A power structure over the Grothendieck ring of varieties

Abstract

Let ℛ be either the Grothendieck semiring (semigroup with multiplication) of complex quasi-projective varieties, or the Grothendieck ring of these varieties, or the Grothendieck ring localized by the class double-struck L of the complex affine line. We define a power structure over these (semi)rings. This means that, for a power series A(t) = 1 + ∑ i=1∞ [Ai]ti with the coefficients [Ai] from ℛ and for [M] ∈ ℛ, there is defined a series (A(t))[M], also with coefficients from ℛ, so that all the usual properties of the exponential function hold. In the particular case A(t) = (1 - t)-1, the series (A(t))[M] is the motivic zeta function introduced by M. Kapranov. As an application we express the generating function of the Hilbert scheme of points, 0-dimensional subschemes, on a surface as an exponential of the surface.