REMARKS ON ABSOLUTE DERHAM AND ABSOLUTE HODGE CYCLES

Let X be a smooth proper variety over a field k of characteristic zero. If alpha is-an-element-of F(i) H(DR)2 i (X/k) is the class of an algebraic cycle, then it has to be an absolute de Rham cycle of moderate growth, a notion we define and compare to Deligne's notion of an absolute Hodge cycle. In fact, absolute Hodge implies absolute de Rham, while it's not clear why the moderate growth is implied by the absoluteness condition.