Remarks on degenerations of hyper-Kähler manifolds
Annales de l'Institut Fourier, Volume 68 (2018) no. 7, pp. 2837-2882.

Using the Minimal model program, any degeneration of K-trivial varieties can be arranged to be in a Kulikov type form, i.e. with trivial relative canonical divisor and mild singularities. In the hyper-Kähler setting, we can then deduce a finiteness statement for the monodromy acting on H 2 , once one knows that one component of the central fiber is not uniruled. Independently of this, using deep results from the theory of hyper-Kähler manifolds, we prove that a finite monodromy projective degeneration of hyper-Kähler manifolds has a smooth filling (after base change and birational modifications). As a consequence of these two results, we prove a generalization of Huybrechts’ theorem about birational versus deformation equivalence, allowing singular central fibers. As an application, we give simple proofs for the deformation type of certain explicit models of projective hyper-Kähler manifolds. In a slightly different direction, we establish some basic properties (dimension and rational homology type) for the dual complex of a Kulikov type degeneration of hyper-Kähler manifolds.

Comme conséquence du Programme du modèle minimal, toute dégénérescence de variétés projectives à fibré canonique trivial admet une forme de Kulikov, c’est-à-dire que les singularités de la fibre centrale sont modérées et le fibré canonique relatif est trivial. Dans le cas hyper-kählérien, on en déduit un résultat de finitude pour l’action de monodromie sur H 2 , dès qu’on sait qu’une composante de la fibre centrale n’est pas uniréglée. Nous montrons par ailleurs, en utilisant des résultats puissants de la théorie des variétés hyper-kählériennes, qu’une dégénérescence de variétés hyper-kählériennes à monodromie finie sur H 2 admet un remplissage lisse, c’est-à-dire, après changement de base, un modèle birationnel à fibre centrale lisse. Combinant ces deux résultats, nous obtenons une version du théorème de Huybrechts sur l’équivalence birationnelle et le type de déformations, valable pour les familles à fibre centrale singulière. Ce résultat nous permet de retrouver de façon simple le type de déformations de la plupart des modèles projectifs connus de variétés hyper-kählériennes. Dans une direction différente, nous établissons des résultats basiques (dimension et type d’homotopie rationnelle) concernant le complexe dual de la dégénérescence de Kulikov d’une variété hyper-kählérienne.

Published online:
DOI: 10.5802/aif.3228
Classification: 14B05,  14D05,  14J32,  14E99
Keywords: Hyper-Kähler manifold, degeneration, deformations, Torelli theorem
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Kollár, János; Laza, Radu; Saccà, Giulia; Voisin, Claire. Remarks on degenerations of hyper-Kähler manifolds. Annales de l'Institut Fourier, Volume 68 (2018) no. 7, pp. 2837-2882. doi : 10.5802/aif.3228. https://aif.centre-mersenne.org/articles/10.5802/aif.3228/

[1] Addington, Nicolas; Lehn, Manfred On the symplectic eightfold associated to a Pfaffian cubic fourfold, J. Reine Angew. Math., Tome 731 (2017), pp. 129-137 | Zbl: 1373.14007

[2] Allcock, Daniel; Carlson, James A.; Toledo, Domingo The moduli space of cubic threefolds as a ball quotient, Mem. Am. Math. Soc., Tome 985 (2011), xii+70 pages | Zbl: 1211.14002

[3] Arapura, Donu; Bakhtary, Parsa; Włodarczyk, Jarosław Weights on cohomology, invariants of singularities, and dual complexes, Math. Ann., Tome 357 (2013) no. 2, pp. 513-550 | Zbl: 1280.14005

[4] Beauville, Arnaud Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differ. Geom., Tome 18 (1983) no. 4, pp. 755-782 | Zbl: 0537.53056

[5] Beauville, Arnaud Symplectic singularities, Invent. Math., Tome 139 (2000) no. 3, pp. 541-549 | Zbl: 0958.14001

[6] Beauville, Arnaud; Donagi, Ron La variété des droites d’une hypersurface cubique de dimension 4, C. R. Math. Acad. Sci. Paris, Tome 301 (1985) no. 14, pp. 703-706 | Zbl: 0602.14041

[7] Birkar, Caucher; Cascini, Paolo; Hacon, Christopher D.; McKernan, James Existence of minimal models for varieties of log general type, J. Am. Math. Soc., Tome 23 (2010) no. 2, pp. 405-468 | Zbl: 1210.14019

[8] Bishop, Errett sets, Conditions for the analyticity of certain, Mich. Math. J., Tome 11 (1964), pp. 289-304 | Zbl: 0143.30302

[9] Bogomolov, Fedor A. On the cohomology ring of a simple hyper-Kähler manifold (on the results of Verbitsky), Geom. Funct. Anal., Tome 6 (1996) no. 4, pp. 612-618 | Zbl: 0862.53050

[10] Burns, Jr Dan; Rapoport, Michael On the Torelli problem for kählerian K-3 surfaces, Ann. Sci. Éc. Norm. Supér., Tome 8 (1975) no. 2, pp. 235-273 | Zbl: 0324.14008

[11] Clemens, C. Herbert Degeneration of Kähler manifolds, Duke Math. J., Tome 44 (1977) no. 2, pp. 215-290 | Zbl: 0353.14005

[12] Clemens, C. Herbert; Griffiths, Phillip A. The intermediate Jacobian of the cubic threefold, Ann. Math., Tome 95 (1972), pp. 281-356 | Zbl: 0214.48302

[13] Collino, Alberto The fundamental group of the Fano surface. I, II, Algebraic threefolds (Varenna, 1981) (Lecture Notes in Mathematics) Tome 947, Springer, 1982, p. 209-218, 219–220 | Zbl: 0492.14031

[14] Debarre, Olivier; Voisin, Claire Hyper-Kähler fourfolds and Grassmann geometry, J. Reine Angew. Math., Tome 649 (2010), pp. 63-87 | Zbl: 1417.14028

[15] Deligne, Pierre Théorie de Hodge. II, Publ. Math., Inst. Hautes Étud. Sci., Tome 40 (1971), pp. 5-57 | Zbl: 0219.14007

[16] Deligne, Pierre Théorie de Hodge. III, Publ. Math., Inst. Hautes Étud. Sci., Tome 44 (1974), pp. 5-77 | Zbl: 0237.14003

[17] de Fernex, Tommaso; Kollár, János; Xu, Chen Yang The dual complex of singularities, Higher dimensional algebraic geometry (Advanced Studies in Pure Mathematics) Tome 74, Mathematical Society of Japan, 2017, pp. 103-130 | Zbl: 1388.14107

[18] Friedman, Robert Simultaneous resolution of threefold double points, Math. Ann., Tome 274 (1986) no. 4, pp. 671-689 | Zbl: 0576.14013

[19] Friedman, Robert; Morrison, David R. The birational geometry of degenerations: an overview, Birational geometry of degenerations (Progress in Mathematics) Tome 29, Birkhäuser, 1981, pp. 1-32 | Zbl: 0508.14024

[20] Fujino, Osamu What is log terminal?, Flips for 3-folds and 4-folds (Oxford Lecture Series in Mathematics and its Applications) Tome 35, Oxford University Press, 2007, pp. 49-62 | Zbl: 1286.14024

[21] Fujino, Osamu Semi-stable minimal model program for varieties with trivial canonical divisor, Proc. Japan Acad., Ser. A, Tome 87 (2011) no. 3, pp. 25-30 | Zbl: 1230.14016

[22] Greb, Daniel; Kebekus, Stefan; Kovács, Sándor J. Differential forms on log canonical spaces, Publ. Math., Inst. Hautes Étud. Sci., Tome 114 (2011), pp. 87-169 | Zbl: 1258.14021

[23] Greb, Daniel; Lehn, Christian; Rollenske, Sönke Lagrangian fibrations on hyperkähler manifolds – on a question of Beauville, Ann. Sci. Éc. Norm. Supér., Tome 46 (2013) no. 3, pp. 375-403 | Zbl: 1281.32016

[24] Griffiths, Phillip A. On the periods of certain rational integrals. I, II, Ann. Math., Tome 90 (1969), p. 460-495, 496–541 | Zbl: 0215.08103

[25] Griffiths, Phillip A.; Schmid, Wilfried Recent developments in Hodge theory: a discussion of techniques and results, Discrete subgroups of Lie groups and applications to moduli (ATA Institute of Fundamental Research Studies in Mathematics) Tome 7, Oxford University Press; Tata Institute of Fundamental Research, 1975, pp. 31-127 | Zbl: 0355.14003

[26] Gross, Mark; Siebert, Bernd From real affine geometry to complex geometry, Ann. Math., Tome 174 (2011) no. 3, pp. 1301-1428 | Zbl: 1266.53074

[27] Gross, Mark; Wilson, Pelham M. H. Large complex structure limits of K3 surfaces, J. Differ. Geom., Tome 55 (2000) no. 3, pp. 475-546 | Zbl: 1027.32021

[28] Gulbrandsen, Martin G.; Halle, Lars H.; Hulek, Klaus A GIT construction of degenerations of Hilbert schemes of points (2016) (https://arxiv.org/abs/1604.00215, to appear in Doc. Math.)

[29] Gulbrandsen, Martin G.; Halle, Lars H.; Hulek, Klaus; Zhang, Ziyu The geometry of degenerations of Hilbert schemes of points (2018) (https://arxiv.org/abs/1802.00622)

[30] Halle, Lars H.; Nicaise, Johannes Motivic zeta functions of degenerating Calabi-Yau varieties, Math. Ann., Tome 370 (2018) no. 3-4, pp. 1277-1320 | Zbl: 06826708

[31] Hassett, Brendan Special cubic fourfolds, Compos. Math., Tome 120 (2000) no. 1, pp. 1-23 | Zbl: 0956.14031

[32] Huybrechts, Daniel Compact hyper-Kähler manifolds: basic results, Invent. Math., Tome 135 (1999) no. 1, pp. 63-113 | Zbl: 0953.53031

[33] Huybrechts, Daniel A global Torelli theorem for hyperkähler manifolds [after M. Verbitsky], Séminaire Bourbaki. Volume 2010/2011 (Astérisque) Tome 348, Société Mathématique de France, 2012, pp. 375-403 | Zbl: 1272.32014

[34] Hwang, Jun-Muk Base manifolds for fibrations of projective irreducible symplectic manifolds, Invent. Math., Tome 174 (2008) no. 3, pp. 625-644 | Zbl: 1161.14029

[35] Kawamata, Yujiro Flops connect minimal models, Publ. Res. Inst. Math. Sci., Tome 44 (2008) no. 2, pp. 419-423 | Zbl: 1145.14014

[36] Kawamata, Yujiro; Matsuda, Katsumi; Matsuki, Kenji Introduction to the minimal model problem, Algebraic geometry (Sendai, 1985) (Advanced Studies in Pure Mathematics) Tome 10, American Mathematical Society, 1985, pp. 283-360 | Zbl: 0672.14006

[37] Kollár, János Singularities of the minimal model program, Cambridge Tracts in Mathematics, Tome 200, Cambridge University Press, 2013, x+370 pages (with a collaboration of Sándor Kovács) | Zbl: 1282.14028

[38] Kollár, János; Kovács, Sándor J. Log canonical singularities are Du Bois, J. Am. Math. Soc., Tome 23 (2010) no. 3, pp. 791-813 | Zbl: 1202.14003

[39] Kollár, János; Mori, Shigefumi Classification of three-dimensional flips, J. Am. Math. Soc., Tome 5 (1992) no. 3, pp. 533-703 | Zbl: 0773.14004

[40] Kollár, János; Mori, Shigefumi Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, Tome 134, Cambridge University Press, 1998 | Zbl: 0926.14003

[41] Kollár, János; Nicaise, Johannes; Xu, Chen Yang Semi-stable extensions over 1-dimensional bases, Acta Math. Sin., Engl. Ser., Tome 34 (2018) no. 1, pp. 103-113 | Zbl: 06839516

[42] Kollár, János; Xu, Chen Yang The dual complex of Calabi-Yau pairs, Invent. Math., Tome 205 (2016) no. 3, pp. 527-557 | Zbl: 1360.14056

[43] Kontsevich, Maxim; Soibelman, Yan Homological mirror symmetry and torus fibrations, Symplectic geometry and mirror symmetry (Seoul, 2000), World Scientific, 2000, pp. 203-263 | Zbl: 1072.14046

[44] Kontsevich, Maxim; Soibelman, Yan Affine structures and non-Archimedean analytic spaces, The unity of mathematics (Progress in Mathematics) Tome 244, Birkhäuser, 2006, pp. 321-385 | Zbl: 1114.14027

[45] Kovács, Sándor J. Rational, log canonical, Du Bois singularities: on the conjectures of Kollár and Steenbrink, Compos. Math., Tome 118 (1999) no. 2, pp. 123-133 | Zbl: 0962.14011

[46] Kulikov, Viktor S. Degenerations of K3 surfaces and Enriques surfaces, Math. USSR, Izv., Tome 11 (1977) no. 5, pp. 957-989 | Zbl: 0387.14007

[47] Kuznetsov, Alexander G. Derived category of a cubic threefold and the variety V 14 , Tr. Mat. Inst. Steklova, Tome 246 (2004) no. 183, pp. 183-207 | Zbl: 1107.14028

[48] Lai, Ching-Jui Varieties fibered by good minimal models, Math. Ann., Tome 350 (2011) no. 3, pp. 533-547 | Zbl: 1221.14018

[49] Laza, Radu The moduli space of cubic fourfolds via the period map, Ann. Math., Tome 172 (2010) no. 1, pp. 673-711 | Zbl: 1201.14026

[50] Laza, Radu; Saccà, Giulia; Voisin, Claire A hyper-Kähler compactification of the intermediate Jacobian fibration associated with a cubic 4-fold, Acta Math., Tome 218 (2017) no. 1, pp. 55-135 | Zbl: 06826204

[51] Lehn, Christian Twisted cubics on singular cubic fourfolds - on Starr’s fibration, Math. Z., Tome 290 (2018) no. 1-2, pp. 379-388

[52] Lehn, Christian; Lehn, Manfred; Sorger, Christoph; van Straten, Duco Twisted cubics on cubic fourfolds, J. Reine Angew. Math., Tome 731 (2017), pp. 87-128 | Zbl: 1376.53096

[53] Looijenga, Eduard The period map for cubic fourfolds, Invent. Math., Tome 177 (2009) no. 1, pp. 213-233 | Zbl: 1177.32010

[54] Markushevich, Dimitri G.; Tikhomirov, Alexander S. The Abel-Jacobi map of a moduli component of vector bundles on the cubic threefold, J. Algebr. Geom., Tome 10 (2001) no. 1, pp. 37-62 | Zbl: 0987.14028

[55] Miyaoka, Yoichi; Mori, Shigefumi A numerical criterion for uniruledness, Ann. Math., Tome 124 (1986) no. 1, pp. 65-69 | Zbl: 0606.14030

[56] Morgan, John W. Topological triviality of various analytic families, Duke Math. J., Tome 50 (1983) no. 1, pp. 215-225 | Zbl: 0543.14010

[57] Morrison, David R. The Clemens-Schmid exact sequence and applications, Topics in transcendental algebraic geometry (Princeton, 1981/1982) (Annals of Mathematics Studies) Tome 106, Princeton University Press, 1981, pp. 101-119 | Zbl: 0576.32034

[58] Mukai, Shigeru Polarized K3 surfaces of genus thirteen, Moduli spaces and arithmetic geometry (Advanced Studies in Pure Mathematics) Tome 45, Mathematical Society of Japan, 2006, pp. 315-326 | Zbl: 1117.14040

[59] Mustaţă, Mircea; Nicaise, Johannes Weight functions on non-Archimedean analytic spaces and the Kontsevich-Soibelman skeleton, Algebr. Geom., Tome 2 (2015) no. 3, pp. 365-404 | Zbl: 1322.14044

[60] Nagai, Yasunari On monodromies of a degeneration of irreducible symplectic Kähler manifolds, Math. Z., Tome 258 (2008) no. 2, pp. 407-426 | Zbl: 1140.14008

[61] Nagai, Yasunari Gulbrandsen–Halle–Hulek degeneration and Hilbert–Chow morphism (2017) (https://arxiv.org/abs/1709.01240)

[62] Namikawa, Yoshinori Deformation theory of singular symplectic n-folds, Math. Ann., Tome 319 (2001) no. 3, pp. 597-623 | Zbl: 0989.53055

[63] Namikawa, Yoshinori On deformations of -factorial symplectic varieties, J. Reine Angew. Math., Tome 599 (2006), pp. 97-110 | Zbl: 1122.14029

[64] Nicaise, Johannes; Xu, Chen Yang The essential skeleton of a degeneration of algebraic varieties, Am. J. Math., Tome 138 (2016) no. 6, pp. 1645-1667 | Zbl: 1375.14092

[65] O’Grady, Kieran; Rapagnetta, Antonio Lagrangian sheaves on cubic fourfolds (2014) (unpublished manuscript)

[66] Persson, Ulf On degenerations of algebraic surfaces, Mem. Am. Math. Soc., Tome 11 (1977) no. 189 | Zbl: 0368.14008

[67] Persson, Ulf; Pinkham, Henry Degeneration of surfaces with trivial canonical bundle, Ann. Math., Tome 113 (1981) no. 1, pp. 45-66 | Zbl: 0426.14015

[68] Roan, Shi-Shyr Degeneration of K3 and Abelian surfaces (1975) (Ph. D. Thesis)

[69] Schwald, Martin Low degree Hodge theory for klt varieties (2016) (https://arxiv.org/abs/1612.01919)

[70] Shah, Jayant Insignificant limit singularities of surfaces and their mixed Hodge structure, Ann. Math., Tome 109 (1979) no. 3, pp. 497-536 | Zbl: 0414.14022

[71] Shah, Jayant A complete moduli space for K3 surfaces of degree 2, Ann. Math., Tome 112 (1980) no. 3, pp. 485-510 | Zbl: 0412.14016

[72] Shah, Jayant Degenerations of K3 surfaces of degree 4, Trans. Am. Math. Soc., Tome 263 (1981) no. 2, pp. 271-308 | Zbl: 0456.14019

[73] Steenbrink, Joseph H. M. Cohomologically insignificant degenerations, Compos. Math., Tome 42 (1980) no. 3, pp. 315-320 | Zbl: 0428.32017

[74] Strominger, Andrew; Yau, Shing-Tung; Zaslow, Eric Mirror symmetry is T-duality, Nuclear Phys. B, Tome 479 (1996) no. 1, pp. 243-259

[75] Todorov, Andrei N. Moduli of hyper-Kählerian manifolds (1990) (preprint 90-41, http://www.mpim-bonn.mpg.de/preblob/4585)

[76] Verbitsky, Misha Mirror symmetry for hyper-Kähler manifolds, Mirror symmetry, III (Montreal, 1995) (AMS/IP Studies in Advanced Mathematics) Tome 10, American Mathematical Society, 1995, pp. 115-156 | Zbl: 0926.32036

[77] Verbitsky, Misha Cohomology of compact hyper-Kähler manifolds and its applications, Geom. Funct. Anal., Tome 6 (1996) no. 4, pp. 601-611 | Zbl: 0861.53069

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

[79] Voisin, Claire Degenerations de Lefschetz et variations de structures de Hodge, J. Differ. Geom., Tome 31 (1990) no. 2, pp. 527-534 | Zbl: 0663.14005

[80] Wahl, Jonathan M. Equisingular deformations of normal surface singularities. I, Ann. Math., Tome 104 (1976) no. 2, pp. 325-356 | Zbl: 0358.14007

[81] Wang, Chin-Lung On the incompleteness of the Weil-Petersson metric along degenerations of Calabi-Yau manifolds, Math. Res. Lett., Tome 4 (1997) no. 1, pp. 157-171 | Zbl: 0881.32017

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