Remarks on degenerations of hyper-Kähler manifolds  [ Remarques sur les dégénérescences des variétés hyper-kählériennes ]
Annales de l'Institut Fourier, à paraître, 46 p.

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.

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.

Publié le : 2019-03-08
Classification:  14B05,  14D05,  14J32,  14E99
Mots clés: Variété hyper-kählérienne, dégénérescence, déformations, théorème de Torelli
@unpublished{AIF_0__0_0_A36_0,
     author = {Koll\'ar, J\'anos and Laza, Radu and Sacc\`a, Giulia and Voisin, Claire},
     title = {Remarks on degenerations of hyper-K\"ahler manifolds},
     note = {to appear in \emph{Annales de l'Institut Fourier}},
}
Kollár, János; Laza, Radu; Saccà, Giulia; Voisin, Claire. Remarks on degenerations of hyper-Kähler manifolds. Annales de l'Institut Fourier, à paraître, 46 p.

[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), Springer (Lecture Notes in Mathematics) Tome 947 (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, Mathematical Society of Japan (Advanced Studies in Pure Mathematics) Tome 74 (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, Birkhäuser (Progress in Mathematics) Tome 29 (1981), pp. 1-32 | Zbl 0508.14024

[20] Fujino, Osamu What is log terminal?, Flips for 3-folds and 4-folds, Oxford University Press (Oxford Lecture Series in Mathematics and its Applications) Tome 35 (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, Oxford University Press; Tata Institute of Fundamental Research (ATA Institute of Fundamental Research Studies in Mathematics) Tome 7 (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, Société Mathématique de France (Astérisque) Tome 348 (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), American Mathematical Society (Advanced Studies in Pure Mathematics) Tome 10 (1985), pp. 283-360 | Zbl 0672.14006

[37] Kollár, János Singularities of the minimal model program, Cambridge University Press, Cambridge Tracts in Mathematics, Tome 200 (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 University Press, Cambridge Tracts in Mathematics, Tome 134 (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, Birkhäuser (Progress in Mathematics) Tome 244 (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), Princeton University Press (Annals of Mathematics Studies) Tome 106 (1981), pp. 101-119 | Zbl 0576.32034

[58] Mukai, Shigeru Polarized K3 surfaces of genus thirteen, Moduli spaces and arithmetic geometry, Mathematical Society of Japan (Advanced Studies in Pure Mathematics) Tome 45 (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, Brandeis University (USA) (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), American Mathematical Society (AMS/IP Studies in Advanced Mathematics) Tome 10 (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