On a multiplicative version of Mumford's theorem (Q266540)

A theorem of \textit{H. Esnault} et al. [in: Journées de géométrie algébrique d'Orsay, France, juillet 20-26, 1992. Paris: Société Mathématique de France. 227--241 (1993; Zbl 0815.14003)] states the following. Suppose \(X\) is an irreducible smooth projective variety of dimension \(d\) over the complex numbers, and suppose that the intersection map on Chow groups with rational coefficients \(A^{j_1}(V) \otimes \cdots \otimes A^{j_n}(V) \to A^d(V)\) is surjective for some non-empty Zariski-open subset \(V\subseteq X\) and for some positive integers \(j_1,\ldots,j_n\) such that \(j_1+\cdots +j_n = d\). Then the cup-product map \(H^{j_1}(X,O_X)\otimes\cdots\otimes H^{j_n}(X,O_X) \to H^d(X,O_X)\) is surjective. The proof is based on the method of ``decomposition of the diagonal'' due to Bloch and Srinivas. In this paper, the author extends the aforementioned result by showing that if moreover \(X\) satisfies Grothendieck's standard conjectures, then for all positive integers \(j_1,\ldots,j_n\) the cup-product map \(H^{j_1}(X,O_X)\otimes\cdots\otimes H^{j_n}(X,O_X) \to H^j(X,O_X)\), \(j=j_1+\cdots+j_n\), is surjective provided the intersection map \(A^{j_1}(V) \otimes \cdots \otimes A^{j_n}(V) \to A^j(V)\) is surjective for some non-empty Zariski-open subset \(V\subseteq X\).