|Series||Lecture notes in mathematics ; 767, Lecture notes in mathematics (Springer-Verlag) ;, 767.|
|LC Classifications||QA3 .L28 no. 767, QA333 .L28 no. 767|
|The Physical Object|
|Pagination||xii, 284 p. :|
|Number of Pages||284|
|LC Control Number||79024604|
This book contains an exposition of the theory of meromorphic functions and linear series on a compact Riemann surface. Thus the main subject matter consists of holomorphic maps from a compact Riemann surface to complex projective space. Our emphasis is on families of meromorphic functions and holomorphic curves. Families of meromorphic functions on compact Riemann surfaces. Berlin ; New York: Springer-Verlag, (DLC) (OCoLC) Material Type: Document, Internet resource: Document Type: Internet Resource, Computer File: All Authors / Contributors: Makoto Nanba. Kichoon Yang (auth.) This book contains an exposition of the theory of meromorphic functions and linear series on a compact Riemann surface. Thus the main subject matter consists of holomorphic maps from a compact Riemann surface to complex projective space. Our emphasis is on families of meromorphic functions and holomorphic curves. the meromorphic function fand is denoted by degf. Clearly degf 0 for any nontrivial meromorphic function on a compact Riemann surface, and degf= 0 if and only if f is everywhere holomorphic and nowhere zero on M, hence is a nonzero complex constant as an immediate consequence of the maximum modulus theorem.
One prefers to consider compact Riemann surfaces and thus the compact-iﬁcation Cˆ is called the Riemann surface of the curve C. It turns out that all compact Riemann surfaces can be described as com-pactiﬁcations of algebraic curves (see for example [Jos06]). Quotients under group actions Deﬁnition 4. Let ∆be a domain in C. Meromorphic function on Riemann surface. Ask Question Asked 7 years, 4 months ago. It might be good to put where the exercise is from if it is from a book, in case someone else is using it. Meromorphic Function in a Compact Riemann Surface. 2. For example, on every compact Riemann surface of genus there is a meromorphic function which realizes a branched covering with at most sheets. An important place in the theory of meromorphic functions of one complex variable is occupied by value-distribution theory (Nevanlinna theory), which studies the distribution of the roots of the equations,, when approaching the boundary . Show that a space of holomorphic sections of a line bundle is isomorphic to the space of meromorphic functions on a Riemann surface. 1 Finite Number of Global Meromorphic Functions Separating Points and Tangents on a Compact Riemann Surface.
of meromorphic functions from Riemann surfaces. We non-trivially apply this theorem to classify compact, simply-connected Riemann surfaces. Contents 1. Introduction 1 2. Riemann Surfaces and Complex Manifolds 2 Holomophic and Meromorphic Forms 3 The Hodge operator and harmonic forms 4 Proof of Hodge’s Theorem 5 3. meromorphic functions on surfaces. The dimensions of the spaces of functions and forms associated to a divisor are given by the Riemann-Roch theorem, one of the most important results in the theory of compact Riemann surfaces. We prove a ﬁrst version of this theorem in § A second version is proved in §; for that we need. Existence of meromorphic functions on a Riemann surface An existence of non-constant meromorphic functions on an arbitrary compact Riemann surface is a non-trivial and important fact in algebraic geometry, which is used, for example, in the elementary proof of the Riemann-Roch theorem. Elliptic functions and Riemann surfaces played an important role in nineteenth-century mathematics. At the present time there is a great revival of interest in these topics not only for their own sake but also because of their applications to so many areas of mathematical research from group theory and number theory to topology and differential equations.5/5(1).