Finiteness of polarized K3 surfaces and hyperkähler manifolds

Huybrechts, Daniel

doi:10.5802/ahl.6

Huybrechts, Daniel

Finiteness of polarized K3 surfaces and hyperkähler manifolds

Annales Henri Lebesgue, Volume 1 (2018), pp. 227-248.

Metadata

DOI10.5802/ahl.6

Keywords K3 surfaces, cone conjecture, moduli spaces

Abstract

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.

References

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

[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

[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 | DOI | Zbl

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

[Bea04] Beauville, Arnaud Fano threefolds and $K 3$ surfaces, The Fano conference (Torino, 2002), UniversitÃ di Torino, Dipartimento di Matematica, 2004, pp. 175-184 | Zbl

[BL16] Bakker, Benjamin; Lehn, Christian A global Torelli theorem for singular symplectic varieties (2016) (https://arxiv.org/abs/1612.07894)

[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 | DOI | MR | Zbl

[Bor86] Borcea, Ciprian Diffeomorphisms of a $K 3$ surface, Math. Ann., Volume 275 (1986), pp. 1-4 | DOI | MR | Zbl

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

[Cox89] Cox, David A. Primes of the form $x^{2} + n y^{2}$ . Fermat, class field theory and complex multiplication, John Wiley & Sons, 1989, xi+351 pages | Zbl

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

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

[Efi17] Efimov, Alexander I. Some remarks on $L$ -equivalence of algebraic varieties (2017) (https://arxiv.org/abs/1707.08997) | Zbl

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

[HLOY04] Hosono, Shinobu; Lian, Bong H.; Oguiso, Keiji; Yau, Shing-Tung Fourier–Mukai number of a $K 3$ surface, Algebraic structures and moduli spaces (MontrÃ©al, 2003) (CRM Proceedings & Lecture Notes), Volume 38, American Mathematical Society, 2004, pp. 177-192 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

[HP13] Hulek, Klaus W.; Ploog, David Fourier–Mukai partners and polarised $K 3$ surfaces, Arithmetic and geometry of $K 3$ surfaces and Calabi-Yau threefolds (Totonto, 2011) (Fields Institute Communications), Volume 67, Springer, 2013, pp. 333-365 | DOI | MR | Zbl

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

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

[Huy16] Huybrechts, Daniel Lectures on $K 3$ surfaces, Cambridge Studies in Advanced Mathematics, 158, Cambridge University Press, 2016 (http://www.math.uni-bonn.de/people/huybrech/K3.html) | DOI | MR | Zbl

[Huy17] Huybrechts, Daniel Motives of isogenous $K 3$ surfaces (2017) (https://arxiv.org/abs/1705.04063, to appear in Comm. Math. Helv.) | Zbl

[JL18] Javanpeykar, Ariyan; Loughran, Daniel Arithmetic hyperbolicity and a stacky Chevalley–Weil theorem (2018) (https://arxiv.org/abs/1808.09876)

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

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

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

[Kud13] Kudla, Stephen A note about special cycles on moduli spaces of $K 3$ surfaces, Arithmetic and geometry of $K 3$ surfaces and Calabi-Yau threefolds (Toronto, 2011) (Fields Institute Communications), Volume 67, Springer, 2013, pp. 411-427 | DOI | MR | Zbl

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

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

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

[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) (Advanced Lectures in Mathematics), Volume 42, Higher Education Press, 2015, pp. 157-185 | Zbl

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

[Mau14] Maulik, Davesh Supersingular $K 3$ surfaces for large primes, Duke Math. J., Volume 163 (2014) no. 13, pp. 2357-2425 | DOI | MR | Zbl

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

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

[MY15] Markman, Eyal; Yoshioka, Kota A proof of the Kawamata–Morrison cone conjecture for holomorphic symplectic varieties of ${K 3}^{[n]}$ or generalized Kummer deformation type, Int. Math. Res. Not., Volume 2015 (2015) no. 24, pp. 13563-13574 | DOI | MR | Zbl

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

[Ogu02] Oguiso, Keiji $K 3$ surfaces via almost-primes, Math. Res. Lett., Volume 9 (2002) no. 1, pp. 47-63 | DOI | MR | Zbl

[OS18] Orr, Martin; Skorobogatov, Alexei N. Finiteness theorems for $K 3$ surfaces and abelian varieties of CM type (2018), pp. 1571-1592 | Zbl

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

[Riz10] Rizov, Jordan Kuga–Satake abelian varieties of $K 3$ surfaces in mixed characteristic, J. Reine Angew. Math., Volume 648 (2010), pp. 13-67 | MR | Zbl

[Ste85] Sterk, Hans Finiteness results for algebraic $K 3$ surfaces, Math. Z., Volume 189 (1985), pp. 507-513 | DOI | MR | Zbl

[Ste08] Stellari, Paolo A finite group acting on the moduli space of $K 3$ surfaces, Trans. Am. Math. Soc., Volume 360 (2008) no. 12, pp. 6631-6642 | DOI | MR | Zbl

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

[Tre15] Tretkoff, Paula $K 3$ surfaces with algebraic period ratios have complex multiplication, Int. J. Number Theory, Volume 11 (2015) no. 5, pp. 1709-1724 | DOI | MR | Zbl

[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 | DOI | Zbl

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

[Ver17] Verbitsky, Misha Ergodic complex structures on hyperkahler manifolds: an erratum (2017) (https://arxiv.org/abs/1708.05802) | Zbl

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

[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 | DOI | MR | Zbl

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