Varieties with very little transcendental cohomology

Given a complex smooth projective algebraic variety X, we define a natural number called the motivic dimension µ(X) which is zero precisely when all the cohomology of X is generated by algebraic cycles. In general, it gives a measure of the amount of transcendental cohomology of X. Alternatively, µ(X) may be rather loosely thought of as measuring the complexity of the motive of X, with Tate motives having µ = 0, motives of curves having µ ≤ 1 and so on. Our interest in this notion stems from the relation to the Hodge conjecture: it is easy to see that it holds for X whenever µ(X) ≤ 3. This paper contains a number of estimates of µ; some elementary, some less so. With these estimates in hand, we conclude this paper by checking or rechecking this conjecture in a number of examples: uniruled fourfolds, rationally connected fivefolds, fourfolds fibred by surfaces with pg = 0, Hilbert schemes of a small number points on surfaces with pg = 0, and generic hypersurfaces. We will work over C. Let H ∗ (−) denote singular cohomology with rational coefficients. The motivic dimension of a smooth projective variety X can be defined