22,113 research outputs found
Total Domishold Graphs: a Generalization of Threshold Graphs, with Connections to Threshold Hypergraphs
A total dominating set in a graph is a set of vertices such that every vertex
of the graph has a neighbor in the set. We introduce and study graphs that
admit non-negative real weights associated to their vertices such that a set of
vertices is a total dominating set if and only if the sum of the corresponding
weights exceeds a certain threshold. We show that these graphs, which we call
total domishold graphs, form a non-hereditary class of graphs properly
containing the classes of threshold graphs and the complements of domishold
graphs, and are closely related to threshold Boolean functions and threshold
hypergraphs. We present a polynomial time recognition algorithm of total
domishold graphs, and characterize graphs in which the above property holds in
a hereditary sense. Our characterization is obtained by studying a new family
of hypergraphs, defined similarly as the Sperner hypergraphs, which may be of
independent interest.Comment: 19 pages, 1 figur
Oka manifolds: From Oka to Stein and back
Oka theory has its roots in the classical Oka-Grauert principle whose main
result is Grauert's classification of principal holomorphic fiber bundles over
Stein spaces. Modern Oka theory concerns holomorphic maps from Stein manifolds
and Stein spaces to Oka manifolds. It has emerged as a subfield of complex
geometry in its own right since the appearance of a seminal paper of M. Gromov
in 1989.
In this expository paper we discuss Oka manifolds and Oka maps. We describe
equivalent characterizations of Oka manifolds, the functorial properties of
this class, and geometric sufficient conditions for being Oka, the most
important of which is Gromov's ellipticity. We survey the current status of the
theory in terms of known examples of Oka manifolds, mention open problems and
outline the proofs of the main results.
In the appendix by F. Larusson it is explained how Oka manifolds and Oka
maps, along with Stein manifolds, fit into an abstract homotopy-theoretic
framework.
The article is an expanded version of the lectures given by the author at the
Winter School KAWA-4 in Toulouse, France, in January 2013. A more comprehensive
exposition of Oka theory is available in the monograph F. Forstneric, Stein
Manifolds and Holomorphic Mappings (The Homotopy Principle in Complex
Analysis), Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, 56,
Springer-Verlag, Berlin-Heidelberg (2011).Comment: With an appendix by Finnur Larusson. To appear in Ann. Fac. Sci.
Toulouse Math. (6), vol. 22, no. 4. This version is identical with the
published tex
The Oka principle for sections of subelliptic submersions
Let X and Y be complex manifolds. One says that maps from X to Y satisfy the
Oka principle if the inclusion of the space of holomorphic maps from X to Y
into the space of continuous maps is a weak homotopy equivalence. In 1957 H.
Grauert proved the Oka principle for maps from Stein manifolds to complex Lie
groups and homogeneous spaces, as well as for sections of fiber bundles with
homogeneous fibers over a Stein base. In 1989 M. Gromov extended Grauert's
result to sections of submersions over a Stein base which admit dominating
sprays over small open sets in the base; for proof see [F. Forstneric and J.
Prezelj: Oka's principle for holomorphic fiber bundles with sprays, Math. Ann.
317 (2000), 117-154, and the preprint math.CV/0101040].
In this paper we prove the Oka principle for maps from Stein manifolds to any
complex manifold Y that admits finitely many sprays which together dominate at
every point of Y (such manifold is called subelliptic). The class of
subelliptic manifolds contains all the elliptic ones, as well as complements of
closed algebraic subvarieties of codimension at least two in a complex
projective space or a complex Grassmanian. We also prove the Oka principle for
removing intersections of holomorphic maps with closed complex subvarieties A
of the target manifold Y, provided that the source manifold is Stein and the
manifolds Y and Y\A are subelliptic.Comment: Revised versio
Holomorphic submersions from Stein manifolds
In this paper we prove results on the existence and homotopy classification
of holomorphic submersions from Stein manifolds to other complex manifolds. We
say that a complex manifold Y satisfies Property S_n for some integer n bigger
or equal the dimension of Y if every holomorphic submersion from a compact
convex set in C^n of a certain special type to Y can be uniformly approximated
by holomorphic submersions from C^n to Y. Assuming this condition we prove the
following. A continuous map f from an n-dimensional Stein manifold X to Y is
homotopic to a holomorphic submersions of X to Y if and only if there exists a
fiberwise surjective complex vector bundle map from TX to TY covering f. We
also prove results on the homotopy classification of holomorphic submersions.
We show that Property S_n is satisfied when n>dim Y and Y is any of the
following manifolds: a complex Euclidean space, a complex projective space or
Grassmanian, a Zariski open set in any of the above whose complement does not
contain any complex hypersurfaces, a complex torus, a Hopf manifold, a
non-hyperbolic Riemann surface, etc. In the case when Y is a complex Euclidean
space the main result of this paper was obtained in [arXiv:math.CV/0211112].Comment: Annales Inst. Fourier, to appea
Contact orderability up to conjugation
We study in this paper the remnants of the contact partial order on the
orbits of the adjoint action of contactomorphism groups on their Lie algebras.
Our main interest is a class of non-compact contact manifolds, called convex at
infinity.Comment: 28 pages, 1 figur
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