TY - JOUR
T1 - REMARKS ON DEGENERATIONS OF HYPER-KÄHLER MANIFOLDS
AU - Kollár, János
AU - Laza, Radu
AU - Saccà, Giulia
AU - Voisin, Claire
N1 - Funding Information:
The first author (JK) was partially supported by NSF grant DMS-1362960. He also acknowledges the support of Collège de France and FSMP that facilitated the meeting of the other co-authors and the work on this project. The second author (RL) was partially supported by NSF grants DMS-1254812 and DMS-1361143. He is also grateful to ENS and Olivier Debarre for hosting him during his sabbatical year, and to Simons Foundation and FSMP for fellowship support. The third author (GS) also would like to thank Collège de France for hospitality.
Publisher Copyright:
© 2018 Association des Annales de l'Institut Fourier. All rights reserved.
PY - 2018
Y1 - 2018
N2 - Using the Minimal model program, any degeneration of Ktrivial 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 H2, 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.
AB - Using the Minimal model program, any degeneration of Ktrivial 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 H2, 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.
KW - deformations
KW - degeneration
KW - Hyper-Kähler manifold
KW - Torelli theorem
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M3 - Article
AN - SCOPUS:85147350193
SN - 0373-0956
VL - 68
SP - 2837
EP - 2882
JO - Annales de l'Institut Fourier
JF - Annales de l'Institut Fourier
IS - 7
ER -