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The Kuga-Satake variety of an abelian surface. (English) Zbl 0559.14031
Let X be a compact Kähler manifold of complex dimension 2 with \(\dim_{{\mathbb{C}}}H^{2,0}(X)=1.\) For a subgroup N of the Néron-Severi group NS(X) and a nonzero element \(\eta \in H_ 4(X,{\mathbb{Q}})\), the author defines the so-called Kuga-Satake variety (actually a torus) KS(X,N,\(\eta)\). He describes concretely the Kuga-Satake varieties KS(X,N,\(\eta)\) up to isogeny for complex tori X or Kummer surfaces X. Moreover, using these results, he shows some relations between the simplicity of X, the Picard number \(\rho\) (X) and the endomorphism ring \(End^ 0(X)\), for an abelian surface X. For Kuga-Satake varieties, one can refer to I. Satake [Nagoya Math. J. 27, 435-466 (1966) and 31, 295-296 (1968; Zbl 0154.206)], M. Kuga and I. Satake [Math. Ann. 169, 239-242 (1967; Zbl 0221.14019)] and P. Deligne [Invent. Math. 15, 206-226 (1972; Zbl 0219.14022)].
Reviewer: T.Sekiguchi

MSC:
14K30 Picard schemes, higher Jacobians
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
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