  Let M be a quasiprojective algebraic manifold with K_M=0 and G a finite
automorphism group of M acting trivially on the canonical class K_M; for
example, a subgroup G of SL(n,C) acting on C^n in the obvious way. We aim to
study the quotient variety X=M/G and its resolutions Y -> X (especially under
the assumption that Y has K_Y=0) in terms of G-equivariant geometry of M. At
present we know 4 or 5 quite different methods of doing this, taken from string
theory, algebraic geometry, motives, moduli, derived categories, etc.
  For G in SL(n,C) with n=2 or 3, we obtain several methods of cobbling
together a basis of the homology of Y consisting of algebraic cycles in
one-to-one correspondence with the conjugacy classes or the irreducible
representations of G.
