Tankeev, S. G. 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 \(X_{\eta}\otimes_{\kappa(\eta)}\mathbb{C}\).” Reviewer: Olaf Teschke (Berlin) 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 Keywords:standard conjecture; abelian variety; minimal Néron model; imaginary quadratic field; Hodge conjecture PDFBibTeX XMLCite \textit{S. G. Tankeev}, Math. Notes 109, No. 3, 498--499 (2021; Zbl 1465.14014); translation from Mat. Zametki 109, No. 3, 479--480 (2021) Full Text: DOI 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.