Power Structure over the Grothendieck Ring of Varieties and Generating Series of Hilbert Schemes of Points

摘要

The Grothendieck semiring S_0(V_C) of complex quasi-projective varieties is the semigroup generated by isomorphism classes [X] of such varieties modulo the relation [X] = [X - Y] + [Y] for a Zariski closed subvariety Y is contained in X; the multiplication is defined by the Cartesian product: [X_1] · [X_2] = [X_1 x X_2]. The Grothendieck ring K_0(V_C) is the group generated by these classes with the same relation and the same multiplication. Let L ∈ K_0(V_C) be the class of the complex affine line, and let K_0( V_C)[L~(-1)] be the localization of Grothendieck ring K_0(V_C) with respect to L. A power structure over a (semi)ring R (as in [10]) is a map (1 + T · R[[T]]) x R → 1 + T · R[[T]]: (A(T),m) ∣→ (A(T))~m (A(T) = 1 + a_1T + a_2T~2 + ···, a_i ∈ R, m ∈ R) such that all usual properties of the exponential function hold. Over a ring R, a finitely determined (in a natural sense that we shall describe) power structure is defined by a pre-λ-ring structure on R (see [12]). Described in [10] is a power structure over each of the (semi)rings just defined. They are connected with the pre-λ-ring structure on the Grothendieck ring K_0(V_C) defined by the Kapranov zeta function.