25 research outputs found
Elliptic characterization and localization of Oka manifolds
We prove that Gromov's ellipticity condition characterizes
Oka manifolds. This characterization gives another proof of the fact that
subellipticity implies the Oka property, and affirmative answers to Gromov's
conjectures. As another application, we establish the localization principle
for Oka manifolds, which gives new examples of Oka manifolds. In the appendix,
it is also shown that the Oka property is not a bimeromorphic invariant.Comment: 15 page
Oka properties of complements of holomorphically convex sets
Our main theorem states that the complement of a compact holomorphically
convex set in a Stein manifold with the density property is an Oka manifold.
This gives a positive answer to the well-known long-standing problem in Oka
theory whether the complement of a polynomially convex set in
is Oka. Furthermore, we obtain new examples of nonelliptic Oka
manifolds which negatively answer Gromov's question. The relative version of
the main theorem is also proved. As an application, we show that the complement
of a totally real affine subspace is
Oka if and .Comment: 15 page
Dense holomorphic curves in spaces of holomorphic maps and applications to universal maps
We study when there exists a dense holomorphic curve in a space of
holomorphic maps from a Stein space. We first show that for any bounded convex
domain and any connected complex manifold , the
space contains a dense holomorphic disc. Our second
result states that is an Oka manifold if and only if for any Stein space
there exists a dense entire curve in every path component of
.
In the second half of this paper, we apply the above results to the theory of
universal functions. It is proved that for any bounded convex domain
, any fixed-point-free automorphism of and
any connected complex manifold , there exists a universal map .
We also characterize Oka manifolds by the existence of universal maps.Comment: 15 page
Elliptic characterization and unification of Oka maps
We generalize our elliptic characterization of Oka manifolds to Oka maps. The
generalized characterization can be considered as an affirmative answer to the
relative version of Gromov's conjecture. As an application, we unify previously
known Oka principles for submersions; namely the Gromov type Oka principle for
subelliptic submersions and the Forstneri\v{c} type Oka principle for
holomorphic fiber bundles with CAP fibers. We also establish the localization
principle for Oka maps which gives new examples of Oka maps.Comment: 19 page
Surjective morphisms onto subelliptic varieties
We prove that every smooth subelliptic variety admits a surjective morphism
from an affine space. This result gives partial answers to the questions of
Arzhantsev and Forstneri\v{c}. As an application, we characterize open images
of morphisms between affine spaces. We also obtain the jet interpolation
theorem for morphisms from zero-dimensional subschemes of affine varieties to
smooth subelliptic varieties.Comment: 7 page
An implicit function theorem for sprays and applications to Oka theory
We solve fundamental problems in Oka theory by establishing an implicit
function theorem for sprays. As the first application of our implicit function
theorem, we obtain an elementary proof of the fact that approximation yields
interpolation. This proof and L\'{a}russon's elementary proof of the converse
give an elementary proof of the equivalence between approximation and
interpolation. The second application concerns the Oka property of a blowup. We
prove that the blowup of an algebraically Oka manifold along a smooth algebraic
center is Oka. In the appendix, equivariantly Oka manifolds are characterized
by the equivariant version of Gromov's condition , and the
equivariant localization principle is also given.Comment: 9 page
Oka complements of countable sets and non-elliptic Oka manifolds
We study the Oka properties of complements of closed countable sets in
which are not necessarily discrete. Our main result
states that every tame closed countable set in with a
discrete derived set has an Oka complement. As an application, we obtain
non-elliptic Oka manifolds which negatively answer a long-standing question of
Gromov. Moreover, we show that these examples are not even weakly subelliptic.
It is also proved that every finite set in a Hopf manifold has an Oka
complement and an Oka blowup.Comment: 7 page
Oka tubes in holomorphic line bundles
Let be a positive holomorphic line bundle on a Grassmann manifold of
dimension . We show that for every semipositive hermitian metric on
, the disc bundle is an Oka manifold, while
the complementary tube
is pseudoconvex and Kobayashi hyperbolic. When the base manifold is a
projective space, we also show that the punctured disc bundle
is an Oka manifold. These phenomena
contribute to the heuristic principle that Oka properties are related to metric
positivity of complex manifolds
Thom's jet transversality theorem for regular maps
We establish Thom's jet transversality theorem for regular maps from an
affine algebraic manifold to an algebraic manifold satisfying a suitable
flexibility condition. It can be considered as the algebraic version of
Forstneri\v{c}'s jet transversality theorem for holomorphic maps from a Stein
manifold to an Oka manifold. Our jet transversality theorem implies genericity
theorems for regular maps of maximal ranks. As an application, it follows that
every connected compact locally flexible manifold is the image of a holomorphic
submersion from an affine space. We also show that every algebraically
degenerate subvariety of codimension at least two in a locally flexible
manifold has an Oka complement.Comment: 11 page