Dicritical holomorphic flows on Stein manifolds

Inventiones Mathematicae, 1982

2007

Let $V$ be an irreducible complex analytic space of dimension two with normal singularities and $\vr:\mathbb{C^*}\times V\to V$ a holomorphic action of the group $\mathbb{C^*}$ on $V$. Denote by $\fa_\vr$ the foliation on $V$ induced by $\vr$. The leaves of this foliation are the one-dimensional orbits of $\vr$. %and its singularities are the fixed points of $\vr$. We will assume that there exists a \emph{dicritical} singularity $p\in V$ for the $\bc^*$-action, i.e. for some neighborhood $p\in W\subset V$ there are infinitely many leaves of $\mathcal {F}_\vr|_{W}$ accumulating only at $p$. The closure of such a local leaf is an invariant local analytic curve called a \emph{separatrix} of $\mathcal{F}_\vr$ through $p$. In \cite{Orlik} Orlik and Wagreich studied the 2-dimensional affine algebraic varieties embedded in $\mathbb{C}^{n+1}$, with an isolated singularity at the origin, that are invariant by an effective action of the form $\sigma_Q(t,(z_{0},...,z_{n}))=(t^{q_{0}}z_{0},..., t^{q_{n}}z_{n})$ where $Q=(q_0,...,q_n) \in\mathbb N^{n+1}$, i.e. all $q_{i}$ are positive integers. Such actions are called \emph{good} actions. In particular they classified the algebraic surfaces embedded in $\mathbb{C}^{3}$ endowed with such an action. It is easy to see that any good action on a surface embedded in $\mathbb{C}^{n+1}$ has a dicritical singularity at $0\in\mathbb{C}^{n+1}$. Conversely, it is the purpose of this paper to show that good actions are the models for analytic $\mathbb{C^*}$-actions on Stein analytic spaces of dimension two with a dicritical singularity.

Inventiones Mathematicae, 1996

manuscripta mathematica, 2009

We study and classify actions of the complex multiplicative group on a nonsingular Stein surface with an isolated nondicritical singularity. We prove that the corresponding foliation exhibits a holomorphic first integral of a type F = f n g m where f and g are global holomorphic functions and n, m ∈ N. Under some additional conditions on the functions f and g we prove analytic linearization for the action. Our results can be viewed as extension of the original work of Masakazu Suzuki.

An open Riemann surface is called parabolic in case every bounded subharmonic function on it reduces to a constant. Several authors introduced seemingly different analogs of this notion for Stein manifolds of arbitrary dimension. In the first part of this note we compile these notions of parabolicity and give some immediate relations among these different definitions. In section 3 we relate some of these notions to the linear topological type of the Fréchet space of analytic functions on the given manifold. In sections 4 and 5 we look at some examples and show, for example, that the complement of the zero set of a Weierstrass polynomial possesses a continuous plurisubharmonic exhaustion function that is maximal off a compact subset.

Meccanica, 2003

The dynamical system or flowż = f (z), where f is holomorphic on C, is considered. The behaviour of the flow at critical points coincides with the behaviour of the linearization when the critical points are non-degenerate: there is no center-focus dichotomy. Periodic orbits about a center have the same period and form an open subset. The flow has no limit cycles in simply connected regions. The advance mapping is holomorphic where the flow is complete. The structure of the separatrices bounding the orbits surrounding a center is determined. Some examples are given including the following: if a quartic polynomial system has 4 distinct centers, then they are collinear.

Journal of Geometric Analysis, 2006

We prove that a pseudoholomorphic diffeomorphism between two almost complex manifolds with boundaries satisfying some pseudoconvexity type conditions cannot map a pseudoholomorphic disc in the boundary to a single point. This can be viewed as an almost complex analogue of a well known theorem of J.E.Fornaess.

Publications Mathematiques De L Ihes, 1978

The integrals of the differential equations defined by a holomorphic vector field F on a complex manifold are complex curves parametrized by C. The corresponding action of C is called a holomorphic flow and the complex curves are its orbits. These orbits, in general two-dimensional real surfaces, form a foliation ~-(F) with singularities at the zeroes of the vector field F. We study the topology of such foliations ~(F), in particular near a singularity. A simple example on C 2, which is rather general from the point of view of topology as we will see later, is given by the differential equations in complex numbers:

Mathematische Annalen, 1996

2013

In this note, we consider the linear topological invariant e for Fréchet spaces of global analytic functions on Stein manifolds. We show that O (M) ; for a Stein manifold M; enjoys the property e if and only if every compact subset of M lies in a relatively compact sublevel set of a bounded plurisubharmonic function de ned on M: We also look at some immediate implications of this characterization. 1. Introduction Spaces of analytic functions, regarded as an important class of nuclear Fréchet spaces contributed amply to the development of the structure theory of Fréchet spaces. A profound example is the pioneering result of Dragilev [6] on the absoluteness of bases in the space of analytic functions on the unit disc with the usual topology. This paved the way to the far-reaching theorem of Dynin-Mitiagin [7] on the absoluteness of bases in every nuclear Fréchet space. Many more examples could readily be provided. Of course this in‡uence has not been one-sided. Techniques and concepts...