Mathematics – Algebraic Geometry
Scientific paper
1999-11-22
Mathematics
Algebraic Geometry
20 pages, uses Latex and bourbaki.cls. S\'eminaire Bourbaki, 52\`eme ann\'ee, novembre 1999, no. 867, to appear in Ast\'erisqu
Scientific paper
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.
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