Grothendieck ring of varieties with finite groups actions
The orbifold Euler characteristic was introduced in papers of L.Dixon, J.Harvey, C.Vafa, and E.Witten. The orbifold Euler characteristic (as well as (orbifold) Euler) characteristics of higher orders defined by M.Atiyah, J.Bryan and J.Fulman, H.Tamanoi) are defined, in particular, for topological spaces with finite group actions. For a finite group G these characteristics can be considered as (additive) functions on the Grothendieck ring of complex quasi-projective varieties with G-actions. They are not ring homomorphisms from this Grothendieck ring to the ring of integers. We define a Grothendieck ring of varieties with finite groups actions and show that the orbifold Euler characteristic and the Euler characteristics of higher orders can be defined as homomorphisms from this ring to the ring of integers.
One has the Macdonald equation for the generating series of the Euler characteristics of symmetric powers of a space. Macdonald type equations for other invariants with values in rings different from the ring of integers can be formulated in terms of power structures over the rings or in terms of lambda-structures on them. Macdonald type equations for generalized (orbifold) Euler characteristics (with values in the Grothendieck ring K_0(Var_C) of complex quasi-projective varieties) are formulated in terms of a power structure over K_0(Var_C). We describe two natural \lambda-structures on the Grothendieck ring of varieties with finite groups actions and the corresponding power structures over it and show that one of these power structures is "effective" in a certain sense.
The talk is based on a joint work with I.Luengo and A.Melle.