Reduced products and sheaves of metric structures

We introduce reduced products in continuous logic, and provide ways to establish satisfaction in the reduced product based on the satisfaction in its fibers, and conversely, on the lines that Horn and Chang did for reduced products in classical logic. We also consider a definition of a sheaf of metric structures, endow its stalks with a metric prestructure, and determine what filters can be used so that a collection of reduced products of arbitrary structures forms a sheaf on a given index space.     

The Atiyah-Segal completion theorem for C*-algebras

We generalize the Atiyah-Segal completion theorem to C ^*-algebras as follows. Let A be a C ^*-algebra with a continuous action of the compact Lie group G. If K _* ^G (A) is finitely generated as an R(G)-module, or under other suitable restrictions, then the I(G)-adic completion K _* ^G (A) is isomorphic to RK _*([A C(EG)]^G), where RK _* is representable K-theory for - C ^*-algebras and EG is a classifying space for G. As a corollary, we show that if and are homotopic actions of G, and if K _*(C ^* (G,A,)) and K _*(C ^* (G,A,)) are finitely generated, then K _*(C ^*(G,A,))K_*(C ^* (G,A,)). We give examples to show that this isomorphism fails without the completions. However, we prove that this isomorphism does hold without the completions if the homotopy is required to be norm continuous.This work was partially supported by an NSF Graduate Fellowship and by an NSF Postdoctoral Fellowship.     

On the Least Number of Fixed Points of an Equivariant Map

The problem on the least number of fixed points of an equivariant map of a compact polyhedron on which a finite group acts is considered. For such a map, the least number of fixed points and the least number of fixed orbits are estimated in terms of invariants of the type of Nielsen numbers. The estimates obtained are sharp. The results are similar to those of P. Wong, but their assumptions are essentially weaker. Some notations are refined. The proofs are constructive.     

The Bass-Heller-Swan formula for the equivariant topological Whitehead group

In the first part of this paper, a geometric definition of theK-theory equivariant nilpotent groups is given. For a finite groupG, the Nil-groups are defined as functors from the category ofG-spaces andG-homotopy classes ofG-maps to Abelian groups. In the nonequivariant case, these groups are isomorphic to the classical algebraic Nil-groups.In the second part, the Bass-Heller-Swan formula is proved for the equivariant topological Whitehead group. The main result of this work is that ifX is a compactG-ANR andG acts trivially onS ¹, then

Some remarks on tautological sheaves on Hilbert schemes of points on a surface

We study tautological sheaves on the Hilbert scheme of points on a smooth quasi-projective algebraic surface by means of the Bridgeland–King–Reid transform. We obtain Brion–Danila’s Formulas for the derived direct image of tautological sheaves or their double tensor product for the Hilbert–Chow morphism; as an application we compute the cohomology of the Hilbert scheme with values in tautological sheaves or in their double tensor product, thus generalizing results previously obtained for tautological bundles.      

Discrete Symmetries of Systems of Isomonodromic Deformations of Second-Order Fuchsian Differential Equations

We compute the discrete affine group of Schlesinger transformations for isomonodromic deformations of a Fuchsian system of second-order differential equations. These transformations are treated as isomorphisms between the moduli spaces of logarithmic sl(2)-connections with given eigenvalues of the residues on ¹. The discrete structure is computed with the use of the modification technique for bundles with connections. The result generalizes the well-known classical computations of symmetries of the hypergeometric equation, the Heun equation, and the sixth Painlevé equation.     

Stability of O(p + 1) × O(p + 1)-Invariant Hypersurfaces with Zero Scalar Curvature in Euclidean Space

Equivariant geometry methods are used to study and classify zero scalarcurvature O(p + 1) × O(p + 1)-invariant hypersurfaces in ^{2p + 2}with p > 1. First of all the O(p + 1) × O(p + 1)-invariant hypersurfaces are classified according to their profile curves, and it is shown that there are complete and embedded examples. Next the Morse index of the complete examples is studied, in particular, it is proved that there exist globally stable examples.These stable examples provide counter-examples, in odddimensions greater than or equal to nine, to a Bernstein-type conjecture in the stable class, for immersions with zero scalar curvature.     

Toric co-Higgs sheaves

We characterise and investigate co-Higgs sheaves and associated algebraic and combinatorial invariants on toric varieties. In particular, we compute explicit examples.     

Surjectivity for Hamiltonian -spaces in -theory

Let be a compact connected Lie group, and a Hamiltonian -space with proper moment map . We give a surjectivity result which expresses the -theory of the symplectic quotient in terms of the equivariant -theory of the original manifold , under certain technical conditions on . This result is a natural -theoretic analogue of the Kirwan surjectivity theorem in symplectic geometry. The main technical tool is the -theoretic Atiyah-Bott lemma, which plays a fundamental role in the symplectic geometry of Hamiltonian -spaces. We discuss this lemma in detail and highlight the differences between the -theory and rational cohomology versions of this lemma.

We also introduce a -theoretic version of equivariant formality and prove that when the fundamental group of is torsion-free, every compact Hamiltonian -space is equivariantly formal. Under these conditions, the forgetful map is surjective, and thus every complex vector bundle admits a stable equivariant structure. Furthermore, by considering complex line bundles, we show that every integral cohomology class in admits an equivariant extension in .

     

Splitting criterion for reflexive sheaves

We give a criterion for a reflexive sheaf to split into a direct sum of line bundles.