Titlebook: Compact Riemann Surfaces; Raghavan Narasimhan Book 1992 Springer Basel AG 1992 Finite.Fundamental theorem of calculus.Morphism.algebra.dif

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https://doi.org/10.1007/978-3-531-91370-4 give here is due to Henrik Martens [12]. There are more “geometric” proofs, some of which will be found in Griffiths-Harris [9] or Arbarello-Cornalba-Griffiths-Harris [10]. We begin with a general fact about complex tori.

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The Sheaf of Germs of Holomorphic Functions,ch pairs (., .) and (., .) are said to be equivalent, and define the same germ of holomorphic function at a, if there exists an open neighbourhood . of ., . ? . ∩ ., such that . = .. An equivalence class is called a germ of holomorphic function at .; the class of a pair (.) is called the germ of . a

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The Riemann Surface of an Algebraic Function,, then .is a finite covering (of .-sheets). In particular, π. (.. ? .) has only finitely many connected components. Moreover, if . is a connected component of .’, then π’|. is again a covering, and so maps . onto P. ? .. Hence .’ has only finitely many connected components. (We shall see below that

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https://doi.org/10.1007/978-3-663-11402-4f ., . ? . ∩ ., such that . = .. An equivalence class is called a germ of holomorphic function at .; the class of a pair (.) is called the germ of . at . and denoted by ... The value at . of .. is defined by ..(.) .(.) for any pair (.) defining ...

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,Das europ?ische Mehrebenensystem,ine (or even vector) bundle on ?. is holomorphically trivial. Let . be a trivialisation. If λ ∈ Λ and . ∈ ?., then the isomorphisms . differ by multiplication by a constant since . if we denote this constant by φλ(.), then for λ ∈ Λ, . →φ.(.) is a holomorphic function without zeros, and we have, for λ, . ∈ Λ,

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The Sheaf of Germs of Holomorphic Functions,f ., . ? . ∩ ., such that . = .. An equivalence class is called a germ of holomorphic function at .; the class of a pair (.) is called the germ of . at . and denoted by ... The value at . of .. is defined by ..(.) .(.) for any pair (.) defining ...

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