By Simon Donaldson

The idea of Riemann surfaces occupies a truly precise position in arithmetic. it's a end result of a lot of conventional calculus, making fabulous connections with geometry and mathematics. it truly is an incredibly valuable a part of arithmetic, wisdom of that's wanted by way of experts in lots of different fields. It presents a version for a great number of more moderen advancements in parts together with manifold topology, worldwide research, algebraic geometry, Riemannian geometry, and various subject matters in mathematical physics.

This graduate textual content on Riemann floor conception proves the basic analytical effects at the lifestyles of meromorphic services and the Uniformisation Theorem. The procedure taken emphasises PDE equipment, appropriate extra as a rule in international research. the relationship with geometric topology, and specifically the function of the mapping classification crew, is additionally defined. To this finish, a few extra refined themes were incorporated, in comparison with conventional texts at this point. whereas the therapy is novel, the roots of the topic in conventional calculus and complicated research are stored good in mind.

Part I units up the interaction among advanced research and topology, with the latter taken care of informally. half II works as a speedy first path in Riemann floor thought, together with elliptic curves. The middle of the booklet is contained partially III, the place the basic analytical effects are proved. Following this part, the rest of the textual content illustrates numerous aspects of the extra complex idea.

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The speculation of Riemann surfaces occupies a really certain position in arithmetic. it's a end result of a lot of conventional calculus, making miraculous connections with geometry and mathematics. it's a really worthwhile a part of arithmetic, wisdom of that's wanted via experts in lots of different fields.

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**Sample text**

In the case of Riemann surfaces it turns out that, for rather deep reasons (Rado’s Theorem), this hypothesis is unnecessary—it is a consequence of the deﬁnition as we have stated it. See the discussion in Chapter 10. 1 Examples First examples First, any open set in C is naturally a Riemann surface. Familiar examples are the unit disc D = {z : |z| < 1} and the upper half-plane H = {w ∈ C : Im(w) > 0}. These Riemann surfaces are equivalent, via the well-known map z= w−i . w+i Next we consider the Riemann sphere S 2 .

11). Case 2(i). Let the component of X to which we attach the strip be A = A \ disc. 12). 13). Case 2(ii). Here the distinction between the ‘twisted’ and ‘untwisted’ attachments is more subtle. 14 Joining the attaching regions corresponding to the attaching regions. 14). Then we say the twisted case is when the union of the attached strip and a strip about in A form a Möbius band, and that the untwisted case is when the union forms an ordinary band. A little thought shows that the operation takes A \ disc \ disc to A T 2 \ disc in the untwisted case and to A K in the twisted case.

Proposition 5. Let X and Y be connected Riemann surfaces and F : X → Y a non-constant holomorphic map. For each point x in X, there is a unique integer k = kx ≥ 1 such that we can ﬁnd charts around x in X and F (x) in Y in which F is represented by the map z → zk. To spell out the statement in more detail, we mean that there are a chart ˜ ψ) about x ∈ X with ψ(x) = 0 ∈ U˜ ⊂ C and a chart (V, V, ˜ φ) about (U, U, F (x) ∈ Y with φ(F (x)) = 0 ∈ V˜ ⊂ C, such that the composite φ ◦ F ◦ ψ −1 is equal to the map z → zk on its domain of deﬁnition.