Elliptic Operators on Manifolds with Singularities and K-Homology

Elliptic operators on smooth compact manifolds are classified by K-homology. We prove that a similar classification is valid also for manifolds with simplest singularities: isolated conical points and edges. The main ingredients of the proof of these results are: Atiyah–Singer difference construction in the noncommutative case and Poincaré isomorphism in K-theory for (our) singular manifolds. As an application we give a formula in topological terms for the obstruction to Fredholm problems on manifolds with edges.Mathematics Subject Classification (2000): 58J05(Primary), 19K33 35S35 47L15(Secondary)(Received: June 2004)     

The topological types of hypersurface simple K3 singularities

We give a result that relates the diffeomorphism type of the link of a non-degenerate semi-quasi-homogeneous hypersurface simple K3 singularity with the singularities of the normal K  3 surface that appears as the exceptional divisor of the resolution of the singularity. As a result, we show that the links are diffeomorphic to the connected sum of copies of S²×S³$S^2\times S^3$. Moreover, we also show that the topological types of hypersurface simple K3 singularities defined by non-degenerate semi-quasi-homogeneous polynomials are all different.     

Orthogonally Additive Polynomials over C(K) are Measures—A Short Proof

We give a simple proof of the fact that orthogonally additive polynomials on C(K) are represented by regular Borel measures over K. We also prove that the Aron-Berner extension preserves this class of polynomials.     

A Direct Proof that Solutions of the Six Painlevé Equations Have No Movable Singularities Except Poles

The Painlevé property of an nth-order differential equation is that no solution has any movable singularities other than poles. This property is strongly indicative of complete integrability (the existence of n ? 1 integrals). However, the usual technique employed to test for the Painlevé property seeks only movable algebraic (or logarithmic) singularities. More general singularities are ignored. But, the six standard Painlevé equations are known to have no such singularities. Painlevé's proof of this is long and laborious; we give here a direct proof.     

Resolution of null fiber and conormal bundles on the Lagrangian Grassmannian

We study the null fiber of a moment map related to dual pairs. We construct an equivariant resolution of singularities of the null fiber, and get conormal bundles of closed K_\mathbbC{K_{\mathbb{C}}} -orbits in the Lagrangian Grassmannian as categorical quotients. The conormal bundles thus obtained turn out to be a resolution of singularities of the closure of nilpotent K_\mathbbC{K_{\mathbb{C}}} -orbits, which is a “quotient” of the resolution of the null fiber.     

A Geometric Characterization of Viable Sets for Controlled Degenerate Diffusions

For a general controlled diffusion process and an arbitrary closed set K we study the viability, or weak invariance, or controlled invariance, of K, that is, the existence of a control for each initial point in K keeping the trajectory forever in K. By viscosity solutions methods we prove a simple necessary and sufficient condition involving only a deterministic second-order normal cone to K and the data of the diffusion process. We also give an extension to stochastic differential games.     

Thom polynomials, symmetries and incidences of singularities

As an application of the generalized Pontryagin-Thom construction [RSz] here we introduce a new method to compute cohomological obstructions of removing singularities — i.e. Thom polynomials [T]. With the aid of this method we compute some sample results, such as the Thom polynomials associated to all stable singularities of codimension ≤8 between equal dimensional manifolds, and some other Thom polynomials associated to singularities of maps N ⁿ ?P ^n+k for k>0. We also give an application by reproving a weak form of the multiple point formulas of Herbert and Ronga ([H], [Ro2]). As a byproduct of the theory we define the incidence class of singularities, which — the author believes — may turn to be an interesting, useful and simple tool to study incidences of singularities. Oblatum 4-II-1999 & 19-VII-2000?Published online: 30 October 2000     

A general method for construction of (<Emphasis Type="Italic">t</Emphasis>, <Emphasis Type="Italic">n</Emphasis>)-threshold visual secret sharing schemes for color images

This paper is concerned with the construction of basis matrices of visual secret sharing schemes for color images under the (t, n)-threshold access structure, where n ≥ t ≥ 2 are arbitrary integers. We treat colors as elements of a bounded semilattice and regard stacking two colors as the join of the two corresponding elements. We generate n shares from a secret image with K colors by using K matrices called basis matrices. The basis matrices considered in this paper belong to a class of matrices each element of which is represented by a homogeneous polynomial of degree n. We first clarify a condition such that the K matrices corresponding to K homogeneous polynomials become basis matrices. Next, we give an algebraic scheme for the construction of basis matrices. It is shown that under the (t, n)-threshold access structure we can obtain K basis matrices from appropriately chosen K − 1 homogeneous polynomials of degree n by using simple algebraic operations. In particular, we give basis matrices that are unknown so far for the cases of t = 2, 3 and n − 1.     

A Quillen-type spectral sequence for the K-theory of varieties with isolated singularities

We construct a spectral sequence to compute the algebraic K-theory of any quasiprojective scheme X, when X has isolated singularities, using an explicit flasque resolution of the K-theory sheaves. This is a generalization of Quillen's construction for nonsingular varieties. The explicit resolution makes it possible to relate K-theory to intersection theory on singular schemes.Partially supported by NSF grants.Dedicated to A. Grothendieck on his sixtieth birthday     

Some Remarks about the FM-partners of <Emphasis Type="Italic">K</Emphasis>3 Surfaces with Picard Numbers 1 and 2

In this paper we describe some results about K3 surfaces with Picard number 1 and 2. In particular, we give a new simple proof of a theorem due to Oguiso which shows that, given an integer N, there is a K3 surface with Picard number 2 and at least N non-isomorphic FM-partners. We describe also the Mukai vectors of the moduli spaces associated to the FM-partners of K3 surfaces with Picard number 1.     

Homogeneous spaces with polar isotropy

 We consider homogeneous spaces G/K with G a simple compact Lie group, endowed with an arbitrary G-invariant Riemannian metric. We classify those spaces where the action of K on G/K is polar and show that such spaces are locally symmetric. Moreover we give a classification of pairs (G,K) with G compact semisimple such that K has polar linear isotropy representation. Received: 16 May 2002 / Revised version: 8 November 2002 Published online: 3 March 2003 Mathematics Subject Classification (2000): 53C35, 57S15     

Einstein metrics and complex singularities

This paper is concerned with the construction of special metrics on non-compact 4-manifolds which arise as resolutions of complex orbifold singularities. Our study is close in spirit to the construction of the hyperkähler gravitational instantons, but we focus on a different class of singularities. We show that any resolution X of an isolated cyclic quotient singularity admits a complete scalar-flat Kähler metric (which is hyperkähler if and only if K_X is trivial), and that if K_X is strictly nef, then X also admits a complete (non-Kähler) self-dual Einstein metric of negative scalar curvature. In particular, complete self-dual Einstein metrics are constructed on simply-connected non-compact 4-manifolds with arbitrary second Betti number.Deformations of these self-dual Einstein metrics are also constructed: they come in families parameterized, roughly speaking, by free functions of one real variable.All the metrics constructed here are toric (that is, the isometry group contains a 2-torus) and are essentially explicit. The key to the construction is the remarkable fact that toric self-dual Einstein metrics are given quite generally in terms of linear partial differential equations on the hyperbolic plane.     

On potentially <Emphasis Type="Italic">H</Emphasis>-graphic sequences

For given a graph H, a graphic sequence π = (d ₁, d ₂,..., d _n) is said to be potentially H-graphic if there is a realization of π containing H as a subgraph. In this paper, we characterize the potentially (K ₅ − e)-positive graphic sequences and give two simple necessary and sufficient conditions for a positive graphic sequence π to be potentially K ₅-graphic, where K _r is a complete graph on r vertices and K _r-e is a graph obtained from K _r by deleting one edge. Moreover, we also give a simple necessary and sufficient condition for a positive graphic sequence π to be potentially K ₆-graphic. Project supported by National Natural Science Foundation of China (No. 10401010).     

Semialgebraic sets and some versions of Tarski-Seidenberg-MacIntyre theorem

LetK be a local field of characteristic zero. We give a new definition of semialgebraic sets, which is then ettended to subsets of K-points of algebraicK-varieties. For algebraically closed fields, this notion is shown to be analogous to the concept of a constructible set. This follows from the various versions of the title theorem, according to which a projection .of any semiaigebraic set is again a semialgebraic set. One of such versions isTHEOREM 3. LetN be a quasiprojective algebraicK-variety andp: N → M a regularK-rational map ofN into the projectiveK-varietyM. Under this map~ then, the imageN(K) is a semialgebraic set inM(K). Using Hironaka's results on the resolution of singularities, we obtain a new proof of Theorem 3. Translated fromAlgebra i Logika, Vol. 34, No. 3, pp. 329-346, May-June, 1995. Supported by the Russian Foundation for Fundamental Research, grant No. 93-011-01520.     

K-theory of blow-ups and vector bundles on the cone over a surface

TheK-theory of the derived categories as well as someK-theoretic invariants associated to the resolution of singularities are applied in order to compute theK ₀-groups of a variety with finitely many singular points. Explicit computations are given in order to determine theK ₀ and the relative CH* groups of the affine cone over a nonsingular surface of degreed in ^d+1 not contained in any hyperplane.     

Saddle singularities of complexity 1 of integrable Hamiltonian systems

Properties of saddle singularities of rank 0 and complexity 1 for integrable Hamiltonian systems are studied. An invariant (f _n -graph) solving the problem of semi-local classification of saddle singularities of rank 0 for an arbitrary complexity was constructed earlier by the author. In this paper, a more simple form of the invariant for singularities of complexity 1 is obtained and some properties of such singularities are described in algebraic terms. In addition, the paper contains a list of saddle singularities of complexity 1 for systems with three degrees of freedom.     

Elliptic Equations with Multi-Singular Inverse-Square Potentials and Critical Nonlinearity

ABSTRACT

This article deals with a class of nonlinear elliptic equations involving a critical power-nonlinearity as well as a potential featuring multiple inverse square singularities. We show that existence of solutions heavily depends on the strength and the location of the singularities. We associate to the problem the corresponding Rayleigh quotient and give both sufficient and necessary conditions on masses and location of singularities for the minimum to be achieved. Both the cases of whole ? ^N and bounded domains are taken into account.     

KSBA surfaces with elliptic quotient singularities, π1 = 1, pg = 0, and K2 = 1, 2

Among log canonical surface singularities, those which have a rational homology disk smoothing are the cyclic quotient singularities \(\frac{1}{{{n^2}}}\left( {1,na - 1} \right)\) with gcd(a, n) = 1, and three distinguished elliptic quotient singularities. We show the existence of smoothable KSBA normal surfaces with π₁ = 1, p_g = 0, and K² = 1, 2 for each of these three singularities. We also give a list of new (and old) normal surface singularities in smoothable KSBA surfaces for invariants π₁ = 1, p_g = 0, and K² = 1, 2, 3, 4.     

(K, K′)-quasiconformal Harmonic Mappings

We prove that a harmonic diffeomorphism between two Jordan domains with C ² boundaries is a (K, K′) quasiconformal mapping for some constants K ≥ 1 and K′ ≥ 0 if and only if it is Lipschitz continuous. In this setting, if the domain is the unit disk and the mapping is normalized by three boundary points condition we give an explicit Lipschitz constant in terms of simple geometric quantities of the Jordan curve which surrounds the codomain and (K, K′). The results in this paper generalize and extend several recently obtained results.     

Weighted approximation of functions by Szász--Mirakyan-type operators

Summary We give error estimates for the weighted approximation of functions with singularities at the endpoints on the semiaxis by some modifications of Sz\'asz--Mirakyan operators. To do so, we define a new weighted modulus of smoothness and prove its equivalence to the weighted K-functional. Also, the class of functions for which the modified Sz\'asz--Mirakyan operator can be defined will be extended to a much wider set than for the original operator.