On representation theory and the cohomology rings of irreducible compact hyperkähler manifolds of complex dimension four

 Access to full text

 Full (PDF)

1. [1] F. Bogomolov: “Kähler manifolds with trivial canonical class”, Izv. Akad. Nauk. SSSR Ser. Mat., Vol. 38, (1974), pp. 11–21. 
2. [2] F. Bogomolov: “The decomposition of Kähler manifolds with a trivial canonical class”, Mat. Sb. (N.S.), Vol. 38, (1974), pp. 573–575. 
3. [3] F. Bogomolov: “Hamiltonian Kählerian manifolds”, Dolk. Akad. Nauk. SSSR, Vol. 243, (1978), pp. 1101–1104. 
4. [4] F. Bogomolov: “On the cohomology ring of a simple hyperkähler manifold (on the results of Verbisky)”, Geom. Funct. Anal., Vol. 6, (1996), pp. 612–618. http://dx.doi.org/10.1007/BF02247113 Zbl0862.53050
5. [5] A. Beauville: “Variétés Kähleriennes dont la premirè classe de Chern est nulle”, J. Differential Geometry, Vol. 18, (1983), pp. 755–782. Zbl0537.53056
6. [6] C. Chevalley: The Algebraic Theory of Spinors, Columbia University Press, New York, 1954. 
7. [7] D. Guan: “Toward a classification of compact complex homogeneous spaces”, preprint, 1998. 
8. [8] Z. Guan: “Toward a classification of almost homogeneous manifolds I-linearization of the singular extremal rays”, International J. Math., Vol. 8, (1997), pp. 999–1014. http://dx.doi.org/10.1142/S0129167X97000470 Zbl0907.14015
9. [9] D. Guan: “Examples of compact holomorphic symplectic manifolds which are not Kählerian II”, Invent Math., Vol. 121, (1995), pp. 135–145. http://dx.doi.org/10.1007/BF01884293 Zbl0827.32026
10. [10] D. Guan: “Examples of compact holomorphic symplectic manifolds which are not Kählerian III”, Intern. J. of Math., Vol. 6, (1995), pp. 709–718. http://dx.doi.org/10.1142/S0129167X95000298 Zbl0857.32018
11. [11] D. Guan: “On Riemann-Roch formula and bounds of the Betti numbers of irreducible compact hyperkähler manifold of complex dimension four”, first version in 1999, a part of paper appeared in Math. Res. Letters, Vol. 8, (2001), pp. 663–669. Zbl1011.53039
12. [12] J. Humphreys: Introduction to Lie Algebras and Representation Theory, GTM 9, Springer-Verlag, New York, 1987. Zbl0254.17004
13. [13] M. Knus, A. Merkurjev, M. Rost, J. Tignol: The Book of Involutions, Colloquium Publications 44, AMS, Providence, Rhode Island, USA, 1998. 
14. [14] E. Looijenga and V. Lunts: “A Lie algebra attached to a projective variety”, Invent. Math., Vol. 129, (1997), pp. 361–412. http://dx.doi.org/10.1007/s002220050166 Zbl0890.53030
15. [15] K. O'Grady: “Desingularized moduli spaces of sheaves on a K3”, J. Reine Angew. Math., Vol. 512, (1999), pp. 49–117. Zbl0928.14029
16. [16] K. O'Grady: “A new six dimensional irreducible symplectic variety”, preprint, 2000. 
17. [17] W. Scharlau: “Quadratic and Hermitian Forms”, Grundlehren der mathematischen Wissenschaften 270, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1985. 
18. [18] S. Salamon: “Riemannian Geometry and Holonomy Groups”, Pitman Research Notes in Mathematics, Series 201, 1989. Zbl0685.53001
19. [19] S. Salamon: “On the cohomology of Kähler and hyperkähler manifolds”, Topology, Vol. 35, (1996), pp. 137–155. http://dx.doi.org/10.1016/0040-9383(95)00006-2 
20. [20] J. Tits: “Représentations linéaires irréducibles d'un groupe réductif sur un corps quelconque”, J. reine angew. Math., Vol. 247, (1971), pp. 196–220. Zbl0227.20015
21. [21] M. Verbitsky: “Action of the Lie algebra of SO(5) on the cohomology of a hyperkähler manifold”, Functional Anal. appl., Vol. 24, (1991), pp. 229–230. http://dx.doi.org/10.1007/BF01077967 Zbl0717.53041
22. [22] M. Verbitsky: “Cohomology of compact hyperkähler manifolds and its applications”, Geom. Funct. Anal., Vol. 6, (1996), pp. 601–611. http://dx.doi.org/10.1007/BF02247112 Zbl0861.53069
23. [23] M. Verbitsky and D. Kaledin: Hyperkähler Manifolds, International Press, Somerville, Massachusetts, USA, 1999. 
24. [24] S. Yau: “On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation I”, Comm. Pure Appl. Math., Vol. 31, (1978), pp. 339–411. Zbl0369.53059