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NO COHOMOLOGICALLY TRIVIAL NONTRIVIAL AUTOMORPHISM OF GENERALIZED KUMMER MANIFOLDS

Part of: Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  05 November 2018

KEIJI OGUISO
KEIJI OGUISO*
Affiliation:
Mathematical Sciences, the University of Tokyo, Meguro Komaba 3-8-1, Tokyo, Japan email oguiso@ms.u-tokyo.ac.jp Korea Institute for Advanced Study, Hoegiro 87, Seoul, 133-722, Korea
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Abstract

For a hyper-Kähler manifold deformation equivalent to a generalized Kummer manifold, we prove that the action of the automorphism group on the total Betti cohomology group is faithful. This is a sort of generalization of a work of Beauville and a more recent work of Boissière, Nieper-Wisskirchen, and Sarti, concerning the action of the automorphism group of a generalized Kummer manifold on the second cohomology group.

MSC classification

Primary: 14J50: Automorphisms of surfaces and higher-dimensional varieties 14J40: $n$-folds ($n>4$)
Type
Article
Information
Nagoya Mathematical Journal , Volume 239 , September 2020 , pp. 110 - 122
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Copyright
© 2018 Foundation Nagoya Mathematical Journal

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Footnotes

The author is supported by JSPS Grant-in-Aid (S) No 25220701, JSPS Grant-in-Aid (S) 15H05738, JSPS Grant-in-Aid (B) 15H03611, and by KIAS Scholar Program.

References

Barth, W. P., Hulek, K., Peters, C. A. M. and Van de Ven, A., Compact complex surfaces, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics 4, Springer, Berlin, 2004, xii+436 pp.10.1007/978-3-642-57739-0Google Scholar
Beauville, A., Variétés Kähleriennes dont la premiére classe de Chern est nulle, J. Differential Geom. 18 (1983), 755–782.10.4310/jdg/1214438181Google Scholar
Beauville, A., “Some remarks on Kähler manifolds with c 1 = 0”, in Classification of Algebraic and Analytic Manifolds (Katata, 1982), Progr. Math. 39, Birkhäuser Boston, Boston, MA, 1983, 1–26.Google Scholar
Boissière, S., Nieper-Wisskirchen, M. and Sarti, A., Higher dimensional Enriques varieties and automorphisms of generalized Kummer varieties, J. Math. Pure Appl. 95 (2011), 553–563.10.1016/j.matpur.2010.12.003Google Scholar
Burns, D. Jr. and Rapoport, M., On the Torelli problem for kählerian K-3 surfaces, Ann. Sci. Éc. Norm. Supér. 8 (1975), 235–273.10.24033/asens.1287Google Scholar
Camere, C., Symplectic involutions of holomorphic symplectic four-folds, Bull. Lond. Math. Soc. 44 (2012), 687–702.Google Scholar
Debarre, O., On the Euler characteristic of generalized Kummer varieties, Amer. J. Math. 121 (1999), 577–586.Google Scholar
Dolgachev, I., On automorphisms of Enriques surfaces, Invent. Math. 76 (1984), 163–177.Google Scholar
Göttsche, L., Hilbert Schemes of Zero-Dimensional Subschemes of Smooth Varieties, Lecture Notes in Mathematics 1572, Springer, Berlin, 1994, x+196 pp.Google Scholar
Göttsche, L. and Soergel, W., Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces, Math. Ann. 296 (1993), 235–245.Google Scholar
Gross, M., Huybrechts, D. and Joyce, D., Calabi–Yau Manifolds and Related Geometries. Lectures from the Summer School held in Nordfjordeid, June 2001, Universitext, Springer, Berlin, 2003.Google Scholar
Huybrechts, D., Compact hyper-Kähler manifolds: basic results, Invent. Math. 135 (1999), 63–113; Erratum: “Compact hyper-Kähler manifolds: basic results”, Invent. Math. 152 (2003), 209–212.10.1007/s002220050280Google Scholar
Hassett, B. and Tschinkel, Y., Hodge theory and Lagrangian planes on generalized Kummer fourfolds, Mosc. Math. J. 13 (2013), 33–56.10.17323/1609-4514-2013-13-1-33-56Google Scholar
Huybrechts, D., A global Torelli theorem for hyperkaehler manifolds (after Verbitsky), Séminaire Bourbaki: Vol. 2010/2011. Exposés 1027–1042, Astérisque 348 (2012), 375–403.Google Scholar
Katsura, T., The unirationality of certain elliptic surfaces in characteristic p, Tohoku Math. J. (2) 36(2) (1984), 217–231.10.2748/tmj/1178228849Google Scholar
Markman, E., “A survey of Torelli and monodromy results for holomorphic-symplectic varieties”, in Complex and Differential Geometry, Springer Proc. Math. 8, Springer, Heidelberg, 2011, 257–322.Google Scholar
Markman, E. and Mehrotra, S., Hilbert schemes of K3 surfaces are dense in moduli, Math. Nachr. 290 (2017), 876–884.Google Scholar
Mongardi, G., On natural deformations of symplectic automorphisms of manifolds of K3[n] type, C. R. Math. Acad. Sci. Paris 351(13–14) (2013), 561–564.Google Scholar
Mukai, S., Numerically trivial involutions of Kummer type of an Enriques surface, Kyoto J. Math. 50 (2010), 889–902.10.1215/0023608X-2010-017Google Scholar
Mukai, S. and Namikawa, Y., Automorphisms of Enriques surfaces which act trivially on the cohomology groups, Invent. Math. 77 (1984), 383–397.Google Scholar
Pjateckii-Shapiro, I. I. and Shafarevich, I. R., Torelli’s theorem for algebraic surfaces of type K3, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 530–572.Google Scholar
Steenbrink, J. H. M., Mixed Hodge Structure on the Vanishing Cohomology, Real and Complex Singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976), Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, 525–563.Google Scholar
Verbitsky, M., Mapping class group and a global Torelli theorem for hyperkähler manifolds, Duke Math. J. 162 (2013), 2929–2986.10.1215/00127094-2382680Google Scholar