Duke Mathematical Journal

Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of abelian varieties

Daniel Greb, Stefan Kebekus, and Thomas Peternell

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Given a quasiprojective variety $X$ with only Kawamata log terminal singularities, we study the obstructions to extending finite étale covers from the smooth locus $X_{\mathrm{reg}}$ of $X$ to $X$ itself. A simplified version of our main results states that there exists a Galois cover $Y\rightarrow X$, ramified only over the singularities of $X$, such that the étale fundamental groups of $Y$ and of $Y_{\mathrm{reg}}$ agree. In particular, every étale cover of $Y_{\mathrm{reg}}$ extends to an étale cover of $Y$. As a first major application, we show that every flat holomorphic bundle defined on $Y_{\mathrm{reg}}$ extends to a flat bundle that is defined on all of $Y$. As a consequence, we generalize a classical result of Yau to the singular case: every variety with at worst terminal singularities and with vanishing first and second Chern class is a finite quotient of an abelian variety. As a further application, we verify a conjecture of Nakayama and Zhang describing the structure of varieties that admit polarized endomorphisms.

Article information

Source
Duke Math. J. Volume 165, Number 10 (2016), 1965-2004.

Dates
Received: 7 July 2014
Revised: 28 July 2015
First available in Project Euclid: 5 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1459878278

Digital Object Identifier
doi:10.1215/00127094-3450859

Mathematical Reviews number (MathSciNet)
MR3522654

Subjects
Primary: 14J17: Singularities [See also 14B05, 14E15]
Secondary: 14B05: Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx] 14E30: Minimal model program (Mori theory, extremal rays) 14B25: Local structure of morphisms: étale, flat, etc. [See also 13B40]

Keywords
minimal model program algebraic fundamental group KLT singularities flat vector bundles torus quotients polarized endomorphisms

Citation

Greb, Daniel; Kebekus, Stefan; Peternell, Thomas. Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of abelian varieties. Duke Math. J. 165 (2016), no. 10, 1965--2004. doi:10.1215/00127094-3450859. https://projecteuclid.org/euclid.dmj/1459878278.


Export citation

References

  • [1] P. Aluffi, Classes de Chern des variétés singulières, revisitées, C. R. Math. Acad. Sci. Paris 342 (2006), 405–410.
  • [2] F. Ambro, The moduli $b$-divisor of an lc-trivial fibration, Compos. Math. 141 (2005), 385–403.
  • [3] S. Bando and Y.-T. Siu, “Stable sheaves and Einstein-Hermitian metrics” in Geometry and Analysis on Complex Manifolds, World Sci., River Edge, N.J., 1994, 39–50.
  • [4] M. C. Beltrametti and A. J. Sommese, The Adjunction Theory of Complex Projective Varieties, De Gruyter Exp. Math. 16, De Gruyter, Berlin, 1995.
  • [5] F. Catanese, Q.E.D. for algebraic varieties, with an appendix by S. Rollenske, J. Differential Geom. 77 (2007), 43–75.
  • [6] P. Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Math. 163, Springer, Berlin, 1970.
  • [7] G. Dethloff and H. Grauert, “Seminormal complex spaces” in Several Complex Variables, VII, Encyclopaedia Math. Sci. 74, Springer, Berlin, 1994, 183–220.
  • [8] A. Dimca, Singularities and Topology of Hypersurfaces, Universitext, Springer, New York, 1992.
  • [9] S. K. Donaldson, Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. Lond. Math. Soc. (3) 50 (1985), 1–26.
  • [10] J.-M. Drézet, “Luna’s slice theorem and applications” in Algebraic Group Actions and Quotients, Hindawi Publ. Corp., Cairo, 2004, 39–89.
  • [11] S. Druel, The Zariski-Lipman conjecture for log canonical spaces, Bull. Lond. Math. Soc. 46 (2014), 827–835.
  • [12] H. Flenner, Restrictions of semistable bundles on projective varieties, Comment. Math. Helv. 59 (1984), 635–650.
  • [13] M. Goresky and R. D. MacPherson, Stratified Morse Theory, Ergeb. Math. Grenzgeb. (3) 14, Springer, Berlin, 1988.
  • [14] P. Graf, Bogomolov–Sommese vanishing on log canonical pairs, J. Reine Angew. Math. 702 (2015), 109–142.
  • [15] H. Grauert and R. Remmert, Coherent Analytic Sheaves, Grundlehren Math. Wiss. 265, Springer, Berlin, 1984.
  • [16] D. Greb, S. Kebekus, S. J. Kovács, and T. Peternell, Differential forms on log canonical spaces, Publ. Math. Inst. Hautes Études Sci. 114 (2011), 87–169.
  • [17] D. Greb, S. Kebekus, and T. Peternell, “Singular spaces with trivial canonical class,” to appear in Minimal Models and Extremal Rays: Proceedings of the Conference in Honor of Shigefumi Mori’s 60th Birthday, preprint, arXiv:1110.5250v3 [math.AG].
  • [18] D. Greb, S. Kebekus, and T. Peternell, Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of Abelian varieties, preprint, arXiv:1307.5718v3 [math.AG].
  • [19] A. Grothendieck, Éléments de géométrie algébrique, IV: Étude locale des schémas et des morphismes de schémas. Troisiéme partie, Inst. Hautes Études Sci. Publ. Math. 28 (1966).
  • [20] A. Grothendieck, Représentations linéaires et compactification profinie des groupes discrets, Manuscripta Math. 2 (1970), 375–396.
  • [21] A. Grothendieck, Revêtements étales et groupe fondamental (SGA 1), Lecture Notes in Math. 224, Springer, Berlin, 1971.
  • [22] A. Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2), Doc. Math. (Paris) 4, Soc. Math. France, Paris, 2005.
  • [23] R. V. Gurjar and D.-Q. Zhang, $\pi_{1}$ of smooth points of a log del Pezzo surface is finite, I, J. Math. Sci. Univ. Tokyo 1 (1994), 137–180.
  • [24] C. D. Hacon, J. McKernan, and C. Xu, ACC for log canonical thresholds, Ann. of Math. (2) 180 (2014), 523–571.
  • [25] R. Hartshorne, Algebraic Geometry, Grad. Texts in Math. 52, Springer, New York, 1977.
  • [26] R. Hartshorne, Stable reflexive sheaves, Math. Ann. 254 (1980), 121–176.
  • [27] A. Hatcher, Algebraic Topology, Cambridge Univ. Press, Cambridge, 2002.
  • [28] D. Huybrechts and M. Lehn, The Geometry of Moduli Spaces of Sheaves, 2nd ed., Cambridge Math. Libr., Cambridge Univ. Press, Cambridge, 2010.
  • [29] S. Kebekus and S. J. Kovács, The structure of surfaces and threefolds mapping to the moduli stack of canonically polarized varieties, Duke Math. J. 155 (2010), 1–33.
  • [30] S. Kobayashi, Differential Geometry of Complex Vector Bundles, Publ. Math. Soc. Japan 15, Princeton Univ. Press, Princeton, 1987.
  • [31] J. Kollár, Shafarevich maps and automorphic forms, M. B. Porter Lectures, Princeton Univ. Press, Princeton, 1995.
  • [32] J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, Cambridge Tracts in Math. 134, Cambridge Univ. Press, Cambridge, 1998.
  • [33] A. Lascoux and M. Berger, Variétés Kähleriennes compactes, Lecture Notes in Math. 154, Springer, Berlin, 1970.
  • [34] R. D. MacPherson, Chern classes for singular algebraic varieties, Ann. of Math. (2) 100 (1974), 423–432.
  • [35] J. S. Milne, Étale Cohomology, Princeton Math. Ser. 33, Princeton Univ. Press, Princeton, 1980.
  • [36] Y. Miyaoka, “The Chern classes and Kodaira dimension of a minimal variety” in Algebraic Geometry (Sendai, 1985), Adv. Stud. Pure Math. 10, North-Holland, Amsterdam, 1987, 449–476.
  • [37] N. Nakayama and D.-Q. Zhang, Polarized endomorphisms of complex normal varieties, Math. Ann. 346 (2010), 991–1018.
  • [38] L. Ribes and P. Zalesskii, Profinite Groups, 2nd ed., Ergeb. Math. Grenzgeb. (3) 40, Springer, Berlin, 2010.
  • [39] H. Rossi, Picard variety of an isolated singular point, Rice Univ. Studies 54 (1968), 63–73.
  • [40] N. I. Shepherd-Barron and P. M. H. Wilson, Singular threefolds with numerically trivial first and second Chern classes, J. Algebraic Geom. 3 (1994), 265–281.
  • [41] C. T. Simpson, Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 5–95.
  • [42] C. T. Simpson, Moduli of representations of the fundamental group of a smooth projective variety, II, Inst. Hautes Études Sci. Publ. Math. 80 (1994), 5–79.
  • [43] S. Takayama, Simple connectedness of weak Fano varieties, J. Algebraic Geom. 9 (2000), 403–407.
  • [44] K. Uhlenbeck and S.-T. Yau, On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Comm. Pure Appl. Math. 39 (1986), S257–S293.
  • [45] B. A. F. Wehrfritz, Infinite Linear Groups. An Account of the Group-Theoretic Properties of Infinite Groups of Matrices, Ergeb. Mat. Grenzgeb. 76, Springer, New York, 1973.
  • [46] C. Xu, Finiteness of algebraic fundamental groups, Compos. Math. 150 (2014), 409–414.
  • [47] S. T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I, Comm. Pure Appl. Math. 31 (1978), 339–411.
  • [48] Q. Zhang, Rational connectedness of log $\mathbf{Q}$-Fano varieties, J. Reine Angew. Math. 590 (2006), 131–142.