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Looking for a better counterexample

This post is doubling my old question asked on Mathoverflow.

Let X be a smooth projective complex variety of dimension d, and let Y\subset X be an ample irreducible divisor. Recall that a cohomology class b\in H^k(X,\mathbf{Q}) is called Y-primitive, if b\smile c_1(Y)^{d-k+1} =0. Here, c_1(Y)\in H^{2}(X,\mathbf{Q}) is the first Chern class of the line bundle associated to Y. We denote by H^k_{\mathrm{prim}}(X,\mathbf{Q}) \subset H^k(X,\mathbf{Q}) the subset of primitive elements.

Let denote …