Let G ⊂ SL(n, C) be a finite subgroup and ϕ: Y → X = C n /G any resolution of singularities of the quotient space. We prove that crepant exceptional prime divisors of Y correspond one-to-one with "junior" conjugacy classes of G. When n = 2 this is a version of the McKay correspondence (with irreducible representations of G replaced by conjugacy classes). In the case n = 3, a resolution with K Y = 0 is known to exist by work of Roan and others; we prove the existence of a basis of H * (Y, Q) by algebraic cycles in one-to-one correspondence with conjugacy classes of G. Our treatment leaves lots of open problems. Contents1. Statement of the results 2. Proofs 3. Examples 4. Discussion References Statement of the resultsLet G ⊂ SL(n, C) be a finite subgroup and X = C n /G the quotient space, an affine variety with K X = 0. A crepant resolution f : Y → X is a resolution of singularities such that K Y = f * K X = 0. A crepant resolution does not necessarily exist in dimension ≥ 4 (see 4.5); it is known to exist in dimension 2 (classical, Du Val), and in dimension 3 by work of a number of people (see for example [Markushevich], [Ito1-3] and [Roan]), but the proofs are computational rather than conceptual.This paper contributes some very easy remarks to the following question raised by [Dixon-Harvey-Vafa-Witten], worked out by [Hirzebruch-Höfer], and now famous among algebraic geometers as the "Physicists' Euler number conjecture". See for example [Roan] for the background.Conjecture 1.1. G ⊂ SL(n, C) is a finite subgroup, X = C n /G the quotient space and f : Y → X a crepant resolution. Then there exists a basis of H * (Y, Q) consisting of algebraic cycles in one-to-one correspondence with conjugacy classes of G.It is an elementary fact that Y has no odd-dimensional cohomology, and that H 2i (Y, Q) is spanned by algebraic cycles (see 4.1).

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The temperature dependences of 'H spin-lattice relaxation times T 1 at the Larmor frequencies of 10.5, 16.0,20.0, and 45.5 MHz, and of IH NMR second moments M 2 were determined for (pyH)AuCI 4 and (pyH)AuBr4' where pyH + indicates a pyridinium ion. The small M 2 data less than 1 G 2 obtained above ca. 380 K indicated that the cations perform reorientational motion rapidly enough about its pseudohexad C;;axis existing at the center of the cation and perpendicular to its plane. Below 140 K, both complexes yielded rigid lattice M 2 values of the cation. It is interesting features for these complexes that motional narrowing for the NMR line occurs over an extremely wide range of temperature, the log T, vs. T-' plots are asymmetric about the lH T, minimum with a gentler gradient on the low temperature side, and the minima are extraordinarily long. These unusual results were interpreted by assuming the C;; reorientation of the cations having electric dipoles among nonequivalent in-plane orientations. Three potential barriers to the C;; reorientation were determined as 21.8, 17.2,and 13.5 kJ mol-I for (pyH) AuCI 4, and 22.2, 17.4,and 13.7 kJ mol~I for (pyH)AuBr4' The fade-out phenomenon of 35C1 NQR signals observed for (pyH)AuCI 4 at ca. 230 K when the sample temperature was lowered is also discussed, by referring to the motion of the pyridinium cations.

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Given a scheme Y equipped with a collection of globally generated vector bundles E 1 , . . . , E n , we study the universal morphism from Y to a fine moduli space M(E) of cyclic modules over the endomorphism algebra of E :This generalises the classical morphism to the linear series of a basepoint-free line bundle on a scheme. We describe the image of the morphism and present necessary and sufficient conditions for surjectivity in terms of a recollement of a module category. When the morphism is surjective, this gives a fine moduli space interpretation of the image, and as an application we show that for a small, finite subgroup G ⊂ GL(2, k), every sub-minimal partial resolution of A 2 k /G is isomorphic to a fine moduli space M(E C ) where E C is a summand of the bundle E defining the reconstruction algebra. We also consider applications to Gorenstein affine threefolds, where Reid's recipe sheds some light on the classes of algebra from which one can reconstruct a given crepant resolution.B Alastair Craw a.craw@bath.ac.uk