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)].


14H40 Jacobians, Prym varieties
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
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[1] Arnaud Beauville, Prym varieties and the Schottky problem, Invent. Math. 41 (1977), no. 2, 149-196. · Zbl 0333.14013
[2] Pierre Deligne, James S. Milne, Arthur Ogus, and Kuang-yen Shih, Hodge cycles, motives, and Shimura varieties, Lecture Notes in Mathematics, no. 900, Springer-Verlag, Berlin, 1982. · Zbl 0465.00010
[3] A. Grothendieck, Hodge’s general conjecture is false for trivial reasons, Topology 8 (1969), 299-303. · Zbl 0177.49002
[4] Joe Harris and David Mumford, On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 (1982), no. 1, 23-88, With an appendix by William Fulton. · Zbl 0506.14016
[5] Roger Howe, Remarks on classical invariant theory, Trans. Amer. Math. Soc. 313 (1989), no. 2, 539-570. · Zbl 0674.15021
[6] -, Erratum to: “Remarks on classical invariant theory”, Trans. Amer. Math. Soc. 318 (1990), no. 2, 823.
[7] David Mumford, Prym varieties. I, Contributions to analysis (a collection of papers dedicated to Lipman Bers), 325-350, Academic Press, New-York, 1974. · Zbl 0299.14018
[8] Gian Pietro Pirola, Base number theorem for abelian varieties. An infinitesimal approach, Math. Ann. 282 (1988), no. 3, 361-368. · Zbl 0625.14024
[9] Hermann Weyl, The Classical Groups. Their Invariants and Representations, Princeton University Press, Princeton, N.J., 1939. · Zbl 0020.20601
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