Kähler spaces with zero first Chern class: Bochner principle, Albanese map and fundamental groups
-
Benoît Claudon
,
Patrick Graf
and
Philipp Naumann
Abstract
Let X be a compact Kähler space with klt singularities and vanishing first Chern class. We prove the Bochner principle for holomorphic tensors on the smooth locus of X: any such tensor is parallel with respect to the singular Ricci-flat metrics. As a consequence, after a finite quasi-étale cover X splits off a complex torus of the maximum possible dimension. We then proceed to decompose the tangent sheaf of X according to its holonomy representation. In particular, we classify those X which have strongly stable tangent sheaf: up to quasi-étale covers, these are either irreducible Calabi–Yau or irreducible holomorphic symplectic. As an application of these results, we show that if X has dimension four, then it satisfies Campana’s Abelianity Conjecture.
Funding source: Agence Nationale de la Recherche
Award Identifier / Grant number: ANR–16–CE40–0008
Award Identifier / Grant number: ANR–16–CE40–0011
Funding statement: Henri Guenancia was partially supported by the ANR project GRACK. Benoît Claudon was partially supported by the ANR projects Foliage ANR–16–CE40–0008 and Hodgefun ANR–16–CE40–0011.
Acknowledgements
Henri Guenancia and Benoît Claudon would like to thank Stéphane Druel and Matei Toma for several enlightening discussions. Patrick Graf and Philipp Naumann would like to thank Mihai Păun and Thomas Peternell for sharing their insight with them. The authors would like to thank an anonymous referee for his/her careful reading and for valuable comments.
References
[1] M. Artin, Algebraic approximation of structures over complete local rings, Publ. Math. Inst. Hautes Études Sci. 36, (1969), 23–58. 10.1007/BF02684596Search in Google Scholar
[2] B. Bakker, H. Guenancia and C. Lehn, Algebraic approximation and the decomposition theorem for Kähler Calabi–Yau varieties, Invent. Math. (2022), 10.1007/s00222-022-01096-y. 10.1007/s00222-022-01096-ySearch in Google Scholar
[3] C. Bănică and O. Stănăşilă, Algebraic methods in the global theory of complex spaces, John Wiley & Sons, New York 1976. Search in Google Scholar
[4] A. Beauville, Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differential Geom. 18 (1983), no. 4, 755–782. 10.4310/jdg/1214438181Search in Google Scholar
[5] C. Birkar, P. Cascini, C. D. Hacon and J. McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), no. 2, 405–468. 10.1090/S0894-0347-09-00649-3Search in Google Scholar
[6] S. Boucksom, P. Eyssidieux and V. Guedj, Introduction, An introduction to the Kähler–Ricci flow, Lecture Notes in Math. 2086, Springer, Cham (2013), 1–6. 10.1007/978-3-319-00819-6_1Search in Google Scholar
[7]
F. Campana,
On twistor spaces of the class
[8] F. Campana, Fundamental group and positivity of cotangent bundles of compact Kähler manifolds, J. Algebraic Geom. 4 (1995), no. 3, 487–502. Search in Google Scholar
[9] F. Campana, Orbifolds, special varieties and classification theory, Ann. Inst. Fourier (Grenoble) 54 (2004), no. 3, 499–630. 10.5802/aif.2027Search in Google Scholar
[10] F. Campana, The Bogomolov–Beauville–Yau decomposition for KLT projective varieties with trivial first Chern class—without tears, Bull. Soc. Math. France 149 (2021), no. 1, 1–13. 10.24033/bsmf.2823Search in Google Scholar
[11] F. Campana, B. Claudon and P. Eyssidieux, Représentations linéaires des groupes kählériens: Factorisations et conjecture de Shafarevich linéaire, Compos. Math. 151 (2015), no. 2, 351–376. 10.1112/S0010437X14007751Search in Google Scholar
[12] J. Cao, H. Guenancia and M. Păun, Variation of singular Kähler-Einstein metrics: Positive Kodaira dimension, J. reine angew. Math. 779 (2021), 1–36. 10.1515/crelle-2021-0028Search in Google Scholar
[13] H. Cartan, Quotient d’un espace analytique par un groupe d’automorphismes, Algebraic geometry and topology, Princeton University, Princeton (1957), 90–102. 10.1515/9781400879915-007Search in Google Scholar
[14] C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778–782. 10.2307/2372597Search in Google Scholar
[15] J.-P. Demailly, Mesures de Monge–Ampère et caractérisation géométrique des variétés algébriques affines, Mém. Soc. Math. France (N. S.) (1985), no. 19, 124. 10.24033/msmf.320Search in Google Scholar
[16] G. Dethloff and H. Grauert, Seminormal complex spaces, Several complex variables. VII, Encyclopaedia Math. Sci. 74, Springer, Berlin (1994), 183–220. 10.1007/978-3-662-09873-8_5Search in Google Scholar
[17] S. Druel, The Zariski–Lipman conjecture for log canonical spaces, Bull. Lond. Math. Soc. 46 (2014), no. 4, 827–835. 10.1112/blms/bdu040Search in Google Scholar
[18] S. Druel, A decomposition theorem for singular spaces with trivial canonical class of dimension at most five, Invent. Math. 211 (2018), no. 1, 245–296. 10.1007/s00222-017-0748-ySearch in Google Scholar
[19] P. Eyssidieux, V. Guedj and A. Zeriahi, Singular Kähler–Einstein metrics, J. Amer. Math. Soc. 22 (2009), no. 3, 607–639. 10.1090/S0894-0347-09-00629-8Search in Google Scholar
[20] A. Fujiki, On automorphism groups of compact Kähler manifolds, Invent. Math. 44 (1978), no. 3, 225–258. 10.1007/BF01403162Search in Google Scholar
[21] M. Goresky and R. MacPherson, Stratified Morse theory, Ergeb. Math. Grenzgeb. (3) 14, Springer, Berlin 1988. 10.1007/978-3-642-71714-7Search in Google Scholar
[22] P. Graf, Algebraic approximation of Kähler threefolds of Kodaira dimension zero, Math. Ann. 371 (2018), no. 1–2, 487–516. 10.1007/s00208-017-1577-4Search in Google Scholar
[23] P. Graf and T. Kirschner, Finite quotients of three-dimensional complex tori, Ann. Inst. Fourier (Grenoble) 70 (2020), no. 2, 881–914. 10.5802/aif.3326Search in Google Scholar
[24] P. Graf and S. J. Kovács, An optimal extension theorem for 1-forms and the Lipman–Zariski conjecture, Doc. Math. 19 (2014), 815–830. 10.4171/dm/465Search in Google Scholar
[25] H. Grauert and R. Remmert, Coherent analytic sheaves, Grundlehren Math. Wiss. 265, Springer, Berlin 1984. 10.1007/978-3-642-69582-7Search in Google Scholar
[26] D. Greb, H. Guenancia and S. Kebekus, Klt varieties with trivial canonical class: Holonomy, differential forms, and fundamental groups, Geom. Topol. 23 (2019), no. 4, 2051–2124. 10.2140/gt.2019.23.2051Search in Google Scholar
[27] D. Greb, S. Kebekus, S. J. Kovács and T. Peternell, Differential forms on log canonical spaces, Publ. Math. Inst. Hautes Études Sci. 114 (2011), 1–83. 10.1007/s10240-011-0036-0Search in Google Scholar
[28] D. Greb, S. Kebekus and T. Peternell, Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of abelian varieties, Duke Math. J. 165 (2016), no. 10, 1965–2004. 10.1215/00127094-3450859Search in Google Scholar
[29] D. Greb, S. Kebekus and T. Peternell, Singular spaces with trivial canonical class, Minimal models and extremal rays (Kyoto 2011), Adv. Stud. Pure Math. 70, Mathematical Society of Japan, Tokyo (2016), 67–113. 10.2969/aspm/07010067Search in Google Scholar
[30] H. Guenancia, Semistability of the tangent sheaf of singular varieties, Algebr. Geom. 3 (2016), no. 5, 508–542. 10.14231/AG-2016-024Search in Google Scholar
[31] A. Höring and T. Peternell, Algebraic integrability of foliations with numerically trivial canonical bundle, Invent. Math. 216 (2019), no. 2, 395–419. 10.1007/s00222-018-00853-2Search in Google Scholar
[32] Y. Kawamata, Characterization of abelian varieties, Compos. Math. 43 (1981), no. 2, 253–276. Search in Google Scholar
[33] Y. Kawamata, Minimal models and the Kodaira dimension of algebraic fiber spaces, J. reine angew. Math. 363 (1985), 1–46. 10.1515/crll.1985.363.1Search in Google Scholar
[34] S. Kebekus and C. Schnell, Extending holomorphic forms from the regular locus of a complex space to a resolution of singularities, J. Amer. Math. Soc. 34 (2021), no. 2, 315–368. 10.1090/jams/962Search in Google Scholar
[35] J. Kollár, Shafarevich maps and plurigenera of algebraic varieties, Invent. Math. 113 (1993), no. 1, 177–215. 10.1007/BF01244307Search in Google Scholar
[36] J. Kollár, Lectures on resolution of singularities, Ann. of Math. Stud. 166, Princeton University, Princeton 2007. Search in Google Scholar
[37] J. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Math. 134, Cambridge University, Cambridge 1998. 10.1017/CBO9780511662560Search in Google Scholar
[38] J. Mather, Notes on topological stability, Bull. Amer. Math. Soc. (N. S.) 49 (2012), no. 4, 475–506. 10.1090/S0273-0979-2012-01383-6Search in Google Scholar
[39] J. N. Mather, Stratifications and mappings, Dynamical systems Bahia State University, Salvador (1973), 195–232. 10.1016/B978-0-12-550350-1.50021-7Search in Google Scholar
[40] M. Păun, Regularity properties of the degenerate Monge–Ampère equations on compact Kähler manifolds, Chin. Ann. Math. Ser. B 29 (2008), no. 6, 623–630. 10.1007/s11401-007-0457-8Search in Google Scholar
[41] G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274–304. 10.4153/CJM-1954-028-3Search in Google Scholar
[42] S. Takayama, Local simple connectedness of resolutions of log-terminal singularities, Internat. J. Math. 14 (2003), no. 8, 825–836. 10.1142/S0129167X0300196XSearch in Google Scholar
[43] G. Tian, Canonical metrics in Kähler geometry, Lectures in Math. ETH Zürich, Birkhäuser, Basel 2000. 10.1007/978-3-0348-8389-4Search in Google Scholar
[44] J.-C. Tougeron, Idéaux de fonctions différentiables. I, Ann. Inst. Fourier (Grenoble) 18 (1968), no. 1, 177–240. 10.1007/978-3-662-59320-2Search in Google Scholar
[45] L. D. Tráng and B. Teissier, Variétés polaires locales et classes de Chern des variétés singulières, Ann. of Math. (2) 114 (1981), no. 3, 457–491. 10.2307/1971299Search in Google Scholar
[46] J. Varouchas, Kähler spaces and proper open morphisms, Math. Ann. 283 (1989), no. 1, 13–52. 10.1007/BF01457500Search in Google Scholar
[47] C. Voisin, Hodge theory and complex algebraic geometry. I, Cambridge Stud. Adv. Math. 76, Cambridge University, Cambridge 2002. 10.1017/CBO9780511615344Search in Google Scholar
[48] B. Wang, Torsion points on the cohomology jump loci of compact Kähler manifolds, Math. Res. Lett. 23 (2016), no. 2, 545–563. 10.4310/MRL.2016.v23.n2.a12Search in Google Scholar
[49] C. Xu, Finiteness of algebraic fundamental groups, Compos. Math. 150 (2014), no. 3, 409–414. 10.1112/S0010437X13007562Search in Google Scholar
[50] 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), no. 3, 339–411. 10.1002/cpa.3160310304Search in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Simultaneous supersingular reductions of CM elliptic curves
- Almost all entries in the character table of the symmetric group are multiples of any given prime
- K-stability of cubic fourfolds
- Nguyen’s tridents and the classification of semigraphical translators for mean curvature flow
- A description of monodromic mixed Hodge modules
- The Lawson surfaces are determined by their symmetries and topology
- CMC hypersurfaces with bounded Morse index
- On Montgomery’s pair correlation conjecture: A tale of three integrals
- Kähler spaces with zero first Chern class: Bochner principle, Albanese map and fundamental groups
Articles in the same Issue
- Frontmatter
- Simultaneous supersingular reductions of CM elliptic curves
- Almost all entries in the character table of the symmetric group are multiples of any given prime
- K-stability of cubic fourfolds
- Nguyen’s tridents and the classification of semigraphical translators for mean curvature flow
- A description of monodromic mixed Hodge modules
- The Lawson surfaces are determined by their symmetries and topology
- CMC hypersurfaces with bounded Morse index
- On Montgomery’s pair correlation conjecture: A tale of three integrals
- Kähler spaces with zero first Chern class: Bochner principle, Albanese map and fundamental groups
