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On the standard conjecture for compactifications of Néron models of three-dimensional abelian varieties with multiplications in an imaginary quadratic field. (English. Russian original) Zbl 1465.14014

Math. Notes 109, No. 3, 498-499 (2021); translation from Mat. Zametki 109, No. 3, 479-480 (2021).
The author announces the following main theorem:
“Theorem. Grothendieck’s standard conjecture \(B(X)\) of Lefschetz type holds for a smooth complex projective four-dimensional variety \(X\) fibered by abelian varieties (possibly with degeneracies) over a smooth projective curve if the endomorphism ring of the generic geometric fiber is the order of an imaginary quadratic field and the Hodge conjecture about algebraic cycles holds for the Cartesian square of the generic geometric fiber \(X_{\eta}\otimes_{\kappa(\eta)}\mathbb{C}\).”

MSC:

14C25 Algebraic cycles
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14F25 Classical real and complex (co)homology in algebraic geometry
14J30 \(3\)-folds
14K99 Abelian varieties and schemes
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References:

[1] Grothendieck, A., Algebraic Geometry, 0, 193-199 (1969) · Zbl 0201.23301
[2] Lieberman, D., Amer. J. Math., 90, 2, 366-374 (1968) · Zbl 0159.50501 · doi:10.2307/2373533
[3] Tankeev, S. G., Izv. Math., 75, 5, 1047-1062 (2011) · Zbl 1234.14009 · doi:10.1070/IM2011v075n05ABEH002563
[4] Tankeev, S. G., On the standard conjecture for projective compactifications of Néron models of \(3\)-dimensional Abelian varieties, Izv. Math., 85, 1, 0 (2021) · Zbl 1467.14108 · doi:10.1070/IM9005
[5] Tankeev, S. G., Sibirsk. Èlektron. Mat. Izv., 17, 89-125 (2020) · Zbl 1433.14006 · doi:10.33048/semi.2020.17.008
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