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The Hodge conjecture for general Prym varieties. (English) Zbl 1060.14044

From the introduction: The authors show that the Mumford-Tate group \(H(P)\) of a general Prym variety \(P\) of a double cover \(C\to D\) of projective curves is isomorphic to the full symplectic group \(\text{Sp} (2g)\); where the class in \(\Lambda^2 H^1(P(\mathbb{C}), \mathbb{Q})=H^2(P (\mathbb{C}),\mathbb{Q})\) which is stabilized by this group is the first Chern class of the natural polarization on the Prym variety. Invariant theory [see H. Weyl, “The classical groups, their invariants and representations” (1939; Zbl 0020.20601)] then implies that the only Hodge cycles on \(P\) are powers (under cup-product) of this polarization class. In particular the authors obtain the Hodge conjecture for \(P\) as a consequence of this result. As a particular case, the Néron-Severi group of a general Prym variety is \(\mathbb{Z}\). This was proved earlier by S. Pirola [Math. Ann. 282, 361–368 (1988; Zbl 0625.14024)].

MSC:

14H40 Jacobians, Prym varieties
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
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References:

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