130 research outputs found
Cohomology of bundles on homological Hopf manifold
We discuss the properties of complex manifolds having rational homology of
including those constructed by Hopf, Kodaira and
Brieskorn-van de Ven. We extend certain previously known vanishing properties
of cohomology of bundles on such manifolds.As an application we consider
degeneration of Hodge-deRham spectral sequence in this non Kahler setting.Comment: To appear in Proceedings of 2007 conference on Several complex
variables and Complex Geometry. Xiamen. Chin
Verdier specialization via weak factorization
Let X in V be a closed embedding, with V - X nonsingular. We define a
constructible function on X, agreeing with Verdier's specialization of the
constant function 1 when X is the zero-locus of a function on V. Our definition
is given in terms of an embedded resolution of X; the independence on the
choice of resolution is obtained as a consequence of the weak factorization
theorem of Abramovich et al. The main property of the specialization function
is a compatibility with the specialization of the Chern class of the complement
V-X. With the definition adopted here, this is an easy consequence of standard
intersection theory. It recovers Verdier's result when X is the zero-locus of a
function on V. Our definition has a straightforward counterpart in a motivic
group. The specialization function and the corresponding Chern class and
motivic aspect all have natural `monodromy' decompositions, for for any X in V
as above. The definition also yields an expression for Kai Behrend's
constructible function when applied to (the singularity subscheme of) the
zero-locus of a function on V.Comment: Minor revision. To appear in Arkiv f\"or Matemati
Finite Temperature Time-Dependent Effective Theory for the Phase Field in two-dimensional d-wave Neutral Superconductor
We derive finite temperature time-dependent effective actions for the phase
of the pairing field, which are appropriate for a 2D electron system with both
non-retarded d- and s-wave attraction. As for s-wave pairing the d-wave
effective action contains terms with Landau damping, but their structure
appears to be different from the s-wave case due to the fact that the Landau
damping is determined by the quasiparticle group velocity v_{g}, which for
d-wave pairing does not have the same direction as the non-interacting Fermi
velocity v_{F}. We show that for d-wave pairing the Landau term has a linear
low temperature dependence and in contrast to the s-wave case are important for
all finite temperatures. A possible experimental observation of the phase
excitations is discussed.Comment: 23 pages, RevTeX4, 10 EPS figures; final version to appear in PR
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
Affine modifications and affine hypersurfaces with a very transitive automorphism group
We study a kind of modification of an affine domain which produces another
affine domain. First appeared in passing in the basic paper of O. Zariski
(1942), it was further considered by E.D. Davis (1967). The first named author
applied its geometric counterpart to construct contractible smooth affine
varieties non-isomorphic to Euclidean spaces. Here we provide certain
conditions which guarantee preservation of the topology under a modification.
As an application, we show that the group of biregular automorphisms of the
affine hypersurface given by the equation
where acts transitively on the
smooth part reg of for any We present examples of such
hypersurfaces diffeomorphic to Euclidean spaces.Comment: 39 Pages, LaTeX; a revised version with minor changes and correction
Singularities and Topology of Meromorphic Functions
We present several aspects of the "topology of meromorphic functions", which
we conceive as a general theory which includes the topology of holomorphic
functions, the topology of pencils on quasi-projective spaces and the topology
of polynomial functions.Comment: 21 pages, 1 figur
On the Number of Zeros of Abelian Integrals: A Constructive Solution of the Infinitesimal Hilbert Sixteenth Problem
We prove that the number of limit cycles generated by a small
non-conservative perturbation of a Hamiltonian polynomial vector field on the
plane, is bounded by a double exponential of the degree of the fields. This
solves the long-standing tangential Hilbert 16th problem. The proof uses only
the fact that Abelian integrals of a given degree are horizontal sections of a
regular flat meromorphic connection (Gauss-Manin connection) with a
quasiunipotent monodromy group.Comment: Final revisio
On Multi-Index Filtrations Associated to Weierstraß Semigroups
This paper is a survey on the main techniques involved in the computation of the Weierstraß semigroup at several points of curves defined over perfect fields, with special emphasis on the case of two points. Some hints about the usage of some packages of the computer algebra software Singular are also given; these are however only valid for curves defined over Fp with p a prime number
Classification of singular Q-homology planes. I. Structure and singularities
A Q-homology plane is a normal complex algebraic surface having trivial
rational homology. We obtain a structure theorem for Q-homology planes with
smooth locus of non-general type. We show that if a Q-homology plane contains a
non-quotient singularity then it is a quotient of an affine cone over a
projective curve by an action of a finite group respecting the set of lines
through the vertex. In particular, it is contractible, has negative Kodaira
dimension and only one singular point. We describe minimal normal completions
of such planes.Comment: improved results from Ph.D. thesis (University of Warsaw, 2009), 25
pages, to appear in Israel J. Mat
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