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A power structure over a ring is a method to give sense to expressions of theform $(1+a_1t+a_2t^2+\ldots)^m$, where $a_i$, $i=1, 2,\ldots$, and $m$ areelements of the ring. The (natural) power structure over the Grothendieck ringof complex quasi-projective varieties appeared to be useful for a number ofapplications. We discuss new examples of $\lambda$- and power structures oversome Grothendieck rings of varieties. The main example is for the Grothendieckring of maps of complex quasi-projective varieties. We describe two natural$\lambda$-structures on it which lead to the same power structure. We show thatthis power structure is effective. In the terms of this power structure wewrite some equations containing classes of Hilbert-Chow morphisms. We describesome generalizations of this construction for maps of varieties with someadditional structures.

Power structure over the Grothendieck ring of maps