NDLI: Boundaries of Holomorphic 1-Chains Within Holomorphic Line Bundles over $\mathbb{CP}^{1}$

Content Provider	Springer Nature Link
Author	Walker, Ronald A.
Copyright Year	2009
Abstract	We show that boundaries of holomorphic 1-chains within holomorphic line bundles of $\mathbb{CP}^{1}$ can be characterized using a single generating function of Wermer moments.In the case of negative line bundles, a rationality condition on the generating function plus the vanishing moment condition together form an equivalent condition for bounding. We provide some examples which reveal that the vanishing moment condition is not sufficient by itself. These examples also can be used to demonstrate one point of caution about the use of birational maps in this topic.In the case of positive line bundles, where the vanishing moment condition vacuously holds, boundaries of holomorphic 1-chains can be characterized using the aforementioned rationality condition modulo a series of polynomial terms whose degrees are dependent on the degree of the line bundle.As a side point with potential independent interest, we show for any meromorphic function that rationality with prescribed bounds on degree is equivalent to the satisfaction of a particular determinantal differential equation.
Starting Page	226
Ending Page	241
Page Count	16
File Format	PDF
ISSN	10506926
Journal	Journal of Geometric Analysis
Volume Number	20
Issue Number	1
e-ISSN	1559002X
Language	English
Publisher	Springer-Verlag
Publisher Date	2009-08-18
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Boundaries of holomorphic chains Analytic varieties Line bundles Cauchy-type integral Wermer moments Rationality Analytic subsets and submanifolds Integration on analytic sets and spaces, currents Holomorphic bundles and generalizations Global Analysis and Analysis on Manifolds Dynamical Systems and Ergodic Theory Abstract Harmonic Analysis Fourier Analysis Convex and Discrete Geometry Differential Geometry
Content Type	Text
Resource Type	Article
Subject	Geometry and Topology

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