31 research outputs found
Absolute neighbourhood retracts and spaces of holomorphic maps from Stein manifolds to Oka manifolds
The basic result of Oka theory, due to Gromov, states that every continuous
map from a Stein manifold to an elliptic manifold can be deformed
to a holomorphic map. It is natural to ask whether this can be done for all
at once, in a way that depends continuously on and leaves fixed if it
is holomorphic to begin with. In other words, is \scrO(S,X) a deformation
retract of \scrC(S,X)? We prove that it is if has a strictly
plurisubharmonic Morse exhaustion with finitely many critical points; in
particular, if is affine algebraic. The only property of used in the
proof is the parametric Oka property with approximation with respect to finite
polyhedra, so our theorem holds under the weaker assumption that is an Oka
manifold. Our main tool, apart from Oka theory itself, is the theory of
absolute neighbourhood retracts. We also make use of the mixed model structure
on the category of topological spaces.Comment: Version 2: A few very minor improvements to the exposition. Version
3: Another few very minor improvements to the exposition. To appear in
Proceedings AM
Mapping Cylinders and the Oka Principle
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.Comment: New results included in version
Holomorphic functions of slow growth on nested covering spaces of compact manifolds
Let Y be an infinite covering space of a projective manifold M in P^N of
dimension n geq 2. Let C be the intersection with M of at most n-1 generic
hypersurfaces of degree d in P^N. The preimage X of C in Y is a connected
submanifold. Let phi be the smoothed distance from a fixed point in Y in a
metric pulled up from M. Let O_phi(X) be the Hilbert space of holomorphic
functions f on X such that f^2 e^(-phi) is integrable on X, and define O_phi(Y)
similarly. Our main result is that (under more general hypotheses than
described here) the restriction O_phi(Y) to O_phi(X) is an isomorphism for d
large enough.
This yields new examples of Riemann surfaces and domains of holomorphy in C^n
with corona. We consider the important special case when Y is the unit ball B
in C^n, and show that for d large enough, every bounded holomorphic function on
X extends to a unique function in the intersection of all the nontrivial
weighted Bergman spaces on B. Finally, assuming that the covering group is
arithmetic, we establish three dichotomies concerning the extension of bounded
holomorphic and harmonic functions from X to B
Smooth toric varieties are Oka
In this brief note, we show that every smooth toric variety over the field of
complex numbers is an Oka manifold.Comment: Version 2: A remark corrected, a reference updated. Version 3: Two
references corrected to match the published version of the monograph [2]
(previous references were to a preliminary version of [2]
Eight lectures on Oka manifolds
Over the past decade, the class of Oka manifolds has emerged from Gromov's
seminal work on the Oka principle. Roughly speaking, Oka manifolds are complex
manifolds that are the target of "many" holomorphic maps from affine spaces.
They are "dual" to Stein manifolds and "opposite" to Kobayashi-hyperbolic
manifolds. The prototypical examples are complex homogeneous spaces, but there
are many other examples: there are many ways to construct new Oka manifolds
from old. The class of Oka manifolds has good formal properties, partly
explained by a close connection with abstract homotopy theory. These notes were
prepared for lectures given at the Institute of Mathematics of the Chinese
Academy of Sciences in Beijing in May 2014. They are meant to give an
accessible introduction, not to all of Oka theory, but more specifically to Oka
manifolds, how they arise and what we know about them.Comment: A few improvements in version