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NO COHOMOLOGICALLY TRIVIAL NONTRIVIAL AUTOMORPHISM OF GENERALIZED KUMMER MANIFOLDS

  • KEIJI OGUISO (a1) (a2)
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.

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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.

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[BHPV04] 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.
[Be83] Beauville, A., Variétés Kähleriennes dont la premiére classe de Chern est nulle , J. Differential Geom. 18 (1983), 755782.
[Be83-2] 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, 126.
[BNS11] Boissière, S., Nieper-Wisskirchen, M. and Sarti, A., Higher dimensional Enriques varieties and automorphisms of generalized Kummer varieties , J. Math. Pure Appl. 95 (2011), 553563.
[BR75] Burns, D. Jr. and Rapoport, M., On the Torelli problem for kählerian K-3 surfaces , Ann. Sci. Éc. Norm. Supér. 8 (1975), 235273.
[Ca12] Camere, C., Symplectic involutions of holomorphic symplectic four-folds , Bull. Lond. Math. Soc. 44 (2012), 687702.
[De10] Debarre, O., On the Euler characteristic of generalized Kummer varieties , Amer. J. Math. 121 (1999), 577586.
[Do84] Dolgachev, I., On automorphisms of Enriques surfaces , Invent. Math. 76 (1984), 163177.
[Go94] Göttsche, L., Hilbert Schemes of Zero-Dimensional Subschemes of Smooth Varieties, Lecture Notes in Mathematics 1572 , Springer, Berlin, 1994, x+196 pp.
[GS93] Göttsche, L. and Soergel, W., Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces , Math. Ann. 296 (1993), 235245.
[GHJ03] 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.
[Hu99] Huybrechts, D., Compact hyper-Kähler manifolds: basic results , Invent. Math. 135 (1999), 63113; Erratum: “Compact hyper-Kähler manifolds: basic results”, Invent. Math. 152 (2003), 209–212.
[HT13] Hassett, B. and Tschinkel, Y., Hodge theory and Lagrangian planes on generalized Kummer fourfolds , Mosc. Math. J. 13 (2013), 3356.
[Hu12] Huybrechts, D., A global Torelli theorem for hyperkaehler manifolds (after Verbitsky), Séminaire Bourbaki: Vol. 2010/2011. Exposés 1027–1042 , Astérisque 348 (2012), 375403.
[Ka84] Katsura, T., The unirationality of certain elliptic surfaces in characteristic p , Tohoku Math. J. (2) 36(2) (1984), 217231.
[Ma11] 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, 257322.
[MM17] Markman, E. and Mehrotra, S., Hilbert schemes of K3 surfaces are dense in moduli , Math. Nachr. 290 (2017), 876884.
[Mo13] Mongardi, G., On natural deformations of symplectic automorphisms of manifolds of K3[n] type , C. R. Math. Acad. Sci. Paris 351(13–14) (2013), 561564.
[Mu10] Mukai, S., Numerically trivial involutions of Kummer type of an Enriques surface , Kyoto J. Math. 50 (2010), 889902.
[MN84] Mukai, S. and Namikawa, Y., Automorphisms of Enriques surfaces which act trivially on the cohomology groups , Invent. Math. 77 (1984), 383397.
[PS71] 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), 530572.
[St77] 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, 525563.
[Ve13] Verbitsky, M., Mapping class group and a global Torelli theorem for hyperkähler manifolds , Duke Math. J. 162 (2013), 29292986.
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Nagoya Mathematical Journal
  • ISSN: 0027-7630
  • EISSN: 2152-6842
  • URL: /core/journals/nagoya-mathematical-journal
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