By T. E. Venkata Balaji
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Extra resources for An Introduction to Families, Deformations and Moduli
A subgroup G of the Lie group W is said to be discrete if the underlying subset of G is a discrete subset of the underlying topological space of W . Since such a discrete subgroup G acts properly discontinuously without fixed points on W , the natural map φ : W −→ W/G becomes a holomorphic covering with deck transformation group G as seen in the previous theorem. If G is normal, W/G naturally becomes a complex Lie group and the map φ a morphism of Lie groups. 1 Example (Tori as Quotients). The complex vector space Cn is a complex commutative Lie group with vector addition as group operation.
Is invariant under conjugation by a M¨obius transformation. The following lemma clarifies the relationship between such properties and the nature of the fixed points. Its proof is a matter of straightforward computation. 1 Lemma. A M¨ obius transformation 1. is parabolic if and only if it has one fixed point; 2. is parabolic if and only if it is conjugate to a translation z → z + a; 3. has two fixed points if and only if it is conjugate to z → λz with λ = 0, 1 and further, in this case, (a) (b) (c) (d) it it it it is is is is loxodromic if and only if |λ| = 1; hyperbolic if and only if λ ∈ R, λ > 0; elliptic if and only if |λ| = 1; elliptic of finite order if and only if λ is a root of unity.
Suppose M is equipped with another complex structure so that it is again a Riemann surface relative to which p becomes a holomorphic map. Then p is again locally biholomorphic relative to the new complex structure. Thus the identity map on M gives a holomorphic isomorphism between the two complex structures on M . Hence any topological covering space of a Riemann surface acquires naturally a unique Riemann surface structure that makes the covering map holomorphic. With the above Riemann surface structure on M , we call p : M −→ M a holomorphic covering of the Riemann surface M.
An Introduction to Families, Deformations and Moduli by T. E. Venkata Balaji