Divisorial Zariski decompositions on compact complex manifolds

Using currents with minimal singularities, we introduce pointwise minimal multiplicities for a real pseudo-effective (1,1)-cohomology class α on a compact complex manifold X, which are the local obstructions to the numerical effectivity of α. The negative part of α is then defined as the real effective divisor N(α) whose multiplicity along a prime divisor D is just the generic multiplicity of α along D, and we get in that way a divisorial Zariski decomposition of α into the sum of a class Z(α) which is nef in codimension 1 and the class of its negative part N(α), which is an exceptional divisor in the sense that it is very rigidly embedded in X. The positive parts Z(α) generate a modified nef cone, and the pseudo-effective cone is shown to be locally polyhedral away from the modified nef cone, with extremal rays generated by exceptional divisors. We then treat the case of a surface and a hyper-Kähler manifold in some detail. Using the intersection form (respectively the Beauville-Bogomolov form), we characterize the modified nef cone and the exceptional divisors. The divisorial Zariski decomposition is orthogonal, and is thus a rational decomposition, which fact accounts for the usual existence statement of a Zariski decomposition on a projective surface, which is thus extended to the hyper-Kähler case. Finally, we explain how the divisorial Zariski decomposition of (the first Chern class of) a big line bundle on a projective manifold can be characterized in terms of the asymptotics of the linear series |kL| as k→∞.     

Open book decompositions for contact structures on Brieskorn manifolds

In this paper, we give an open book decomposition for the contact structures on some Brieskorn manifolds, in particular for the contact structures of Ustilovsky. The decomposition uses right-handed Dehn twists as conjectured by Giroux.

     

Isomorphisms of combinatorial block decompositions of three-dimensional manifolds

For three-dimensional manifolds with the structure of a combinatorial block complex, we construct an invariant that allows one to verify the existence of isomorphisms, between these manifolds. For complexes of small dimensionality, we solve the problem on the possibility of extending the isomorphisms of subcomplexes to those of complexes. Shevchenko National University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal., Vol. 51, No. 4, pp. 568–571, April, 1999.     

Face vectors of simplicial cell decompositions of manifolds

In this paper, we study face vectors of simplicial posets that are the face posets of cell decompositions of topological manifolds without boundary. We characterize all possible face vectors of simplicial posets whose geometric realizations are homeomorphic to the product of spheres. As a corollary, we obtain the characterization of face vectors of simplicial posets whose geometric realizations are odd-dimensional manifolds without boundary.     

Heat extensions,optimal atomic decompositions and Sobolev embeddings in presence of symmetries on manifolds

Our purpose is twofold. We give the heat semi-group characterization of spaces of Besov and Triebel-Lizorkin types defined on Riemannian manifolds of bounded geometry. Moreover we study boundedness and compactness of Sobolev embeddings of the spaces in presence of symmetries. To make it possible we first construct an optimal atomic decomposition for the above function spaces. Received: 4 October 2001; in final form: 13 March 2002 / Published online: 24 February 2003     

Finiteness of polyhedral decompositions of cusped hyperbolic manifolds obtained by the Epstein-Penner's method

Epstein and Penner give a canonical method of decomposing a cusped hyperbolic manifold into ideal polyhedra. The decomposition depends on arbitrarily specified weights for the cusps. From the construction, it is rather obvious that there appear at most a finite number of decompositions if the given weights are slightly changed. However, since the space of weights is not compact, it is not clear whether the total number of such decompositions is finite. In this paper we prove that the number of polyhedral decompositions of a cusped hyperbolic manifold obtained by the Epstein-Penner's method is finite.

     

Skew-product decompositions of Brownian motions on manifolds: A probabilistic aspect of the Lichnerowicz-Szabo theorem

We show that the only compact simply connected manifolds for which the radial part of Brownian motion enjoys the Markov property are compact two points homogeneous spaces, i.e. rank one symmetric spaces.     

Examples of cell-like decompositions of the infinite-dimensional manifolds σ and Σ

Denote by σ the subspace of Hilbert space {(x_i)?l₂:x_i=0 for all but finitely many i}. Examples of cell-like decompositions of σ are constructed that have decomposition spaces that are not homeomorphic to σ. At one extreme is a cell-like decomposition G of σ produced using ghastly finite dimensional examples such that the decomposition space σ?G contains no embedded 2-cell but (σ?G)×$\text{R}$ is homeomorphic to σ. At the other extreme is a cell-like decomposition G of σ satisfying: (a) the nondegeneracy set N_G={g?G:g≠point} consists of countably many arcs (necessarily tame); (b) the nondegeneracy set N_G is a closed subset of the decomposition space σ?G; (c) each map f:B²→σ?G of a 2-cell into σ?G can be approximated arbitrarily closely by an embedding; (d) σ?G is not homeomorphic to σ but (σ?G)×$\text{R}$ is homeomorphic to σ. The fact that both conditions (a) and (b) can be satisfied (and have (d) hold) is directly attributable to σ’s incompleteness as a topological space.     

Perspectives on CUR decompositions

This note discusses an interesting matrix factorization called the CUR Decomposition. We illustrate various viewpoints of this method by comparing and contrasting them in different situations. Additionally, we offer a new characterization of CUR decompositions which synergizes these viewpoints and shows that they are indeed the same in the exact decomposition case.     

Multiscale decompositions on bounded domains

A construction of multiscale decompositions relative to domains is given. Multiscale spaces are constructed on which retain the important features of univariate multiresolution analysis including local polynomial reproduction and locally supported, stable bases.

     

Wavelet decompositions on a manifold

A general method for constructing chains of embedded spline spaces on a smooth (not necessarily compact) manifold is suggested. A wavelet decomposition is obtained for the case of an arbitrary vector space. The results are illustrated by constructing the wavelet decompositon of a chain of embedded spaces of the B _φ-splines of zero order on a smooth manifold. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 346, 2007, pp. 26–38.     

Rainbow decompositions

A rainbow coloring of a graph is a coloring of the edges with distinct colors. We prove the following extension of Wilson's Theorem. For every integer there exists an so that for all , if

then every properly edge-colored contains pairwise edge-disjoint rainbow copies of .

Our proof uses, as a main ingredient, a double application of the probabilistic method.

     

Filtering on manifolds

We coasider a partially observable diffusion process (x _t,y_t)t0 whose unobservable componentx _t lives on a submanifold M ofR ⁿ . We present some general conditions under which the conditional law ofx _t, given the observationsy _s ,s [0,t], admits a density w.r.t. a given measure on M. We characterize the analytical properties of this density by using appropriate Sobolev spaces.Research supported by the Hungarian National Foundation of Scientific Research No. 2290.     

Nets on manifolds

This paper is devoted to investigations on the geometry of multidimensional nets, connected with the thoery of distributions on smooth manifolds, and to the survey of some other papers on multidimensional nets, reviewed in RZh Matematika from 1964 to the present time.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 12, pp. 97–126, 1981.     

Nonsmooth spline-wavelet decompositions on a segment

We consider different methods for constructing a chain of embedded spaces of first order splines (not necessarily smooth and polynomial), based on a local enlargement of an irregular grid. We present the corresponding wavelet decompositions and prove that the decomposition operators are commutative relative to the order of removing grid points. Bibliography: 6 titles.     

Topological Jordan decompositions

The notion of a topological Jordan decomposition of a compact element of a reductive p-adic group has proven useful in many contexts. In this paper, we generalise it to groups defined over fairly general discretely valued fields and prove the usual existence and uniqueness properties, as well as an analogue of a fixed-point result of Prasad and Yu.