Remarks on degenerations of hyper-Kähler manifolds

Kollár, János; Laza, Radu; Saccà, Giulia; Voisin, Claire

doi:10.5802/aif.3228

Kollár, János ¹; Laza, Radu ²; Saccà, Giulia ³; Voisin, Claire ⁴

¹ Princeton University Princeton, NJ 08544 (USA)
² Stony Brook University Stony Brook, NY 11794 (USA)
³ Columbia University New York, NY 10027 (USA)
⁴ Collège de France 3 rue d’Ulm 75005 Paris (France)

Annales de l'Institut Fourier, Volume 68 (2018) no. 7, pp. 2837-2882.

Abstract
Résumé

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: 2019-05-24

DOI: 10.5802/aif.3228

Classification: 14B05, 14D05, 14J32, 14E99
Keywords: Hyper-Kähler manifold, degeneration, deformations, Torelli theorem

Author's affiliations:

Kollár, János ¹; Laza, Radu ²; Saccà, Giulia ³; Voisin, Claire ⁴

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{AIF_2018__68_7_2837_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},
     journal = {Annales de l'Institut Fourier},
     pages = {2837--2882},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {68},
     number = {7},
     year = {2018},
     doi = {10.5802/aif.3228},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3228/}
}

TY  - JOUR
AU  - Kollár, János
AU  - Laza, Radu
AU  - Saccà, Giulia
AU  - Voisin, Claire
TI  - Remarks on degenerations of hyper-Kähler manifolds
JO  - Annales de l'Institut Fourier
PY  - 2018
SP  - 2837
EP  - 2882
VL  - 68
IS  - 7
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3228/
UR  - https://doi.org/10.5802/aif.3228
DO  - 10.5802/aif.3228
LA  - en
ID  - AIF_2018__68_7_2837_0
ER  -

%0 Journal Article
%A Kollár, János
%A Laza, Radu
%A Saccà, Giulia
%A Voisin, Claire
%T Remarks on degenerations of hyper-Kähler manifolds
%J Annales de l'Institut Fourier
%D 2018
%P 2837-2882
%V 68
%N 7
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.3228
%R 10.5802/aif.3228
%G en
%F AIF_2018__68_7_2837_0

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/

References
Cited by

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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), Volume 7, Oxford University Press; Tata Institute of Fundamental Research, 1975, pp. 31-127 | Zbl

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

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

[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., Volume 370 (2018) no. 3-4, pp. 1277-1320 | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

[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

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

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

[46] Kulikov, Viktor S. Degenerations of $K 3$ surfaces and Enriques surfaces, Math. USSR, Izv., Volume 11 (1977) no. 5, pp. 957-989 | Zbl

[47] Kuznetsov, Alexander G. Derived category of a cubic threefold and the variety $V_{14}$ , Tr. Mat. Inst. Steklova, Volume 246 (2004) no. 183, pp. 183-207 | Zbl

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

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

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

[51] Lehn, Christian Twisted cubics on singular cubic fourfolds - on Starr’s fibration, Math. Z., Volume 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., Volume 731 (2017), pp. 87-128 | Zbl

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

[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., Volume 10 (2001) no. 1, pp. 37-62 | Zbl

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

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

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

[58] Mukai, Shigeru Polarized $K 3$ surfaces of genus thirteen, Moduli spaces and arithmetic geometry (Advanced Studies in Pure Mathematics), Volume 45, Mathematical Society of Japan, 2006, pp. 315-326 | Zbl

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

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

[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., Volume 319 (2001) no. 3, pp. 597-623 | Zbl

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

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

[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., Volume 11 (1977) no. 189 | Zbl

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

[68] Roan, Shi-Shyr Degeneration of $K 3$ 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., Volume 109 (1979) no. 3, pp. 497-536 | Zbl

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

[72] Shah, Jayant Degenerations of $K 3$ surfaces of degree $4$ , Trans. Am. Math. Soc., Volume 263 (1981) no. 2, pp. 271-308 | Zbl

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

[74] Strominger, Andrew; Yau, Shing-Tung; Zaslow, Eric Mirror symmetry is $T$ -duality, Nuclear Phys. B, Volume 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), Volume 10, American Mathematical Society, 1995, pp. 115-156 | Zbl

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

[78] 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

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

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

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

Cited by Sources:

ANNALES DE L'INSTITUT FOURIER

ARTICLES TO APPEAR

BROWSE ISSUES

ABOUT THE JOURNAL

EDITORIAL BOARD

SUBMIT A PAPER

SUBSCRIPTION