo: University of Tokyo Press 1975
[21] Verdier, J. L.: Classe d’homologie associ?e a un cycle. In: S?minaire de g?om?trie analytique. Douady-Verdier, Exp. 7, pp. 101-151. Ast?risque36, 37 (1976). Paris: Soc. Math. France 1976
[22] Weil, A.: Vari?t?s K?hl?riennes. Paris: Hermann 1971
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