31 research outputs found
estimates for the operator
This is a survey article about estimates for the
operator. After a review of the basic approach that has come to be called the
"Bochner-Kodaira Technique", the focus is on twisted techniques and their
applications to estimates for , to extension theorems, and
to other problems in complex analysis and geometry, including invariant metric
estimates and the -Neumann Problem.Comment: To appear in Bulletin of Mathematical Science
Analytic inversion of adjunction: L^2 extension theorems with gain
We establish new results on weighted extension of holomorphic top forms
with values in a holomorphic line bundle, from a smooth hypersurface cut out by
a holomorphic function. The weights we use are determined by certain functions
that we call denominators. We give a collection of examples of these
denominators related to the divisor defined by the submanifold.Comment: To Appear in Ann. Inst. Fourie
Algebraic volume density property of affine algebraic manifolds
We introduce the notion of algebraic volume density property for affine
algebraic manifolds and prove some important basic facts about it, in
particular that it implies the volume density property. The main results of the
paper are producing two big classes of examples of Stein manifolds with volume
density property. One class consists of certain affine modifications of \C^n
equipped with a canonical volume form, the other is the class of all Linear
Algebraic Groups equipped with the left invariant volume form.Comment: 35 page
Complete holomorphic vector fields on C^2 whose underlying foliation is polynomial
We extend the classification of complete polynomial vector fields on C^2
given by Marco Brunella (Topology 43(2): 433-445, 2004) to cover the case of
holomorphic (non-polynomial) vector fields whose underlying foliation is
however still polynomial.Comment: The original publication is available at this http URL:
http://www.worldscinet.com/ijm/21/2103/S0129167X102103.htm
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
Entire curves avoiding given sets in C^n
Let be a proper closed subset of and
at most countable (). We give conditions
of and , under which there exists a holomorphic immersion (or a proper
holomorphic embedding) with .Comment: 10 page
Holomorphic automorphisms of Danielewski surfaces II -- structure of the overshear group
We apply Nevanlinna theory for algebraic varieties to Danielewski surfaces
and investigate their group of holomorphic automorphisms. Our main result
states that the overshear group which is known to be dense in the identity
component of the holomorphic automorphism group, is a free amalgamated product.Comment: 24 page
Section Extension from Hyperbolic Geometry of Punctured Disk and Holomorphic Family of Flat Bundles
The construction of sections of bundles with prescribed jet values plays a
fundamental role in problems of algebraic and complex geometry. When the jet
values are prescribed on a positive dimensional subvariety, it is handled by
theorems of Ohsawa-Takegoshi type which give extension of line bundle valued
square-integrable top-degree holomorphic forms from the fiber at the origin of
a family of complex manifolds over the open unit 1-disk when the curvature of
the metric of line bundle is semipositive. We prove here an extension result
when the curvature of the line bundle is only semipositive on each fiber with
negativity on the total space assumed bounded from below and the connection of
the metric locally bounded, if a square-integrable extension is known to be
possible over a double point at the origin. It is a Hensel-lemma-type result
analogous to Artin's application of the generalized implicit function theorem
to the theory of obstruction in deformation theory. The motivation is the need
in the abundance conjecture to construct pluricanonical sections from flatly
twisted pluricanonical sections. We also give here a new approach to the
original theorem of Ohsawa-Takegoshi by using the hyperbolic geometry of the
punctured open unit 1-disk to reduce the original theorem of Ohsawa-Takegoshi
to a simple application of the standard method of constructing holomorphic
functions by solving the d-bar equation with cut-off functions and additional
blowup weight functions