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- W311864900 abstract "In this chapter, we consider a useful generalization of the notion of a holomorphic function, namely, that of a holomorphic section of a holomorphic line bundle. We first consider the basic properties of holomorphic line bundles as well as those of sheaves and divisors. We then proceed with a discussion of the solution of the inhomogeneous Cauchy–Riemann equation with L 2 estimates in this more general setting. In this setting, there is a natural generalization of Theorem 2.9.1 for Hermitian holomorphic line bundles (E,h) with positive curvature; that is, iΘ h >0, where Θ h is a natural generalization of the curvature form $$Theta_{varphi}=partialbar{partial} varphi $$ considered in Sect. 2.8 . In fact, Sects. 3.6–3.9 may be read in place of most of the material in Sects. 2.6 – 2.9 . We then consider applications, mostly to the study of holomorphic line bundles on open Riemann surfaces (holomorphic line bundles on compact Riemann surfaces are considered in greater depth in Chap. 4 ). For example, in Sect. 3.11, we prove that every holomorphic line bundle on an open Riemann surface admits a positive-curvature Hermitian metric (this follows easily from the results of Sect. 2.14 ); and we then obtain a slightly more streamlined proof of (a generalization of) the Mittag-Leffler theorem (Theorem 2.15.1). In Sect. 3.12, we prove the Weierstrass theorem (Theorem 3.12.1), according to which every holomorphic line bundle on an open Riemann surface is actually holomorphically trivial." @default.
- W311864900 created "2016-06-24" @default.
- W311864900 creator A5037944917 @default.
- W311864900 creator A5060334059 @default.
- W311864900 date "2011-01-01" @default.
- W311864900 modified "2023-09-23" @default.
- W311864900 title "The L 2 $$bar{partial}$$ -Method in a Holomorphic Line Bundle" @default.
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- W311864900 doi "https://doi.org/10.1007/978-0-8176-4693-6_3" @default.
- W311864900 hasPublicationYear "2011" @default.
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