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Register a new userStein neighbourhoods of bordered complex curves attached to holomorphically convex sets
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Franc Forstneric
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In this paper we construct open Stein neighbourhoods of compact sets of the form $Acup K$ in a complex space, where $K$ is a compact holomorphically convex set, $A$ is a compact complex curve with boundary $bA$ of class $mathscr C^2$ which may intersect $K$, and $Acap K$ is $mathscr O(A)$-convex.
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We consider a subanalytic subset A of a complex analytic manifold M (when M is viewed as a real manifold) and formulate conditions under which A is a complex analytic subset of M.
In this paper we study the global geometry of the Kobayashi metric on convex sets. We provide new examples of non-Gromov hyperbolic domains in $mathbb{C}^n$ of many kinds: pseudoconvex and non-pseudocon ewline -vex, bounded and unbounded. Our first aim is to prove that if $Omega$ is a bounded weakly linearly convex domain in $mathbb{C}^n,,ngeq 2,$ and $S$ is an affine complex hyperplane intersecting $Omega,$ then the domain $Omegasetminus S$ endowed with the Kobayashi metric is not Gromov hyperbolic (Theorem 1.3). Next we localize this result on Kobayashi hyperbolic convex domains. Namely, we show that Gromov hyperbolicity of every open set of the form $Omegasetminus S,$ where $S$ is relatively closed in $Omega$ and $Omega$ is a convex domain, depends only on that how $S$ looks near the boundary, i.e., whether $S$ near $partialOmega$ (Theorem 1.7). We close the paper with a general remark on Hartogs type domains. The paper extends in an essential way results in [6].
Let E be an elliptic curve without complex multiplication (CM) over a number field K, and let G_E(ell) be the image of the Galois representation induced by the action of the absolute Galois group of K on the ell-torsion subgroup of E. We present two probabilistic algorithms to simultaneously determine G_E(ell) up to local conjugacy for all primes ell by sampling images of Frobenius elements; one is of Las Vegas type and the other is a Monte Carlo algorithm. They determine G_E(ell) up to one of at most two isomorphic conjugacy classes of subgroups of GL_2(Z/ell Z) that have the same semisimplification, each of which occurs for an elliptic curve isogenous to E. Under the GRH, their running times are polynomial in the bit-size n of an integral Weierstrass equation for E, and for our Monte Carlo algorithm, quasi-linear in n. We have applied our algorithms to the non-CM elliptic curves in Cremonas tables and the Stein--Watkins database, some 140 million curves of conductor up to 10^10, thereby obtaining a conjecturally complete list of 63 exceptional Galois images G_E(ell) that arise for E/Q without CM. Under this conjecture we determine a complete list of 160 exceptional Galois images G_E(ell) the arise for non-CM elliptic curves over quadratic fields with rational j-invariants. We also give examples of exceptional Galois images that arise for non-CM elliptic curves over quadratic fields only when the j-invariant is irrational.
A class of maps in a complex Banach space is studied, which includes both unbounded linear operators and nonlinear holomorphic maps. The defining property, which we call {sl pseudo-contractivity}, is introduced by means of the Abel averages of such maps. We show that the studied maps are dissipative in the spirit of the classical Lumer-Phillips theorem. For pseudo-contractive holomorphic maps, we establish the power convergence of the Abel averages to holomorphic retractions.
Polynomial spaces associated to a convex body $C$ in $({bf R}^+)^d$ have been the object of recent studies. In this work, we consider polynomial spaces associated to non-convex $C$. We develop some basic pluripotential theory including notions of $C-$extremal plurisubharmonic functions $V_{C,K}$ for $Ksubset {bf C}^d$ compact. Using this, we discuss Bernstein-Walsh type polynomial approximation results and asymptotics of random polynomials in this non-convex setting.
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