In this paper we present the main developments in Oka theory since the publication of my book Stein Manifolds and Holomorphic Mappings (The Homotopy Principle in Complex Analysis), Second Edition, ...Springer, 2017. We also give several new results, examples and constructions of Oka domains in Euclidean and projective spaces. Furthermore, we show that for n>1 the fibre ℂn in a Stein family can degenerate to a non-Oka fibre, thereby answering a question of Takeo Ohsawa. Several open problems are discussed.
Full text
Available for:
GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
2.
Gromov’s Oka Principle for Equivariant Maps Kutzschebauch, Frank; Lárusson, Finnur; Schwarz, Gerald W.
The Journal of geometric analysis,
06/2021, Volume:
31, Issue:
6
Journal Article
Peer reviewed
Open access
We take the first step in the development of an equivariant version of modern, Gromov-style Oka theory. We define equivariant versions of the standard Oka property, ellipticity, and homotopy Runge ...property of complex manifolds, show that they satisfy all the expected basic properties, and present examples. Our main theorem is an equivariant Oka principle saying that if a finite group
G
acts on a Stein manifold
X
and another manifold
Y
in such a way that
Y
is
G
-Oka, then every
G
-equivariant continuous map
X
→
Y
can be deformed, through such maps, to a
G
-equivariant holomorphic map. Approximation on a
G
-invariant holomorphically convex compact subset of
X
and jet interpolation along a
G
-invariant subvariety of
X
can be built into the theorem. We conjecture that the theorem holds for actions of arbitrary reductive complex Lie groups and prove partial results to this effect.
Full text
Available for:
EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
We survey recent work, published since 2015, on equivariant Oka theory. The main results described in the survey are as follows. Homotopy principles for equivariant isomorphisms of Stein manifolds on ...which a reductive complex Lie group
G
acts. Applications to the linearisation problem. A parametric Oka principle for sections of a bundle
E
of homogeneous spaces for a group bundle
G
, all over a reduced Stein space
X
with compatible actions of a reductive complex group on
E
,
G
, and
X
. Application to the classification of generalised principal bundles with a group action. Finally, an equivariant version of Gromov’s Oka principle based on a notion of a
G
-manifold being
G
-Oka.
Full text
Available for:
EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
Gromov, in his seminal 1989 paper on the Oka principle, introduced the notion of an elliptic manifold and proved that every continuous map from a Stein manifold to an elliptic manifold is homotopic ...to a holomorphic map. We show that a much stronger Oka principle holds in the special case of maps from certain open Riemann surfaces called circular domains into ℂ×ℂ
∗
, namely that every continuous map is homotopic to a proper holomorphic embedding. An important ingredient is a generalization to ℂ×ℂ
∗
of recent results of Wold and Forstnerič on the long-standing problem of properly embedding open Riemann surfaces into ℂ
2
, with an additional result on the homotopy class of the embeddings. We also give a complete solution to a question that arises naturally in Lárusson’s holomorphic homotopy theory, of the existence of acyclic embeddings of Riemann surfaces with abelian fundamental group into 2-dimensional elliptic Stein manifolds.
Full text
Available for:
EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
5.
Five-Dimensional Biquotients Pavlov, A. V.
Siberian mathematical journal,
11/2004, Volume:
45, Issue:
6
Journal Article
Peer reviewed
Full text
Available for:
DOBA, EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, IZUM, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PILJ, PNG, SAZU, SBCE, SBJE, SBMB, UILJ, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
We apply concepts and tools from abstract homotopy theory to complex analysis and geometry, continuing our development of the idea that the Oka Principle is about fibrancy in suitable model ...structures. We explicitly factor a holomorphic map between Stein manifolds through mapping cylinders in three different model structures and use these factorizations to prove implications between ostensibly different Oka properties of complex manifolds and holomorphic maps. We show that for Stein manifolds, several Oka properties coincide and are characterized by the geometric condition of ellipticity. Going beyond the Stein case to a study of cofibrant models of arbitrary complex manifolds, using the Jouanolou Trick, we obtain a geometric characterization of an Oka property for a large class of manifolds, extending our result for Stein manifolds. Finally, we prove a converse Oka Principle saying that certain notions of cofibrancy for manifolds are equivalent to being Stein.
Full text
Available for:
BFBNIB, NMLJ, NUK, PNG, UL, UM, UPUK
In geometric singular perturbation theory, Fenichel manifolds are typically only finitely smooth. In this paper, we prove better local smoothness properties in the analytic setting, under the ...condition that no singularities in the slow flow are present. We also investigate cases where the slow flow has a node or focus, where summability results are obtained. Various techniques are being employed like formal power series methods, majorant equations, Gevrey-asymptotics, and studies in the Borel plane.
We give a characterization of closed, simply connected, rationally elliptic 6-manifolds in terms of their rational cohomology rings and a partial classification of their real cohomology rings. We ...classify rational, real and complex homotopy types of closed, simply connected, rationally elliptic 7-manifolds. We give partial results in dimensions 8 and 9.
Full text
Available for:
EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OBVAL, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
All Gizatullin surfaces that admit such a
C
+
-action, for which the quotient is a
C
1
-fibration with a reduced degenerate fibre, have the density property. This result includes all previously known ...results for the density property of affine surfaces as special cases. We also give a description of the identity component of the group of holomorphic automorphisms of these surfaces.
Full text
Available for:
EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OBVAL, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
We obtain a quantitative cohomological boundedness theorem for closed manifolds receiving entire mappings of bounded mean distortion and finite lower order. We also prove an equidistribution theorem ...for mappings of finite distortion.
Full text
Available for:
EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
PDF
