By Terrence Napier, Mohan Ramachandran

This textbook offers a unified method of compact and noncompact Riemann surfaces from the viewpoint of the so-called L2 $\bar{\delta}$-method. this system is a strong procedure from the idea of a number of complicated variables, and offers for a special method of the essentially various features of compact and noncompact Riemann surfaces.

The inclusion of constant routines operating in the course of the e-book, which bring about generalizations of the most theorems, in addition to the routines integrated in every one bankruptcy make this article excellent for a one- or two-semester graduate path. the must haves are a operating wisdom of ordinary issues in graduate point genuine and complicated research, and a few familiarity of manifolds and differential forms.

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Zj , . . , zn ∈ C, the function ζ → f (z1 , . . , zj −1 , ζ, zj +1 , . . , zn ) is a holomorphic function on the open set {ζ ∈ C | (z1 , . . , zj −1 , ζ, zj +1 , . . , zn ) ∈ } (with the obvious adjustments for j = 1 or n). A C 1 map F = (f1 , . . , fm ) : → Cm is called a holomorphic map if fi is a holomorphic function for each i = 1, . . , m. For each j = 1, . . , n, we set 1 ∂ ∂ ∂ = −i ∂zj 2 ∂x j ∂y j and ∂ 1 ∂ ∂ . = +i ∂ z¯ j 2 ∂x j ∂y j (a) Prove that a C 1 function f on an open set only if ⊂ Cn is holomorphic if and ∂f ≡ 0 ∀j = 1, .

1 Definitions and Examples 29 Fig. 3 A complex torus u1 , u2 ∈ R, the set 1 1 Uz ≡ z + t 1 ξ 1 + t 2 ξ 2 | − < t 1 , t 2 < 2 2 is the image of the open square (u1 − 1/2, u1 + 1/2) × (u2 − 1/2, u2 + 1/2) ⊂ R2 under the real linear isomorphism R2 → C given by (t1 , t2 ) → t1 ξ1 +t2 ξ2 , and hence Uz is a relatively compact neighborhood of z in C (see Fig. 3). It is also easy to check that Uz ∩ (Uz + ( \ {0})) = ∅ and ϒ(U z ) = X. In particular, the openness of ϒ , together with the first equality, implies that ϒ maps Uz homeomorphically onto ϒ(Uz ); and the second equality implies that X is compact.

5 Let X and X be complex 1-manifolds, let : G → G be a biholomorphism of an open set G ⊂ X onto an open set G ⊂ X, let ι : X → X X and ι : X → X X be the inclusion mappings of X and X into the disjoint union X X , let ∼ be the equivalence relation in the disjoint union X X determined by ι (x) ∼ ι( (x)) for every point x ∈ G , and let Y ≡ (X X )/∼ be the associated quotient space. Prove that if there exists a compact set K ⊂ X for which −1 (K ∩G) is not closed in X , then Y is not Hausdorff. 4 Holomorphic Tangent Bundle A complex 1-manifold X has an underlying real 2-dimensional C ∞ structure, and therefore a tangent bundle and a complexified tangent bundle TX: TX→X and (T X)C : (T X)C → X, respectively (see Sect.