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THE KUGA–SATAKE CONSTRUCTION UNDER DEGENERATION

Part of: Families, fibrations

Published online by Cambridge University Press:  17 May 2019

Stefan Schreieder  and
Andrey Soldatenkov
Stefan Schreieder
Affiliation:
Mathematisches Institut, LMU München, Theresienstr. 39, 80333München, Germany ([email protected])
Andrey Soldatenkov
Affiliation:
Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099Berlin, Germany ([email protected])
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Abstract

We extend the Kuga–Satake construction to the case of limit mixed Hodge structures of K3 type. We use this to study the geometry and Hodge theory of degenerations of Kuga–Satake abelian varieties associated with polarized variations of K3 type Hodge structures over the punctured disc.

Keywords

Hodge structureK3 surfaceabelian varietylimit mixed Hodge structure

MSC classification

Primary: 14D06: Fibrations, degenerations 14D07: Variation of Hodge structures
Secondary: 14D05: Structure of families (Picard-Lefschetz, monodromy, etc.)
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 6 , November 2020 , pp. 2165 - 2182
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Copyright
© Cambridge University Press 2019

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Footnotes

The authors are supported by the SFB/TR 45 ‘Periods, Moduli Spaces and Arithmetic of Algebraic Varieties’ of the DFG (German Research Foundation).

References

Groupes de monodromie en géométrie algébrique. I. Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 I), Dirigé par A. Grothendieck. Avec la collaboration de M. Raynaud et D. S. Rim. Lecture Notes in Mathematics, Volume 288. Springer, Berlin, 1972.Google Scholar
Arapura, D., Bakhtary, P. and Włodarczyk, J., Weights on cohomology, invariants of singularities, and dual complexes, Math. Ann. 357 (2013), 513–550.Google Scholar
Baily, W. L. and Borel, A., Compactification of arithmetic quotients of bounded symmetric domains, Ann. of Math. 84 (1966), 442–528.Google Scholar
Borel, A., Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem, J. Differential Geom. 6 (1972), 543–560.Google Scholar
Bosch, S., Lütkebohmert, W. and Raynaud, M., Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Volume 21 (Springer, Berlin, New York, 1990).Google Scholar
Cattani, E., Kaplan, A. and Schmid, W., Degeneration of Hodge structures, Ann. of Math. (2) 123(3) (1986), 457–535.Google Scholar
Clemens, H., The Néron model for families of intermediate Jacobians acquiring “algebraic” singularities, Publ. Math. Inst. Hautes Études Sci. 58 (1983), 5–18.Google Scholar
Deligne, P., La conjecture de Weil pour les surfaces K3, Invent. Math. 15 (1972), 206–226.Google Scholar
Halle, L. H. and Nicaise, J., Motivic zeta functions of abelian varieties, and the monodromy conjecture, Adv. Math. 227 (2011), 610–653.Google Scholar
Halle, L. H. and Nicaise, J., Jumps and monodromy of abelian varieties, Doc. Math. 16 (2011), 937–968.Google Scholar
Huybrechts, D., Lectures on K3 Surfaces, Cambridge Studies in Advanced Mathematics, Volume 158 (Cambridge University Press, Cambridge, 2016).Google Scholar
Jordan, B. W. and Morrison, D. R., On the Néron models of abelian surfaces with quaternionic multiplication, J. Reine Angew. Math. 447 (1994), 1–22.Google Scholar
Kempf, G., Knudsen, F. F., Mumford, D. and Saint-Donat, B., Toroidal Embeddings I, Lecture Notes in Mathematics, Volume 339 (Springer, Berlin, 1973).Google Scholar
Kollár, J., Laza, R., Saccà, G. and Voisin, C., Remarks on degenerations of hyper-Kähler manifolds, Ann. Inst. Fourier. to appear.Google Scholar
Kuga, M. and Satake, I., Abelian varieties attached to polarized K3 surfaces, Math. Ann. 169 (1967), 239–242.Google Scholar
Kurnosov, N., Soldatenkov, A. and Verbitsky, M., Kuga–Satake construction and cohomology of hyperkähler manifolds, Adv. Math. to appear.Google Scholar
Kulikov, V. S., Degenerations of K3 surfaces and Enriques surfaces, Math. USSR Izvestija 11(5) (1977), 957–989.Google Scholar
Morrison, D. R., The Clemens–Schmid exact sequence and applications, in Topics in Transcendental Algebraic Geometry, Annals of Mathematics Studies, Volume 106, pp. 101–119 (Princeton University Press, Princeton, NJ, 1984).Google Scholar
Schmid, W., Variation of Hodge structure: the singularities of the period mapping, Invent. Math. 22 (1973), 211–319.Google Scholar
Saito, M., Admissible normal functions, J. Algebraic Geom. 5 (1996), 235–276.Google Scholar
Simpson, C. T., Moduli of representations of the fundamental group of a smooth projective variety I, Publ. Math. Inst. Hautes Études Sci. 79 (1994), 47–129.Google Scholar
Stewart, A. and Vologodsky, V., Motivic integral of K3 surfaces over a non-archimedean field, Adv. Math. 228 (2011), 2688–2730.Google Scholar
Voisin, C., A generalization of the Kuga–Satake construction, Pure Appl. Math. Q. 1 (2005), 415–439.Google Scholar
Zucker, S., Generalized intermediate Jacobians and the theorem on normal functions, Invent. Math. 33(3) (1976), 185–222.Google Scholar