The projective hull X^ of a subset X in complex projective space P^n is an analogue of the classical polynomial hull of a set in C^n. If X is contained in an affine chart C^n on P^n, then the affine part of X^ is the set of points x in C^n for which there exists a constant M=M_x so that
|p(x)| < M^d sup{|p(y)| : y in X} for all polynomials p of degree less than or equal to d, and any d > 0. Let X^(M) be the set of points x where M_x can be chosen < M. Using an argument of E. Bishop, we show the following. Let G be a compact real analytic curve (not necessarily connected) in C^2. Then for any linear projection p: C^2 --> C^1, the set of points in G^(M) lying above a point z in C^1 is finite for almost all z. Using this, we prove the conjecture that for any compact stable real-analytic curve G in P^n, the set G^-G is a 1-dimensional complex analytic subvariety of P^n-G.
We introduce the notion of a ``projective hull'' for subsets of complex projective varieties, parallel to the idea of the polynomial hull in affine varieties. With this concept, a generalization of J. Wermer's classical theorem on the hull of a curve in $C^n$ is established in the projective setting. The projective hull is shown to have interesting properties and is related to various extremal functions and capacities in pluripotential theory. A main analytic result asserts that for any point x in the projective hull K^ of a compact set K in $P^n$, there exists a positive current T of bidimens…
We prove that if $K$ is a compact subset of an affine variety O = P^n - D (where D is a projective hypersuface), and if K is a compact subset of a closed analytic subvariety V \subset O, then the projective hull K^ of K has the property that K^ \cap O is contained in V. If V is smooth and 1-dimensional, then K^ \cap O is also closed in O. The result has applications to graphs in C^2 of functions in the disk algebra.
The notion of the projective hull of a compact set in a complex projective space was introduced by Harvey and Lawson in 2006. In this paper we describe the projective hull by Poletsky sequences of analytic discs, in analogy to the known descriptions of the holomorphic and the plurisubharmonic hull.
Using a recent result of Lárusson and Poletsky regarding plurisubharmonic subextensions we prove a disc formula for the quasiplurisubharmonic global extremal function for domains in complex projective space. As a corollary we get a characterization of the projective hull for connected compact sets in complex projective space by the existence of analytic discs.
Let X be a complex manifold and c a simple closed curve in X. We address the question: What conditions on c ensure the existence of a 1-dimensional complex subvariety V with boundary c in X. When X = C^n, an answer to this question involves the polynomial hull of gamma. When X = P^n, complex projective space, the projective hull hat{c} of c comes into play. One always has V contained in hat{c}, and for analytic curves they conjecturally coincide. In this paper we establish an approximate analogue of this idea which holds without the analyticity of c. We characterize points in hat{c} as those w…
We give conditions characterizing holomorphic and meromorphic functions in the unit disk of the complex plane in terms of certain weak forms of the maximum principle. Our work is directly inspired by recent results of John Wermer, and by the theory of the projective hull of a compact subset of complex projective space developed by Reese Harvey and Blaine Lawson.
Walter Rudin, in 1966, characterized analytic functions in the unit disk in terms of the maximum principle relative to the boundary. We conjecture a generalization of this result when the disk is replaced by the punctured disk, and we prove a special case of this conjecture, making use of the notion of projective hull in P^n introduced by R. Harvey and B. Lawson.
We consider complex projective space P^{n} and a smooth closed curve gamma in P^{n}. Harvey and Lawson have defined the notion of the projective hull \hat{K} of a compact subset K in P^n. This concept is an analogue of the polynomial hull of compact subsets of C^{n}. In the present note we study the relation between the following two properties of the curve gamma: (1) \hat{gamma} - gamma is a one-dimensional complex analytic subvariety of P^{n} - gamma, and (2) There exists a Stein subdomain of P^{n} which contains the projective hull \hat{gamma} of gamma.
It is shown that a simple closed curve in $\mathbb C^n$ that is a uniform limit of rectifiable simple closed curves each of which has nontrivial polynomial hull has itself nontrivial polynomial hull. In case the limit curve is rectifiable, the hull of the limit is shown to be the limit of the hulls. It is also shown that every rectifiable simple closed curve in $\mathbb C^n$, $n\geq 2$, can be approximated in total variation norm by a polynomially convex, rectifiable simple closed curve that coincides with the original curve except on an arbitrarily small segment. As a corollary, it is shown t…
We prove that if a compact set E in complex Euclidean space is contained in an arc J, then there is a choice of J whose polynomial hull is the union of J and the polynomial hull of E. This strengthens an earlier result of the author. We also correct an inaccuracy in the statement, and fill a gap in the proof, of that earlier result.
It is shown that there exist arcs and simple closed curves in ${\mathbb C}^3$ with nontrivial polynomial hulls that contain no analytic discs. It is also shown that in any bounded Runge domain of holomorphy in ${\mathbb C}^N$ ($N \geq 2$) there exist polynomially convex arcs and simple closed curves of almost full measure. These results, which strengthen earlier results of the author, are obtained as consequences of a general result about polynomial hulls of arcs and simple closed curves through Cantor sets.
This are the notes of a course, given by the first author for the Graduiertenkollegs (=graduate students) at the Ruhr-University Bochum, in December 1997. These lectures pursued two main tasks: FIRST - to give a systematic and self-contained introduction to the Gromov theory of pseudoholomorphic curves. This is done in Chapters I,II,III. SECOND - to explain our join results on envelopes of meromorphy of real surfaces in complex two-dimensional manifolds. We do this in Chapter IV.
A primary goal in this paper is to study the question that asks when a real analytic submanifold $M$ in ${\mathbb{C}}^{n+1}$ bounds a real analytic (up to $M$) Levi-flat hypersurface $\hat{M}$ near $p\in M$ such that $\hat{M}$ is foliated by a family of complex hypersurfaces moving along the normal direction of $M$ at $p$, and gives the invariant local hull of holomorphy of $M$ near $p$. This question is equivalent to the holomorphic flattening problem for $M$ near $p$. We will give an affirmative answer to above question when $p$ is a real complex tangent point with at least one elliptic dire…
The paper investigates the algebraic properties of Banach algebras of complex-valued functions of bounded variation on a finite interval. It is proved that such algebras have Bass stable rank one and are projective free if they do not contain nontrivial idempotents. These properties are derived from a new result on the vanishing of the second Čech cohomology group of the polynomially convex hull of a continuum of a finite linear measure.
The pluripolar hull of a pluripolar set E in $\mathbb{P}^n$ is the intersection of all complete pluripolar sets in $\mathbb{P}^n$ that contain $E$. We prove that the pluripolar hull of each compact pluripolar set in $\mathbb{P}^n$ is $F_σ$. The convergence set of a divergent formal power series $f(z_{0}, \dots,z_{n})$ is the set of all "directions" $ξ\in\mathbb{P}^{n}$ along which $f$ is convergent. We prove that the union of the pluripolar hulls of a countable collection of compact pluripolar sets in $\mathbb{P}^n$ is the convergence set of some divergent series $f$. The convergence sets on $…
Let $P(E)$ be the projectivization of a holomorphic vector bundle $E$ over a compact complex curve $C$. We characterize the existence of an extremal Kähler metric on the ruled manifold $P(E)$ in terms of relative K-polystability and the fact that $E$ decomposes as a direct sum of stable bundles.
In this paper, we study holomorphic curves satisfying the Fubini-Study derivative $\|f'(z)\|=O(|z|^σ)$ for some $σ>-1$ from the viewpoint of Nevanlinna theory
It is an old question how massive polynomial hulls of Cantor sets in $\mathbb{C}^n$ can be. In contrast to expectation e.g. Rudin, Vitushkin and Henkin showed on examples that it can be rather massive. Motivated by problems of holomorphic convexity of subsets of strictly pseudoconvex boundaries and removable singularities the question was asked for Cantor sets in the unit sphere. It was known that tame Cantor sets in the unit sphere are polynomially convex. We give an example of a wild Cantor set in the sphere whose polynomial hull contains a large ball. In some sense this can be opposed to a …
A holomorphic map from the complex line to a complex projective space is called normal (a. k. a. Brody curve) if it is uniformly continuous from the Euclidean metric to the Fubini--Study metric. The paper contains a survey of known results about such maps, as well as some new theorems.
Yan and Chen proved a weak Cartan-type second main theorem for holomorphic curves meeting hypersurfaces in projective space that included truncated counting functions. Here we give an explicit estimate for the level of truncation.
We establish a type of the Picard's theorem for entire curves in $P^n(\mathbb C)$ whose spherical derivative vanishes on the inverse images of hypersurface targets. Then, as a corollary, we prove that there is an union $D$ of finite number of hypersurfaces in the complex projective space $P^n(\mathbb C)$ such that for every entire curve $f$ in $P^n(\mathbb C)$, if the spherical derivative $f^{\#}$ of $f$ is bounded on $ f^{-1}(D)$, then $f^{\#}$ is bounded on the entire complex plane, and hence, $f$ is a Brody curve.
In this article, we prove that there does not exist a family of entire curves in the universal family of hypersurfaces of degree $d\geq 2n$ in the complex projective space ${\mathbb P}^n$. This can be seen as a weak version of the Kobayashi conjecture asserting that a general projective hypersurface of high degree is hyperbolic in the sense of Kobayashi.
Let f be a rational self-map of P^2 which leaves invariant an elliptic curve C with strictly negative transverse Lyapunov exponent. We show that C is an attractor, i.e. it possesses a dense orbit and its basin is of strictly positive measure.
The Hodge spectrum is an important analytic invariant of singularities encoding the Hodge filtration and the monodromy of the Milnor fiber. Explicit formulas exist for only a few cases. In this article the main result is a combinatorial formula for homogeneous polynomials in three variables whose reduced one define projective curve arrangements having only ordinary multiple points.
