×
Campana, F.

Remarks on nilpotent Kähler groups. (Remarques sur les groupes de Kähler nilpotents.) (French) Zbl 0829.32006

Ann. Sci. Éc. Norm. Supér. (4) 28, No. 3, 307-316 (1995).
The properties of the fundamental group \(\pi_1 (X)\) for a Kähler manifold \(X\) are discussed. By \(\text{Alb} (X)\) denote the Albanese manifold of \(X\). If \(G\) is a group, then define \(G_1 = G\), \(G_{n + 1} = [G, G_{n + 1}]\), \(G_\infty = \bigcap_{n \geq 1} G_n\), \(G_n' = \sqrt {G_n}\), \(1 \leq n \leq \infty\). For \(E \subset G\) set \(\sqrt {E} = \{g \in E \mid g^n \in E\) for some integer \(n > 0\}\).
The following main theorem is proved. Theorem: Let \(X\) be a compact Kähler manifold such that the natural mapping \(\lambda : H^2 (\text{Alb}(X); \mathbb{R}) \to H^2 (X,\mathbb{R})\) is injective. Then \(G_2' = G_\infty'\) and the nilpotent completion \(G^{\text{nilp}}\) of the group \(G = \pi_1 (X)\) is abelian. The mapping \(\lambda\) is injective if and only if \(\lambda^{2,0} : \Lambda^2 H^0 (X, \Omega^1_X) \to H^0 (X, \Omega^2_X)\) and \(\lambda^{1,1} : H^0 (X, \Omega^1_X) \otimes H^0 (X, \Omega^1_X) \to H^1 (X, \Omega^1_X)\) are injective.
Reviewer: V.V.Trofimov (Moskva)

Cited in 2 Reviews
Cited in 18 Documents

MSC:

32Q15 Kähler manifolds

Keywords:

Albanese morphism; nilpotent completion; fundamental group; Kähler manifold
Cite Review PDF
Full Text: DOI Numdam EuDML

References:

[1] D. ARAPURA , P. BRESSLER et M. RAMACHANDRAN , On the fundamental group of a compact Kähler manifold (Duke Math. J., vol. 68, 1992 , p. 477-488). Article | MR 94e:57040 | Zbl 0783.57009 · Zbl 0783.57009 · doi:10.1215/S0012-7094-92-06819-0
[2] F. CAMPANA , On twistor spaces of class C (J. Diff. Geom., vol. 33, 1991 , p. 541-549). MR 92g:32059 | Zbl 0694.32017 · Zbl 0694.32017
[3] F. CAMPANA , Remarques sur les groupes de Kähler nilpotents (C.R. Acad. Sci. Paris, vol. 17, 1993 , p. 777-780). MR 94k:32048 | Zbl 0801.53053 · Zbl 0801.53053
[4] J. CARLSON et D. TOLEDO , Notes on nilpotent Kähler group , 1992 .
[5] P. DELIGNE , Théorie de Hodge III (Publi. Math. I.H.E.S., vol. 44, 1974 , p. 5-77). Numdam | MR 58 #16653b | Zbl 0237.14003 · Zbl 0237.14003 · doi:10.1007/BF02685881
[6] DELIGNE - GRIFFITHS - MORGAN - SULLIVAN , Real homotopy theory of Kähler manifolds (Inv. Math., vol. 29, 1975 , p. 245-274). MR 52 #3584 | Zbl 0312.55011 · Zbl 0312.55011 · doi:10.1007/BF01389853
[7] A. FUJIKI , On automorphism groups of compact Kähler manifolds (Inv. Math., vol. 44, 1978 , p. 225-258). MR 58 #1285 | Zbl 0367.32004 · Zbl 0367.32004 · doi:10.1007/BF01403162
[8] FULTON - LAZARSFELD , Connectivity in algebraic geometry (LNM n^\circ 862, p. 26-92). MR 83i:14002 | Zbl 0484.14005 · Zbl 0484.14005
[9] P. GAUDUCHON , Variétés riemanniennes autoduales (Séminaire Bourbaki, Exposé 767, mars 1993 ). Numdam | Zbl 0789.53026 · Zbl 0789.53026
[10] GORESKI - MAC PHERSON , Stratified Morse theory , Springer Verlag, 1988 .
[11] M. GROMOV , Groups of polynomial growth and expanding maps (Publ. Math. I.H.E.S., vol. 53, 1981 , p. 53-73). Numdam | MR 83b:53041 | Zbl 0474.20018 · Zbl 0474.20018 · doi:10.1007/BF02698687
[12] M. GROMOV , Metric invariants of Kähler manifolds (Preprint I.H.E.S., 1992 ). · Zbl 0888.53047
[13] R. HAIN , The Geometry of the Mixed Hodge Structure on the Fundamental Group (Proc. Symp. Pure Math., vol. 46, 1984 , p. 247-282). MR 89g:14010 | Zbl 0654.14006 · Zbl 0654.14006
[14] N. HITCHIN , Kählerian twistor spaces (Proc. Lond. Math. Soc., vol. 43, 1981 , p. 133-150). MR 84b:32014 | Zbl 0474.14024 · Zbl 0474.14024 · doi:10.1112/plms/s3-43.1.133
[15] Y. KAWAMATA , Characterization of abelian varieties (Comp.Math., vol. 43, 1981 , p. 253-276). Numdam | MR 83j:14029 | Zbl 0471.14022 · Zbl 0471.14022
[16] SOMMESE - VAN DE VEN , Homotopy groups of pullbacks of varieties (Nagoya Math. J., vol. 102, 1986 , p. 79-90). Article | MR 87i:14016 | Zbl 0564.14010 · Zbl 0564.14010
[17] J. STALLINGS , Homology and central series of groups (J. Algebra, vol. 2, 1965 , p. 170-181). MR 31 #232 | Zbl 0135.05201 · Zbl 0135.05201 · doi:10.1016/0021-8693(65)90017-7
[18] D. SULLIVAN , Infinitesimal computations in topology (Publ. Math. I.H.E.S., vol. 47, 1977 , p. 269-331). Numdam | MR 58 #31119 | Zbl 0374.57002 · Zbl 0374.57002 · doi:10.1007/BF02684341
[19] C.H. TAUBES , The existence of self dual conformal structures (J. Diff. Geom., vol. 36, 1992 , p. 163-253). MR 93j:53063 | Zbl 0822.53006 · Zbl 0822.53006
[20] K. UENO , Classification theory of algebraic varieties (LNM n^\circ 439, Springer Verlag, 1975 ). Zbl 0299.14007 · Zbl 0299.14007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.