Huybrechts, Daniel
Finiteness of polarized K3 surfaces and hyperkähler manifolds
Annales Henri Lebesgue, Volume 1 (2018), p. 227-246


KeywordsK3 surfaces, cone conjecture, moduli spaces


In the moduli space of polarized varieties (X,L) the same unpolarized variety X can occur more than once. However, for K3 surfaces, compact hyperkähler manifolds, and abelian varieties the ‘orbit’ of X, i.e. the subset {(X i ,L i )X i X}, is known to be finite, which may be viewed as a consequence of the Kawamata–Morrison cone conjecture. In this note we provide a proof of this finiteness not relying on the cone conjecture and, in fact, not even on the global Torelli theorem. Instead, it uses the geometry of the moduli space of polarized varieties to conclude the finiteness by means of Baily–Borel type arguments. We also address related questions concerning finiteness in twistor families associated with polarized K3 surfaces of CM type.


[And96] André, Yves On the Shafarevich and Tate conjectures for hyper-Kähler varieties, Math. Ann., Volume 305 (1996) no. 2, pp. 205-248 | Zbl 0942.14018

[AV16] Amerik, Ekaterina; Verbitsky, Misha Hyperbolic geometry of the ample cone of a hyperkähler manifold, Res. Math. Sci., Volume 3 (2016), 7, 9 pages (Art. ID 7, 9 pages) | Zbl 1348.53057

[AV17] Amerik, Ekaterina; Verbitsky, Misha Morrison–Kawamata cone conjecture for hyperkähler manifolds, Ann. Sci. Éc. Norm. Supér., Volume 50 (2017) no. 4, pp. 973-993 | Zbl 1379.53060

[BB66] Baily, Walter L. Jun.; Borel, Armand Compactification of arithmetic quotients of bounded symmetric domains, Ann. Math., Volume 84 (1966), pp. 442-528 | Zbl 0154.08602

[Bea04] Beauville, Arnaud Fano threefolds and K3 surfaces, The Fano conference (Torino, 2002), Università di Torino, Dipartimento di Matematica (2004), pp. 175-184 | Zbl 1096.14034

[BL16] Bakker, Benjamin; Lehn, Christian A global Torelli theorem for singular symplectic varieties (2016) ( )

[Bor72] Borel, Armand Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem, J. Differ. Geom., Volume 6 (1972), pp. 543-560 | Zbl 0249.32018

[Bor86] Borcea, Ciprian Diffeomorphisms of a K3 surface, Math. Ann., Volume 275 (1986), pp. 1-4 | Zbl 0596.32036

[CGGH83] Carlson, James; Green, Mark; Griffiths, Phillip; Harris, Joe Infinitesimal variations of Hodge structure. I, Compos. Math., Volume 50 (1983), pp. 109-205 | Zbl 0531.14006

[Cox89] Cox, David A. Primes of the form x 2 +ny 2 . Fermat, class field theory and complex multiplication, John Wiley & Sons (1989), xi+351 pages | Zbl 0701.11001

[CS98] Conway, John H.; Sloane, Neil J. A. Sphere packings, lattices and groups, Springer, Grundlehren der Mathematischen Wissenschaften, Volume 290 (1998), lxxiv+703 pages | Zbl 0915.52003

[Dol96] Dolgachev, Igor V. Mirror symmetry for lattice polarized K3 surfaces, J. Math. Sci., New York, Volume 81 (1996) no. 3, pp. 2599-2630 | Zbl 0890.14024

[Efi17] Efimov, Alexander I. Some remarks on L-equivalence of algebraic varieties (2017) ( )

[Hay68] Hayashika, T. A class number associated with a product of elliptic curves, J. Math. Soc. Japan, Volume 20 (1968), pp. 26-43

[HLOY04] Hosono, Shinobu; Lian, Bong H.; Oguiso, Keiji; Yau, Shing-Tung Fourier–Mukai number of a K3 surface, Algebraic structures and moduli spaces (Montréal, 2003), American Mathematical Society (CRM Proceedings & Lecture Notes) Volume 38 (2004), pp. 177-192 | Zbl 1076.14045

[How01] Howe, Everett W. Plane quartics with Jacobians isomorphic to a hyperelliptic Jacobian, Proc. Am. Math. Soc., Volume 129 (2001) no. 6, pp. 1647-1657 | Zbl 0974.14021

[How05] Howe, Everett W. Infinite families of pairs of curves over with isomorphic Jacobians, J. Lond. Math. Soc., Volume 72 (2005) no. 2, pp. 327-350 | Zbl 1093.14041

[HP13] Hulek, Klaus W.; Ploog, David Fourier–Mukai partners and polarised K3 surfaces, Arithmetic and geometry of K3 surfaces and Calabi-Yau threefolds (Totonto, 2011), Springer (Fields Institute Communications) Volume 67 (2013), pp. 333-365 | Zbl 1309.14016

[Huy03] Huybrechts, Daniel Finiteness results for compact hyperkähler manifolds, J. Reine Angew. Math., Volume 558 (2003), pp. 15-22 | Zbl 1042.53032

[Huy16] Huybrechts, Daniel Lectures on K3 surfaces, Cambridge University Press, Cambridge Studies in Advanced Mathematics, Volume 158 (2016) ( ) | Article

[Huy17] Huybrechts, Daniel Motives of isogenous K3 surfaces (2017) (, to appear in Comm. Math. Helv.)

[Huy99] Huybrechts, Daniel Compact hyperkähler manifolds: Basic results, Invent. Math., Volume 135 (1999) no. 1, pp. 63-113 | Zbl 0953.53031

[JL18] Javanpeykar, Ariyan; Loughran, Daniel Arithmetic hyperbolicity and a stacky Chevalley–Weil theorem (2018) ( )

[Kaw97] Kawamata, Yujiro On the cone of divisors of Calabi–Yau fiber spaces, Int. J. Math., Volume 8 (1997) no. 5, pp. 665-687 | Zbl 0931.14022

[KM97] Keel, Seán; Mori, Shigefumi Quotients by groupoids, Ann. Math., Volume 145 (1997) no. 1, pp. 193-213 | Zbl 0881.14018

[Kne02] Kneser, Martin Quadratische Formen, Springer (2002), viii+164 pages | Zbl 1001.11014

[Kud13] Kudla, Stephen A note about special cycles on moduli spaces of K3 surfaces, Arithmetic and geometry of K3 surfaces and Calabi-Yau threefolds (Toronto, 2011), Springer (Fields Institute Communications) Volume 67 (2013), pp. 411-427 | Zbl 1302.14030

[Lan06] Lange, Herbert Principal polarizations on products of elliptic curves, The geometry of Riemann surfaces and abelian varieties. III (Salamanca, 2004), American Mathematical Society (Contemporary Mathematics) Volume 397 (2006), pp. 153-162 | Zbl 1118.14050

[Lan87] Lange, Herbert Abelian varieties with several principal polarizations, Duke Math. J., Volume 55 (1987), pp. 617-628 | Zbl 0657.14023

[LMB00] Laumon, Gérard; Moret-Bailly, Laurent Champs algébriques, Springer, Ergebnisse der Mathematik und ihrer Grenzgebiete, Volume 39 (2000), xii+208 pages | Zbl 0945.14005

[LOP15] Lazić, Vladimir; Oguiso, Keiji; Peternell, Thomas The Morrison–Kawamata cone conjecture and abundance on Ricci flat manifolds, Uniformization, Riemann-Hilbert correspondence, Calabi-Yau manifolds and Picard-Fuchs equations (Stockholm, 2015), Higher Education Press (Advanced Lectures in Mathematics) Volume 42 (2015), pp. 157-185 | Zbl 06999866

[Mar11] Markman, Eyal A survey of Torelli and monodromy results for holomorphic-symplectic varieties, Complex and differential geometry (Leibniz, 2009), Springer (Springer Proceedings in Mathematics) Volume 8 (2011), pp. 257-322 | Zbl 1229.14009

[Mau14] Maulik, Davesh Supersingular K3 surfaces for large primes, Duke Math. J., Volume 163 (2014) no. 13, pp. 2357-2425 | Zbl 1308.14043

[Mil86] Milne, James S. Abelian varieties, Arithmetic geometry (Storrs, 1984), Springer (1986), pp. 103-150 | Zbl 0604.14028

[Mor93] Morrison, David R. Compactifications of moduli spaces inspired by mirror symmetry, Days on algebraic geometry (Orsay,1992), Société Mathématique de France (Astérisque) Volume 218 (1993), pp. 243-271 | Zbl 0824.14007

[MY15] Markman, Eyal; Yoshioka, Kota A proof of the Kawamata–Morrison cone conjecture for holomorphic symplectic varieties of K3 [n] or generalized Kummer deformation type, Int. Math. Res. Not., Volume 2015 (2015) no. 24, pp. 13563-13574 | Zbl 1353.14049

[NN81] Narasimhan, Mudumbai S.; Nori, Madhav V. Polarisations on an abelian variety, Proc. Indian Acad. Sci., Math. Sci., Volume 90 (1981), pp. 125-128 | Zbl 0509.14047

[Ogu02] Oguiso, Keiji K3 surfaces via almost-primes, Math. Res. Lett., Volume 9 (2002) no. 1, pp. 47-63 | Zbl 1043.14010

[OS18] Orr, Martin; Skorobogatov, Alexei N. Finiteness theorems for K3 surfaces and abelian varieties of CM type (2018), pp. 1571-1592 | Zbl 06944055

[PS12] Prendergast-Smith, Artie The cone conjecture for abelian varieties, J. Math. Sci., Tokyo, Volume 19 (2012) no. 2, pp. 243-261 | Zbl 1284.14021

[Riz10] Rizov, Jordan Kuga–Satake abelian varieties of K3 surfaces in mixed characteristic, J. Reine Angew. Math., Volume 648 (2010), pp. 13-67 | Zbl 1208.14031

[Ste08] Stellari, Paolo A finite group acting on the moduli space of K3 surfaces, Trans. Am. Math. Soc., Volume 360 (2008) no. 12, pp. 6631-6642 | Zbl 1151.14028

[Ste85] Sterk, Hans Finiteness results for algebraic K3 surfaces, Math. Z., Volume 189 (1985), pp. 507-513 | Zbl 0545.14032

[Sze99] Szendrői, Balázs Some finiteness results for Calabi–Yau threefolds, J. Lond. Math. Soc., Volume 60 (1999) no. 3, pp. 689-699 | Zbl 0961.14028

[Tre15] Tretkoff, Paula K3 surfaces with algebraic period ratios have complex multiplication, Int. J. Number Theory, Volume 11 (2015) no. 5, pp. 1709-1724 | Zbl 1366.14036

[Ver13] Verbitsky, Misha Mapping class group and a global Torelli theorem for hyperkähler manifolds, Duke Math. J., Volume 162 (2013) no. 15, pp. 2929-2986 | Zbl 1295.53042

[Ver15] Verbitsky, Misha Ergodic complex structures on hyperkähler manifolds, Acta Math., Volume 215 (2015) no. 1, pp. 161-182 | Zbl 1332.53092

[Ver17] Verbitsky, Misha Ergodic complex structures on hyperkahler manifolds: an erratum (2017) ( )

[Vie95] Viehweg, Eckart Quasi-projective moduli for polarized manifolds, Springer, Ergebnisse der Mathematik und ihrer Grenzgebiete, Volume 30 (1995), viii+320 pages | Zbl 0844.14004

[Zar85] Zarhin, Yu. G. A finiteness theorem for unpolarized Abelian varieties over number fields with prescribed places of bad reduction, Invent. Math., Volume 79 (1985), pp. 309-321 | Zbl 0557.14024