The fundamental group of compact Kähler threefolds

Claudon, Benoît¹ ; Höring, Andreas² ; Lin, Hsueh-Yung³

¹ Université Rennes 1, IRMAR-UMR 6625, Rennes, France, Institut Universitaire de France
² Université Côte d’Azur, CNRS, Laboratoire Jean-Alexandre Dieudonné, Nice, France, Institut Universitaire de France
³ Mathematisches Institut der Universität Bonn, Bonn, Germany

Geometry & topology, Tome 23 (2019) no. 7, pp. 3233-3271.

Résumé

Let $X$ be a compact Kähler manifold of dimension three. We prove that there exists a projective manifold $Y$ such that ${\pi }_1\left(X\right)\simeq {\pi }_1\left(Y\right)$. We also prove the bimeromorphic existence of algebraic approximations for compact Kähler manifolds of algebraic dimension $dimX-1$. Together with the work of Graf and the third author, this settles in particular the bimeromorphic Kodaira problem for compact Kähler threefolds.

Détail

DOI : 10.2140/gt.2019.23.3233

Classification : 14D07, 32J17, 32J27, 32Q55
Keywords: fundamental group, compact Kähler manifolds, algebraic approximations, elliptic fibrations

Bibliographie

[1] D Arapura, Homomorphisms between Kähler groups, from: "Topology of algebraic varieties and singularities" (editors J I Cogolludo-Agustín, E Hironaka), Contemp. Math. 538, Amer. Math. Soc. (2011) 95 | DOI

[2] W P Barth, K Hulek, C A M Peters, A Van De Ven, Compact complex surfaces, 4, Springer (2004) | DOI

[3] O Baues, J Riesterer, Virtually abelian Kähler and projective groups, Abh. Math. Semin. Univ. Hambg. 81 (2011) 191 | DOI

[4] A Beauville, Le nombre minimum de fibres singulières d’une courbe stable sur P1, from: "Seminar on pencils of curves of genus at least two", Astérisque 86, Soc. Math. France (1981) 97

[5] J Bingener, On deformations of Kähler spaces, I, Math. Z. 182 (1983) 505 | DOI

[6] K S Brown, Cohomology of groups, 87, Springer (1982) | DOI

[7] Y Brunebarbe, F Campana, Fundamental group and pluridifferentials on compact Kähler manifolds, Mosc. Math. J. 16 (2016) 651 | DOI

[8] N Buchdahl, Algebraic deformations of compact Kähler surfaces, Math. Z. 253 (2006) 453 | DOI

[9] N Buchdahl, Algebraic deformations of compact Kähler surfaces, II, Math. Z. 258 (2008) 493 | DOI

[10] F Campana, Remarques sur le revêtement universel des variétés kählériennes compactes, Bull. Soc. Math. France 122 (1994) 255 | DOI

[11] F Campana, Negativity of compact curves in infinite covers of projective surfaces, J. Algebraic Geom. 7 (1998) 673

[12] F Campana, Orbifolds, special varieties and classification theory, Ann. Inst. Fourier Grenoble 54 (2004) 499 | DOI

[13] F Campana, Orbifoldes géométriques spéciales et classification biméromorphe des variétés kählériennes compactes, J. Inst. Math. Jussieu 10 (2011) 809 | DOI

[14] F Campana, B Claudon, Abelianity conjecture for special compact Kähler 3–folds, Proc. Edinb. Math. Soc. 57 (2014) 55 | DOI

[15] F Campana, B Claudon, P Eyssidieux, Représentations linéaires des groups kählériens et de leurs analogues projectifs, J. Éc. polytech. Math. 1 (2014) 331 | DOI

[16] F Campana, B Claudon, P Eyssidieux, Représentations linéaires des groupes kählériens : factorisations et conjecture de Shafarevich linéaire, Compos. Math. 151 (2015) 351 | DOI

[17] F Campana, T Peternell, Complex threefolds with non-trivial holomorphic 2–forms, J. Algebraic Geom. 9 (2000) 223

[18] F Campana, Q Zhang, Compact Kähler threefolds of π1–general type, from: "Recent progress in arithmetic and algebraic geometry" (editors Y Kachi, S B Mulay, P Tzermias), Contemp. Math. 386, Amer. Math. Soc. (2005) 1 | DOI

[19] B Claudon, Invariance de la Γ–dimension pour certaines familles kählériennes de dimension 3, Math. Z. 266 (2010) 265 | DOI

[20] B Claudon, Smooth families of tori and linear Kähler groups, Ann. Fac. Sci. Toulouse Math. 27 (2018) 477 | DOI

[21] J P Demailly, Mesures de Monge–Ampère et caractérisation géométrique des variétés algébriques affines, 19, Soc. Math. France (1985) 124

[22] H Flenner, S Kosarew, On locally trivial deformations, Publ. Res. Inst. Math. Sci. 23 (1987) 627 | DOI

[23] A Fujiki, Closedness of the Douady spaces of compact Kähler spaces, Publ. Res. Inst. Math. Sci. 14 (1978) 1 | DOI

[24] O Fujino, S Mori, A canonical bundle formula, J. Differential Geom. 56 (2000) 167 | DOI

[25] P Graf, Algebraic approximation of Kähler threefolds of Kodaira dimension zero, Math. Ann. 371 (2018) 487 | DOI

[26] H Grauert, Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann. 146 (1962) 331 | DOI

[27] D Greb, S Kebekus, S J Kovács, Extension theorems for differential forms and Bogomolov–Sommese vanishing on log canonical varieties, Compos. Math. 146 (2010) 193 | DOI

[28] R Hartshorne, Algebraic geometry, 52, Springer (1977) | DOI

[29] A Höring, T Peternell, Minimal models for Kähler threefolds, Invent. Math. 203 (2016) 217 | DOI

[30] B Iversen, Cohomology of sheaves, Springer (1986) | DOI

[31] K Kodaira, On compact complex analytic surfaces, I, Ann. of Math. 71 (1960) 111 | DOI

[32] K Kodaira, On compact analytic surfaces, II, Ann. of Math. 77 (1963) 563 | DOI

[33] K Kodaira, On compact analytic surfaces, III, Ann. of Math. 78 (1963) 1 | DOI

[34] J Kollár, Shafarevich maps and plurigenera of algebraic varieties, Invent. Math. 113 (1993) 177 | DOI

[35] J Kollár, Lectures on resolution of singularities, 166, Princeton Univ. Press (2007)

[36] R Lazarsfeld, Positivity in algebraic geometry, I : Classical setting — line bundles and linear series, 48, Springer (2004) | DOI

[37] H Y Lin, Algebraic approximations of compact Kähler threefolds, preprint (2017)

[38] H Y Lin, Algebraic approximations of fibrations in abelian varieties over a curve, preprint (2018)

[39] N Nakayama, On Weierstrass models, from: "Algebraic geometry and commutative algebra, II" (editors H Hijikata, H Hironaka, M Maruyama, H Matsumura, M Miyanishi, T Oda, K Ueno), Kinokuniya (1988) 405 | DOI

[40] N Nakayama, Global structure of an elliptic fibration, Publ. Res. Inst. Math. Sci. 38 (2002) 451 | DOI

[41] N Nakayama, Local structure of an elliptic fibration, from: "Higher dimensional birational geometry" (editors S Mori, Y Miyaoka), Adv. Stud. Pure Math. 35, Math. Soc. Japan (2002) 185 | DOI

[42] Y Namikawa, Extension of 2–forms and symplectic varieties, J. Reine Angew. Math. 539 (2001) 123 | DOI

[43] E Sernesi, Deformations of algebraic schemes, 334, Springer (2006) | DOI

[44] J P Serre, Sur la topologie des variétés algébriques en caractéristique p, from: "Symposium internacional de topología algebraica International symposium on algebraic topology", Universidad Nacional Autónoma de México (1958) 24

[45] Y T Siu, Strong rigidity for Kähler manifolds and the construction of bounded holomorphic functions, from: "Discrete groups in geometry and analysis" (editor R Howe), Progr. Math. 67, Birkhäuser (1987) 124 | DOI

[46] S Takayama, Local simple connectedness of resolutions of log-terminal singularities, Internat. J. Math. 14 (2003) 825 | DOI

[47] K Ueno, Classification theory of algebraic varieties and compact complex spaces, 439, Springer (1975) | DOI

[48] C Voisin, Théorie de Hodge et géométrie algébrique complexe, 10, Soc. Math. France (2002) | DOI

[49] C Voisin, On the homotopy types of compact Kähler and complex projective manifolds, Invent. Math. 157 (2004) 329 | DOI

[50] C Voisin, On the homotopy types of Kähler manifolds and the birational Kodaira problem, J. Differential Geom. 72 (2006) 43 | DOI

[51] J Włodarczyk, Resolution of singularities of analytic spaces, from: "Proceedings of Gökova Geometry–Topology Conference 2008" (editors S Akbulut, T Önder, R J Stern), GGT (2009) 31

[52] S Zucker, Hodge theory with degenerating coefficients : L2 cohomology in the Poincaré metric, Ann. of Math. 109 (1979) 415 | DOI