Boundaries of holomorphic 1-chains within holomorphic line bundles over ℂℙ 1

Research output: Contribution to journalArticle

Abstract

We show that boundaries of holomorphic 1-chains within holomorphic line bundles of 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.

Original languageEnglish (US)
Pages (from-to)226-241
Number of pages16
JournalJournal of Geometric Analysis
Volume20
Issue number1
DOIs
StatePublished - Jan 1 2010

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Line Bundle
Vanishing Moments
Moment Conditions
Rationality
Generating Function
Birational Maps
Meromorphic Function
Modulo
Sufficient
Differential equation
Moment
Polynomial
Series
Dependent
Term
Demonstrate

All Science Journal Classification (ASJC) codes

  • Geometry and Topology

Cite this

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abstract = "We show that boundaries of holomorphic 1-chains within holomorphic line bundles of 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.",
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Boundaries of holomorphic 1-chains within holomorphic line bundles over ℂℙ 1 . / Walker, Ronald.

In: Journal of Geometric Analysis, Vol. 20, No. 1, 01.01.2010, p. 226-241.

Research output: Contribution to journalArticle

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