An exponential function on the set of varieties

Abstract

Let $\cal R$ be either the Grothendieck semiring (semiring with multiplication) of complex algebraic varieties, or the Grothendieck ring of these varieties, or the Grothendieck ring localized by the class of the complex affine line. We introduce a construction which defines operations of taking powers of series over these (semi)rings. This means that, for a power series $A(t)=1+\sum\limits_{i=1}^\infty A_i t^i$ with the coefficients $A_i$ from $\cal R$ and for $M\in {\cal R}$, there is defined a series $(A(t))^M$ (with coefficients from $\cal R$ as well) so that all the usual properties of the exponential function hold.We also express in these terms the generating function of the Hilbert scheme of points (0-dimensional subschemes) on a surface.