On the symplectic structures on moduli space of stable sheaves over a K3 or abelian surface and on Hilbert scheme of points

Fix a smooth very ample curve C on a K3 or abelian surface X. Let $ \mathcal{M} $ denote the moduli space of pairs of the form (F, s), where F is a stable sheaf over X whose Hilbert polynomial coincides with that of the direct image, by the inclusion map of C in X, of a line bundle of degree d over C, and s is a nonzero section of F. Assume d to be sufficiently large such that F has a nonzero section. The pullback of the Mukai symplectic form on moduli spaces of stable sheaves over X is a holomorphic 2-form on $ \mathcal{M} $. On the other hand, $ \mathcal{M} $ has a map to a Hilbert scheme parametrizing 0-dimensional subschemes of X that sends (F, s) to the divisor, defined by s, on the curve defined by the support of F. We prove that the above 2-form on $ \mathcal{M} $ coincides with the pullback of the symplectic form on the Hilbert scheme.     

Non-divisible cycles on surfaces over local fields

In this paper, we are concerned with the reciprocity map of unramified class field theory for smooth projective surfaces over non-archimedean local fields which do not have potentially good reduction. We will construct two types of smooth projective surfaces whose reciprocity maps modulo positive integers are not injective. The first type is the case where the kernel of the reciprocity map is not divisible. The second is the case where the kernel of the reciprocity map is divisible, but where nevertheless the reciprocity map modulo some integer is not injective.     

Poisson structures on moduli spaces of sheaves over Poisson surfaces

Summary We introduce and study the notion of Poisson surface. We prove that the choice of a Poisson structure on a surfaceS canonically determines a Poisson structure on the moduli space of stable sheaves onS. This result generalizes previous results obtained by Mukai [14], for abelian orK3 surfaces, and by Tyurin [16].Oblatum 13-VI-1994 & 22-III-1995This article was processed by the author using thepjourlm style file from Springer-Verlag     

Poisson type approximations for the Markov binomial distribution

The Markov binomial distribution is approximated by the Poisson distribution with the same mean, by a translated Poisson distribution and by two-parametric Poisson type signed measures. Using an adaptation of Le Cam’s operator technique, estimates of accuracy are proved for the total variation, local and Wasserstein norms. In a special case, asymptotically sharp constants are obtained. For some auxiliary results, we used Stein’s method.     

On some moduli spaces of stable vector bundles on cubic and quartic threefolds

We study certain moduli spaces of stable vector bundles of rank 2 on cubic and quartic threefolds. In many cases under consideration, it turns out that the moduli space is complete and irreducible and a general member has vanishing intermediate cohomology. In one case, all except one component of the moduli space has such vector bundles.     

Components of the stack of torsion-free sheaves of rank 2 on ruled surfaces

Supported in part by NSA research grant MDA904-92-H-3009     

Homotopy groups of moduli spaces of representations

We calculate certain homotopy groups of the moduli spaces for representations of a compact oriented surface in the Lie groups and . Our approach relies on the interpretation of these representations in terms of Higgs bundles and uses Bott-Morse theory on the corresponding moduli spaces.     

Cylindrical fractional Brownian motion in Banach spaces

In this article we introduce cylindrical fractional Brownian motions in Banach spaces and develop the related stochastic integration theory. Here a cylindrical fractional Brownian motion is understood in the classical framework of cylindrical random variables and cylindrical measures. The developed stochastic integral for deterministic operator valued integrands is based on a series representation of the cylindrical fractional Brownian motion, which is analogous to the Karhunen–Loève expansion for genuine stochastic processes. In the last part we apply our results to study the abstract stochastic Cauchy problem in a Banach space driven by cylindrical fractional Brownian motion.     

The moment problem on the Wiener space

Consider an L¹-continuous functional ? on the vector space of polynomials of Brownian motion at given times, suppose ? commutes with the quadratic variation in a natural sense, and consider a finite set of polynomials of Brownian motion at rational times, , mapping the Wiener space to R.In the spirit of Schmüdgen's solution to the finite-dimensional moment problem, we give sufficient conditions under which ? can be written in the form ∫⋅dμ for some probability measure μ on the Wiener space such that μ-almost surely, all the random variables are nonnegative.     

Chen-Ruan cohomology of some moduli spaces of parabolic vector bundles

Let (X,D) be an ?-pointed compact Riemann surface of genus at least two. For each point x∈D, fix parabolic weights such that . Fix a holomorphic line bundle ξ over X of degree one. Let PMξ denote the moduli space of stable parabolic vector bundles, of rank two and determinant ξ, with parabolic structure over D and parabolic weights . The group of order two line bundles over X acts on PMξ by the rule E_∗⊗L?E_∗⊗L. We compute the Chen-Ruan cohomology ring of the corresponding orbifold.     

Representations for the extreme zeros of orthogonal polynomials

We establish some representations for the smallest and largest zeros of orthogonal polynomials in terms of the parameters in the three-terms recurrence relation. As a corollary we obtain representations for the endpoints of the true interval of orthogonality. Implications of these results for the decay parameter of a birth-death process (with killing) are displayed.     

Non-nested configuration of algebraic limit cycles in quadratic systems

This work deals with algebraic limit cycles of planar polynomial differential systems of degree two. More concretely, we show among other facts that a quadratic vector field cannot possess two non-nested algebraic limit cycles contained in different irreducible invariant algebraic curves.     

Centers for a 6-parameter family of polynomial vector fields of arbitrary degree

For all non-negative integers n₁,n₂,n₃,j₁,j₂ and j₃ with nk+jk>1 for k=1,2,3, (nk,jk)≠(nl,jl) if k≠l, j₃=n₃−1 and jk≠nk−1 for k=1,2, we study the center variety of the 6-parameter family of real planar polynomial vector given, in complex notation, by , where z=x+iy and A,B,C∈C\{0}.     

The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems

In this work we study the centers of planar analytic vector fields which are limit of linear type centers. It is proved that all the nilpotent centers are limit of linear type centers and consequently the Poincaré-Liapunov method to find linear type centers can be also used to find the nilpotent centers. Moreover, we show that the degenerate centers which are limit of linear type centers are also detectable with the Poincaré-Liapunov method.     

Chern classes of reflexive sheaves on normal surfaces

We define Chern classes of reflexive sheaves using Wahl's relative local Chern classes of vector bundles. The main result of the paper bounds contributions of singularities of a sheaf to the Riemann–Roch formula. Using it we are able to prove inequality in Wahl's conjecture on relative asymptotic RR formula for rank 2 vector bundles. Moreover, we prove that if Wahl's conjecture is true for a singularity then it is true for any its quotient. This implies Wahl's conjecture for quotient singularities and for quotients of cones over elliptic curves. Received March 2, 1998; in final form March 24, 1999 / Published online September 14, 2000     

The monad of probability measures over compact ordered spaces and its Eilenberg-Moore algebras

The probability measures on compact Hausdorff spaces K form a compact convex subset PK of the space of measures with the vague topology. Every continuous map of compact Hausdorff spaces induces a continuous affine map extending f. Together with the canonical embedding associating to every point its Dirac measure and the barycentric map β associating to every probability measure on PK its barycenter, we obtain a monad (P,ε,β). The Eilenberg-Moore algebras of this monad have been characterised to be the compact convex sets embeddable in locally convex topological vector spaces by Swirszcz [T. Swirszcz, Monadic functors and convexity, Bul. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 22 (1974) 39-42].We generalise this result to compact ordered spaces in the sense of Nachbin [L. Nachbin, Topology and Order, Von Nostrand, Princeton, NJ, 1965. Translated from the 1950 monograph “Topologia e Ordem” (in Portugese). Reprinted by Robert E. Kreiger Publishing Co., Huntington, NY, 1967]. The probability measures form again a compact ordered space when endowed with the stochastic order. The maps ε and β are shown to preserve the stochastic orders. Thus, we obtain a monad over the category of compact ordered spaces and order preserving continuous maps. The algebras of this monad are shown to be the compact convex ordered sets embeddable in locally convex ordered topological vector spaces.This result can be seen as a step towards the characterisation of the algebras of the monad of probability measures on the category of stably compact spaces (see [G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M. Mislove, D.S. Scott, Continuous Lattices and Domains, Encyclopedia Math. Appl., vol. 93, Cambridge University Press, 2003, Section VI-6]).