arXiv subjects

John Wermer

Publications and source records attributed to John Wermer.

6 recordsLinked to original sources

arXiv2008

Analytic Disks and the Projective Hull

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 which lie on a sequence of analytic disks whose boundaries converge down to c. This is in the spirit of work of Poletsky and of Larusson-Sigurdsson, whose work is essential here. The results are applied to construct a remarkable example of a closed curve c in P^2, which is real analytic at all but one point, and for which the closure of hat{c} is W \cup L where L is a projective line and W is an analytic (non-algebraic) subvariety of P^2 - L. Furthermore, hat{c} itself is the union of W with only two points on L.

math.CV↗

arXiv2008

On a Conjecture of Harvey and 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.

math.CV↗

arXiv2007

The Projective Hull of Certain Curves in C^2

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.

math.CV↗

arXiv2007

On the Complement of the Projective Hull in C^n

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.

math.CV↗

arXiv2006

Rudin's Theorem and Projective Hulls

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.

math.CV↗

arXiv2000

Linking numbers and boundaries of varieties

The intersection index at a common point of two analytic varieties of complementary dimensions in $\Bbb C^n$ is positive. This observation, which has been called a ``cornerstone'' of algebraic geometry ([GH, p.~62]), is a simple consequence of the fact that analytic varieties carry a natural orientation. Recast in terms of linking numbers, it is our principal motivation. It implies the following: Let $M$ be a smooth oriented compact 3-manifold in $\Bbb C^3$. Suppose that $M$ bounds a bounded complex 2-variety $V$. Here ``bounds'' means, in the sense of Stokes' theorem, i.e., that ${b[V]}={[M]}$ as currents. Let $A$ be an algebraic curve in $\Bbb C^3$ which is disjoint from M. Consider the linking number ${\rm link}(M,A)$ of $M$ and $A$. Since this linking number is equal to the intersection number (i.e. the sum of the intersection indices) of $V$ and $A$, by the positivity of these intersection indices, we have ${\rm link}(M,A) \geq 0$. The linking number will of course be 0 if $V$ and $A$ are disjoint. (As $A$ is not compact, this usage of ``linking number'' will be clarified later.) This reasoning shows more generally that ${\rm link}(M,A) \geq 0$ if $M$ bounds a positive holomorphic 2-chain. Recall that a {\it holomorphic $k$-chain} in $Ω\subseteq \Bbb C^n$ is a sum $\sum n_j [V_j]$ where $\{V_j\}$ is a locally finite family of irreducible $k$-dimensional subvarieties of $Ω$ and $n_j \in \Bbb Z$ and that the holomorphic 2-chain is {\it positive} if $n_j >0$ for all $j$. Our first result is that, conversely, the nonnegativity of the linking number characterizes boundaries of positive holomorphic 2-chains.

math.CV↗