Fundamental groups of links of isolated singularities

Kapovich, Michael¹ ; Kollár, János²

¹ Department of Mathematics, University of California, Davis, California 95616
² Department of Mathematics, Princeton University, Princeton, New Jersey 08544-1000

Journal of the American Mathematical Society, Tome 27 (2014) no. 4, pp. 929-952

Résumé

We study fundamental groups of projective varieties with normal crossing singularities and of germs of complex singularities. We prove that for every finitely-presented group $G$ there is a complex projective surface $S$ with simple normal crossing singularities only, so that the fundamental group of $S$ is isomorphic to $G$. We use this to construct 3-dimensional isolated complex singularities so that the fundamental group of the link is isomorphic to $G$. Lastly, we prove that a finitely-presented group $G$ is ${\mathbb Q}$-superperfect (has vanishing rational homology in dimensions 1 and 2) if and only if $G$ is isomorphic to the fundamental group of the link of a rational 6-dimensional complex singularity.

Détail

DOI : 10.1090/S0894-0347-2014-00807-9

Bibliographie

[1] Amorã³S, J., Burger, M., Corlette, K., Kotschick, D., Toledo, D. Fundamental groups of compact Kähler manifolds 1996

[2] D. Arapura, P. Bakhatry, J. Wå‚Odarczyk The combinatorial part of the cohomology of a singular variety 2009

[3] Artin, M. Algebraization of formal moduli. II. Existence of modifications Ann. of Math. (2) 1970 88 135

[4] Berrick, A. J. A topologist’s view of perfect and acyclic groups 2002 1 28

[5] Bridson, Martin R., Grunewald, Fritz J. Grothendieck’s problems concerning profinite completions and representations of groups Ann. of Math. (2) 2004 359 373

[6] Cairns, Stewart S. A simple triangulation method for smooth manifolds Bull. Amer. Math. Soc. 1961 389 390

[7] Campbell, Colin M., Havas, George, Ramsay, Colin, Robertson, Edmund F. Nice efficient presentations for all small simple groups and their covers LMS J. Comput. Math. 2004 266 283

[8] Classification of irregular varieties 1992

[9] Corson, Jon Michael Complexes of groups Proc. London Math. Soc. (3) 1992 199 224

[10] Corlette, Kevin, Simpson, Carlos On the classification of rank-two representations of quasiprojective fundamental groups Compos. Math. 2008 1271 1331

[11] Davis, Michael W. The geometry and topology of Coxeter groups 2008

[12] Dimca, Alexandru, Papadima, ŞTefan, Suciu, Alexander I. Topology and geometry of cohomology jump loci Duke Math. J. 2009 405 457

[13] Draniå¡Nikov, A. N., Repovå¡, D. Embeddings up to homotopy type in Euclidean space Bull. Austral. Math. Soc. 1993 145 148

[14] Ferrand, Daniel Conducteur, descente et pincement Bull. Soc. Math. France 2003 553 585

[15] The birational geometry of degenerations 1983

[16] Fortune, Steven Voronoi diagrams and Delaunay triangulations 1997 377 388

[17] Goresky, Mark, Macpherson, Robert Stratified Morse theory 1988

[18] Gordon, Gerald Leonard On a simplicial complex associated to the monodromy Trans. Amer. Math. Soc. 1980 93 101

[19] Grothendieck, Alexander Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (𝑆𝐺𝐴 2) 1968

[20] Griffiths, Phillip, Schmid, Wilfried Recent developments in Hodge theory: a discussion of techniques and results 1975 31 127

[21] Hartshorne, Robin Algebraic geometry 1977

[22] Hatcher, Allen Algebraic topology 2002

[23] Higman, Graham A finitely generated infinite simple group J. London Math. Soc. 1951 61 64

[24] Hirsch, Morris W. Smooth regular neighborhoods Ann. of Math. (2) 1962 524 530

[25] Hopf, Heinz Fundamentalgruppe und zweite Bettische Gruppe Comment. Math. Helv. 1942 257 309

[26] Hausmann, Jean-Claude, Weinberger, Shmuel Caractéristiques d’Euler et groupes fondamentaux des variétés de dimension 4 Comment. Math. Helv. 1985 139 144

[27] Ishii, Shihoko On isolated Gorenstein singularities Math. Ann. 1985 541 554

[28] Kapovich, Michael Conformally flat metrics on 4-manifolds J. Differential Geom. 2004 289 301

[29] Kervaire, Michel A. Smooth homology spheres and their fundamental groups Trans. Amer. Math. Soc. 1969 67 72

[30] Kervaire, Michel A., Milnor, John W. Groups of homotopy spheres. I Ann. of Math. (2) 1963 504 537

[31] Kapovich, Michael, Millson, John J. On representation varieties of Artin groups, projective arrangements and the fundamental groups of smooth complex algebraic varieties Inst. Hautes Études Sci. Publ. Math. 1998

[32] Kollã¡R, Jã¡Nos, Mori, Shigefumi Birational geometry of algebraic varieties 1998

[33] Kollã¡R, Jã¡Nos Shafarevich maps and plurigenera of algebraic varieties Invent. Math. 1993 177 215

[34] Jã¡Nos Kollã¡R Quotients by finite equivalence relations 2008

[35] Jã¡Nos Kollã¡R New examples of terminal and log canonical singularities 2011

[36] Kulikov, Vik. S. Degenerations of 𝐾3 surfaces and Enriques surfaces Izv. Akad. Nauk SSSR Ser. Mat. 1977

[37] Livingston, Charles Four-manifolds of large negative deficiency Math. Proc. Cambridge Philos. Soc. 2005 107 115

[38] Morgan, John W. The algebraic topology of smooth algebraic varieties Inst. Hautes Études Sci. Publ. Math. 1978 137 204

[39] Mumford, David The topology of normal singularities of an algebraic surface and a criterion for simplicity Inst. Hautes Études Sci. Publ. Math. 1961 5 22

[40] Sam Payne Lecture at MSRI 2009

[41] Sam Payne Boundary complexes and weight filtrations 2011

[42] Persson, Ulf On degenerations of algebraic surfaces Mem. Amer. Math. Soc. 1977

[43] Phillips, Anthony Submersions of open manifolds Topology 1967 171 206

[44] Prasad, Gopal, Yeung, Sai-Kee Fake projective planes Invent. Math. 2007 321 370

[45] Simpson, Carlos Local systems on proper algebraic 𝑉-manifolds Pure Appl. Math. Q. 2011 1675 1759

[46] Steenbrink, J. H. M. Mixed Hodge structures associated with isolated singularities 1983 513 536

[47] Stepanov, D. A. A note on resolution of rational and hypersurface singularities Proc. Amer. Math. Soc. 2008 2647 2654

[48] Takayama, Shigeharu Local simple connectedness of resolutions of log-terminal singularities Internat. J. Math. 2003 825 836

[49] Thuillier, Amaury Géométrie toroïdale et géométrie analytique non archimédienne. Application au type d’homotopie de certains schémas formels Manuscripta Math. 2007 381 451

[50] Toledo, Domingo Projective varieties with non-residually finite fundamental group Inst. Hautes Études Sci. Publ. Math. 1993 103 119

[51] Wå‚Odarczyk, Jaroså‚Aw Toroidal varieties and the weak factorization theorem Invent. Math. 2003 223 331