334 research outputs found

### Stein structures and holomorphic mappings

We prove that every continuous map from a Stein manifold X to a complex
manifold Y can be made holomorphic by a homotopic deformation of both the map
and the Stein structure on X. In the absence of topological obstructions the
holomorphic map may be chosen to have pointwise maximal rank. The analogous
result holds for any compact Hausdorff family of maps, but it fails in general
for a noncompact family. Our main results are actually proved for smooth almost
complex source manifolds (X,J) with the correct handlebody structure. The paper
contains another proof of Eliashberg's (Int J Math 1:29--46, 1990) homotopy
characterization of Stein manifolds and a slightly different explanation of the
construction of exotic Stein surfaces due to Gompf (Ann Math 148 (2):619--693,
1998; J Symplectic Geom 3:565--587, 2005). (See also the related preprint
math/0509419).Comment: The original publication is available at http://www.springerlink.co

### Flexibility properties in Complex Analysis and Affine Algebraic Geometry

In the last decades affine algebraic varieties and Stein manifolds with big
(infinite-dimensional) automorphism groups have been intensively studied.
Several notions expressing that the automorphisms group is big have been
proposed. All of them imply that the manifold in question is an
Oka-Forstneri\v{c} manifold. This important notion has also recently merged
from the intensive studies around the homotopy principle in Complex Analysis.
This homotopy principle, which goes back to the 1930's, has had an enormous
impact on the development of the area of Several Complex Variables and the
number of its applications is constantly growing. In this overview article we
present 3 classes of properties: 1. density property, 2. flexibility 3.
Oka-Forstneri\v{c}. For each class we give the relevant definitions, its most
significant features and explain the known implications between all these
properties. Many difficult mathematical problems could be solved by applying
the developed theory, we indicate some of the most spectacular ones.Comment: thanks added, minor correction

### Tameness of holomorphic closure dimension in a semialgebraic set

Given a semianalytic set S in a complex space and a point p in S, there is a
unique smallest complex-analytic germ at p which contains the germ of S, called
the holomorphic closure of S at p. We show that if S is semialgebraic then its
holomorphic closure is a Nash germ, for every p, and S admits a semialgebraic
filtration by the holomorphic closure dimension. As a consequence, every
semialgebraic subset of a complex vector space admits a semialgebraic
stratification into CR manifolds satisfying a strong version of the condition
of the frontier.Comment: Published versio

### Bounded derived categories of very simple manifolds

An unrepresentable cohomological functor of finite type of the bounded
derived category of coherent sheaves of a compact complex manifold of dimension
greater than one with no proper closed subvariety is given explicitly in
categorical terms. This is a partial generalization of an impressive result due
to Bondal and Van den Bergh.Comment: 11 pages one important references is added, proof of lemma 2.1 (2)
and many typos are correcte

### A refined stable restriction theorem for vector bundles on quadric threefolds

Let E be a stable rank 2 vector bundle on a smooth quadric threefold Q in the
projective 4-space P. We show that the hyperplanes H in P for which the
restriction of E to the hyperplane section of Q by H is not stable form, in
general, a closed subset of codimension at least 2 of the dual projective
4-space, and we explicitly describe the bundles E which do not enjoy this
property. This refines a restriction theorem of Ein and Sols [Nagoya Math. J.
96, 11-22 (1984)] in the same way the main result of Coanda [J. reine angew.
Math. 428, 97-110 (1992)] refines the restriction theorem of Barth [Math. Ann.
226, 125-150 (1977)].Comment: Ann. Mat. Pura Appl. 201

### Automorphism covariant representations of the holonomy-flux *-algebra

We continue an analysis of representations of cylindrical functions and
fluxes which are commonly used as elementary variables of Loop Quantum Gravity.
We consider an arbitrary principal bundle of a compact connected structure
group and following Sahlmann's ideas define a holonomy-flux *-algebra whose
elements correspond to the elementary variables. There exists a natural action
of automorphisms of the bundle on the algebra; the action generalizes the
action of analytic diffeomorphisms and gauge transformations on the algebra
considered in earlier works. We define the automorphism covariance of a
*-representation of the algebra on a Hilbert space and prove that the only
Hilbert space admitting such a representation is a direct sum of spaces L^2
given by a unique measure on the space of generalized connections. This result
is a generalization of our previous work (Class. Quantum. Grav. 20 (2003)
3543-3567, gr-qc/0302059) where we assumed that the principal bundle is
trivial, and its base manifold is R^d.Comment: 34 pages, 1 figure, LaTeX2e, minor clarifying remark

### Stein structures: existence and flexibility

This survey on the topology of Stein manifolds is an extract from our recent
joint book. It is compiled from two short lecture series given by the first
author in 2012 at the Institute for Advanced Study, Princeton, and the Alfred
Renyi Institute of Mathematics, Budapest.Comment: 29 pages, 11 figure

### Proof of the Hyperplane Zeros Conjecture of Lagarias and Wang

We prove that a real analytic subset of a torus group that is contained in
its image under an expanding endomorphism is a finite union of translates of
closed subgroups. This confirms the hyperplane zeros conjecture of Lagarias and
Wang for real analytic varieties. Our proof uses real analytic geometry,
topological dynamics and Fourier analysis.Comment: 25 page

### Differential Forms on Log Canonical Spaces

The present paper is concerned with differential forms on log canonical
varieties. It is shown that any p-form defined on the smooth locus of a variety
with canonical or klt singularities extends regularly to any resolution of
singularities. In fact, a much more general theorem for log canonical pairs is
established. The proof relies on vanishing theorems for log canonical varieties
and on methods of the minimal model program. In addition, a theory of
differential forms on dlt pairs is developed. It is shown that many of the
fundamental theorems and techniques known for sheaves of logarithmic
differentials on smooth varieties also hold in the dlt setting.
Immediate applications include the existence of a pull-back map for reflexive
differentials, generalisations of Bogomolov-Sommese type vanishing results, and
a positive answer to the Lipman-Zariski conjecture for klt spaces.Comment: 72 pages, 6 figures. A shortened version of this paper has appeared
in Publications math\'ematiques de l'IH\'ES. The final publication is
available at http://www.springerlink.co

### Positivity of relative canonical bundles and applications

Given a family $f:\mathcal X \to S$ of canonically polarized manifolds, the
unique K\"ahler-Einstein metrics on the fibers induce a hermitian metric on the
relative canonical bundle $\mathcal K_{\mathcal X/S}$. We use a global elliptic
equation to show that this metric is strictly positive on $\mathcal X$, unless
the family is infinitesimally trivial.
For degenerating families we show that the curvature form on the total space
can be extended as a (semi-)positive closed current. By fiber integration it
follows that the generalized Weil-Petersson form on the base possesses an
extension as a positive current. We prove an extension theorem for hermitian
line bundles, whose curvature forms have this property. This theorem can be
applied to a determinant line bundle associated to the relative canonical
bundle on the total space. As an application the quasi-projectivity of the
moduli space $\mathcal M_{\text{can}}$ of canonically polarized varieties
follows.
The direct images $R^{n-p}f_*\Omega^p_{\mathcal X/S}(\mathcal K_{\mathcal
X/S}^{\otimes m})$, $m > 0$, carry natural hermitian metrics. We prove an
explicit formula for the curvature tensor of these direct images. We apply it
to the morphisms $S^p \mathcal T_S \to R^pf_*\Lambda^p\mathcal T_{\mathcal
X/S}$ that are induced by the Kodaira-Spencer map and obtain a differential
geometric proof for hyperbolicity properties of $\mathcal M_{\text{can}}$.Comment: Supercedes arXiv:0808.3259v4 and arXiv:1002.4858v2. To appear in
Invent. mat

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