Alastair King

The McKay correspondence as an equivalence of derived categories

Let G be a finite group of automorphisms of a nonsingular complex threefold M such that the canonical bundle omega_M is locally trivial as a G-sheaf. We prove that the Hilbert scheme Y=GHilb M parametrising G-clusters in M is a crepant resolution of X=M/G and that there is a derived equivalence (Fourier- Mukai transform) between coherent sheaves on Y and coherent G-sheaves on M. This identifies the K theory of Y with the equivariant K theory of M, and thus generalises the classical McKay correspondence. Some higher dimensional extensions are possible.

Periodic algebras which are almost Koszul

The preprojective algebra and the trivial extension algebra of a Dynkin quiver (in bipartite orientation) are very close to being a Koszul dual pair of algebras. In this case the usual duality theory may be adapted to show that each algebra has a periodic bimodule resolution built using the other algebra and some extra data: an algebra automorphism. A general theory of such ‘almost Koszul’ algebras is developed and other examples are given.

Homological algebra of twisted quiver bundles

Several important cases of vector bundles with extra structure (such as Higgs bundles and triples) may be regarded as examples of twisted representations of a finite quiver in the category of sheaves of modules on a variety/manifold/ringed space. It is shown that the category of such representations is an abelian category with enough injectives by the construction of an explicit injective resolution. Using this explicit resolution, a long exact sequence is found that computes the Ext groups in this new category in terms of the Ext groups in the old category. The quiver formulation is directly reflected in the form of the long exact sequence. It is also shown that under suitable circumstances, the Ext groups are isomorphic to certain hypercohomology groups.

Helices and vector bundles : seminaire Rudakov

1. Exceptional collections, mutations and helixes A. N. Rudakov 2. Construction of bundles on an elliptic curve S. A. Kuleshov 3. Computing invariants of exceptional bundles on a quadric S. K. Zube and D. Yu Nogin 4. Exceptional bundles of small rank on P1 x P1 D. Yu Nogin 5. On the functors Ext applied to exceptional bundles on P2 A. I. Bondal and A. L. Gorodentsev 6. Homogeneous bundles A. I. Bondal and M. M. Kapranov 7. Exceptional objects and mutations in derived categories A. L. Gorodentsev 8. Helixes, representations of quivers and Koszul algebras A. I. Bondal 9. Exceptional collections on ruled surfaces A. V. Kvichansky and D. Yu Nogin 10. Exceptional bundles on K3 surfaces S. A. Kuleshov 11. Stability of exceptional bundles on three dimensional projective space S. K. Zube 12. A symmetric helix on the Pluker quadric B. V. Karpov.

On the cohomology ring of the moduli space of rank 2 vector bundles on a curve

Abstract Let N g be the moduli space of stable holomorphic vector bundles of rank 2 and fixed determinant of odd degree over a smooth complex projective curve of genus g. This paper gives a complete and very simple description of the rational cohomology ring H ∗ ( N g ) . A structural formula is proved for H ∗ ( N g ) , which was originally conjectured by Mumford. It is shown that the first relation in genus g between the standard generators satisfies a recurrence relation, first found by Zagier, and that the invariant subring for the mapping class group is a complete intersection ring. A Grobner basis is found for the ideal of invariant relations; this leads to a natural monomial basis for H ∗ ( N g ) .

A functorial construction of moduli of sheaves

We show how natural functors from the category of coherent sheaves on a projective scheme to categories of Kronecker modules can be used to construct moduli spaces of semistable sheaves. This construction simplifies or clarifies technical aspects of existing constructions and yields new simpler definitions of theta functions, about which more complete results can be proved.

Rationality of moduli of vector bundles on curves

The moduli space M(r,d) of stable, rank r, degree d vector bundles on a smooth projective curve of genus g>1 is shown to be birational to M(h,0) x A, where h=hcf(r,d) and A is affine space of dimension (r^2-h^2)(g-1). The birational isomorphism is compatible with fixing determinants in M(r,d) and M(h,0) and we obtain as a corollary that the moduli space of bundles of rank r and fixed determinant of degree d is rational, when r and d are coprime. A key ingredient in the proof is the use of a naturally defined Brauer class for the function field of M(r,d).

Torsion waves in metric-affine field theory

The approach of metric-affine field theory is to define spacetime as a real oriented 4-manifold equipped with a metric and an affine connection. The 10 independent components of the metric tensor and the 64 connection coefficients are the unknowns of the theory. We write the Yang-Mills action for the affine connection and vary it both with respect to the metric and the connection. We find a family of spacetimes which are stationary points. These spacetimes are waves of torsion in Minkowski space. We then find a special subfamily of spacetimes with zero Ricci curvature; the latter condition is the Einstein equation describing the absence of sources of gravitation. A detailed examination of this special subfamily suggests the possibility of using it to model the neutrino. Our model naturally contains only two distinct types of particles which may be identified with left-handed neutrinos and right-handed antineutrinos.

Dimer models and cluster categories of Grassmannians

We associate a dimer algebra A to a Postnikov diagram D (in a disc) corresponding to a cluster of minors in the cluster structure of the Grassmannian (Gr) (k,n). We show that A is isomorphic to the endomorphism algebra of a corresponding Cohen-Macaulay module T over the algebra B used to categorify the cluster structure of (Gr) (k,n) by Jensen-King-Su. It follows that B can be realised as the boundary algebra of A, that is, the subalgebra eAe for an idempotent e corresponding to the boundary of the disc. The construction and proof uses an interpretation of the diagram D, with its associated plabic graph and dual quiver (with faces), as a dimer model with boundary. We also discuss the general surface case, in particular computing boundary algebras associated to the annulus.

A categorification of Grassmannian cluster algebras

We describe a ring whose category of Cohen-Macaulay modules provides an additive categorication the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of k-planes in n-space. More precisely, there is a cluster character dened on the category which maps the rigid indecomposable objects to the cluster variables and the maximal rigid objects to clusters. This is proved by showing that the quotient of this category by a single projective-injective object is Geiss-Leclerc-Schroers category Sub Qk, which categories the coordinate ring of the big cell in this Grassmannian.