[Paper Review] The Projective Hull of Certain Curves in C^2

F. Reese Harvey, H. Blaine Lawson+1 more|arXiv (Cornell University)|Nov 15, 2006

Geometry and complex manifolds8 references3 citations

TL;DR

This paper proves that for a compact, stable, real analytic curve γ in ℂ², the projective hull minus γ is a 1-dimensional complex analytic subvariety of ℙ²−γ. Using Bishop's finiteness argument adapted to projective hulls, it shows that for almost every complex line, the intersection of the hull with the line is countable, leading to the main result that the hull's complement over γ is a union of analytic curves with prescribed boundary behavior on γ.

ABSTRACT

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.

Motivation & Objective

• To prove Conjecture 1.1 that the projective hull of a finite union of simple closed real analytic curves in ℙⁿ minus the curve itself is a 1-dimensional complex analytic subvariety.
• To establish finiteness of the intersection of the projective hull with generic complex lines, under the assumption of stability.
• To show that for stable curves, the set  \widehat{\gamma} - \gamma is a union of irreducible analytic curves with boundary components on γ.
• To extend results from polynomial convexity to projective hulls via adaptation of Bishop’s methods.

Proposed method

• Adapted Errett Bishop’s finiteness argument from polynomial convexity to the setting of projective hulls in ℙⁿ.
• Defined the set  \widehat{X}_M as points x ∈ ℂⁿ for which |p(x)| ≤ M^d sup_X |p| for all polynomials p of degree d.
• Used the notion of bidegree (d,e) for polynomials in two variables to control growth and apply maximum modulus principles.
• Introduced the set S_M(z) = {w ∈ ℂ : |Q(z,w)| ≤ M^{d+e} sup_X |Q| for all Q of bidegree (d,e)} to analyze fibers over z.
• Applied the Federer Flat Support Theorem and King’s structure theorem on currents to analyze the boundary behavior of analytic components.
• Used complexification of real analytic curves to construct embedded Riemann surfaces mapping onto components of the hull.

Experimental results

Research questions

• RQ1 Does the projective hull of a stable, real analytic curve in ℂ² minus the curve itself form a 1-dimensional complex analytic subvariety?
• RQ2 For almost every complex line, is the intersection of the projective hull with the line finite or countable?
• RQ3 Can the structure of the projective hull be decomposed into analytic curves with boundary on components of the original curve?
• RQ4 What is the role of stability (boundedness of the constant C_x) in ensuring the hull is analytic?

Key findings

• For any compact real analytic curve γ ⊂ ℂ² and linear projection π: ℂ² → ℂ, the set  \widehat{\gamma}_M ∩ π^{-1}(z) is finite for almost all z ∈ ℂ.
• The set  \widehat{\gamma}(z) = \bigcup_M \widehat{\gamma}_M(z) is countable for almost all z ∈ ℂ.
• If γ ⊂ ℙⁿ is a finite union of simple closed real analytic curves and is stable, then  \widehat{\gamma} - \gamma is a 1-dimensional complex analytic subvariety of ℙⁿ − γ.
• Each irreducible component of  \widehat{\gamma} - \gamma is either a Riemann surface with boundary mapped to a union of components of γ or an algebraic curve containing the complexification of some γ_k.
• The boundary of the closure of each analytic component V_j is a union of components γ_j of γ, with multiplicity one, and the component is analytically continued via complexification of γ_j.
• The projective hull  \widehat{\gamma} has finite area and finitely many irreducible components, as shown by finite sheeting over generic projections to ℙ¹.