Decomposability of Chow groups implies decomposability of cohomology
Journées de géométrie algébrique d'Orsay - Juillet 1992, Astérisque, no. 218 (1993), pp. 227-241. 
@incollection{AST_1993__218__227_0,
     author = {Esnault, H\'el\`ene and Srinivas, V. and Viehweg, Eckart},
     title = {Decomposability of {Chow} groups implies decomposability of cohomology},
     booktitle = {Journ\'ees de g\'eom\'etrie alg\'ebrique d'Orsay - Juillet 1992},
     series = {Ast\'erisque},
     pages = {227--241},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {218},
     year = {1993},
     language = {en},
     url = {http://www.numdam.org/item/AST_1993__218__227_0/}
}
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Esnault, Hélène; Srinivas, V.; Viehweg, Eckart. Decomposability of Chow groups implies decomposability of cohomology, dans Journées de géométrie algébrique d'Orsay - Juillet 1992, Astérisque, no. 218 (1993), pp. 227-241. http://www.numdam.org/item/AST_1993__218__227_0/

[B] Bloch, S. : On an argument of Mumford in the theory of algebraic cycles, Géométrie Algébrique (Ed.: A. Beauville), Angers 1979, 217-221, Sijthoff & Noordhoff.

[B2] Bloch, S.: Lectures on Algebraic Cycles, Duke Univ. Math. Series IV, Durham, N.C., 1980.

[B3] Bloch, S.: Some elementary theorems about algebraic cycles on abelian varieties, Invent. Math. 37 (1976), 215-228.

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[D] Deligne, P.: Théorie de Hodge II, Publ. Math. I.H.E.S. 40 (1972), 5-57.

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[S] Srinivas, V.: Gysin maps and cycle classes for Hodge cohomology, J. Ind. Acad. Sci. (Math. Sci), 103 (1993), 209 - 247.