5,056 research outputs found
Topological Behavior of Families of Algebraic Curves Continuously Depending on a Parameter Under Certain Conditions.
En trabajos previos del autor, se considera el problema de determinar los tipos topológicos en una familia de curvas algebraicas planas, algebraicamente dependientes de un parámetro. En este trabajo, estos resultados se generalizan, bajo ciertas condiciones, al caso de una familia de curvas algebraicas planas dependientes de forma continua de un parámetro t, que toma valores en un subconjunto de la recta real que es unión de una cantidad finita de intervalos abiertos. Los resultados conducen al cálculo de un polinomio R(t) con la propiedad de que para todo intervalo contenido en U, que no contenga ninguna raíz de R(t), el tipo topológico de la familia no varía. Un caso importante en el que los resultados son aplicables, es el caso en que los coeficientes de las curvas son algebraicamente independientes. Si el número de raíces de R(t) es finito, los tipos topológicos presentes en la familia pueden calcularse mediante métodos bien conocidos.In previous works of the author, the question of computing the different shapes arising in a family of algebraic curves\ud
algebraically depending on a real parameter was addressed. In this work we show how the ideas in these papers can be used to extend the results to\ud
a more general class of families of algebraic curves, namely families not algebraically but just continuously depending on a parameter. These\ud
families correspond to polynomials in the variables x,y whose coefficients are continuous functions of a parameter t taking values in U, where U is in general the union of finitely many open intervals. Under certain conditions, here we provide an algorithm for computing a\ud
univariate real function R(t), with the property that the topology of the family stays invariant along every real interval I contained in U, and \ud
not containing any real root of R(t). In that situation, a partition of the real line where each\ud
element gives rise to a same shape arising in the family, can be computed. Then, these shapes can be described by using well-known methods. An important situation when the method is applicable is the case when the coefficients are algebraically\ud
independent, or can be expressed in terms of algebraically independent functions
Generic behavior of asymptotically holomorphic Lefschetz pencils
We study some asymptotic properties of the sequences of symplectic
Lefschetz pencils constructed by Donaldson. In particular we prove that the
vanishing spheres of these pencils are, for large degree, conjugated under the
action of the symplectomorphism group of the fiber. This implies the
non-existence of homologically trivial vanishing spheres in these pencils.
Moreover we show some basic topological properties of the space of
asymptotically holomorphic transverse sections. These properties allow us to
define a new set of symplectic invariants of the original symplectic structure
Continuity of the Green function in meromorphic families of polynomials
We prove that along any marked point the Green function of a meromorphic
family of polynomials parameterized by the punctured unit disk explodes
exponentially fast near the origin with a continuous error term.Comment: Modified references. Added a corollary about the adelic metric
associated with an algebraic family endowed with a marked poin
Symplectic maps to projective spaces and symplectic invariants
After reviewing recent results on symplectic Lefschetz pencils and symplectic
branched covers of CP^2, we describe a new construction of maps from symplectic
manifolds of any dimension to CP^2 and the associated monodromy invariants. We
also show that a dimensional induction process makes it possible to describe
any compact symplectic manifold by a series of words in braid groups and a word
in a symmetric group.Comment: 39 pages; to appear in Proc. 7th Gokova Geometry-Topology Conferenc
Oka manifolds: From Oka to Stein and back
Oka theory has its roots in the classical Oka-Grauert principle whose main
result is Grauert's classification of principal holomorphic fiber bundles over
Stein spaces. Modern Oka theory concerns holomorphic maps from Stein manifolds
and Stein spaces to Oka manifolds. It has emerged as a subfield of complex
geometry in its own right since the appearance of a seminal paper of M. Gromov
in 1989.
In this expository paper we discuss Oka manifolds and Oka maps. We describe
equivalent characterizations of Oka manifolds, the functorial properties of
this class, and geometric sufficient conditions for being Oka, the most
important of which is Gromov's ellipticity. We survey the current status of the
theory in terms of known examples of Oka manifolds, mention open problems and
outline the proofs of the main results.
In the appendix by F. Larusson it is explained how Oka manifolds and Oka
maps, along with Stein manifolds, fit into an abstract homotopy-theoretic
framework.
The article is an expanded version of the lectures given by the author at the
Winter School KAWA-4 in Toulouse, France, in January 2013. A more comprehensive
exposition of Oka theory is available in the monograph F. Forstneric, Stein
Manifolds and Holomorphic Mappings (The Homotopy Principle in Complex
Analysis), Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, 56,
Springer-Verlag, Berlin-Heidelberg (2011).Comment: With an appendix by Finnur Larusson. To appear in Ann. Fac. Sci.
Toulouse Math. (6), vol. 22, no. 4. This version is identical with the
published tex
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