An Introduction to Families, Deformations and Moduli by T. E. Venkata Balaji

By T. E. Venkata Balaji

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1 Example (Tori as Quotients). The complex vector space Cn is a complex commutative Lie group with vector addition as group operation. Since it is 2n-dimensional as a real vector space, we may fix a basis 22 1. Classification of Annuli and Elliptic Curves ω := w1 , . . , w2n . , a Z-submodule of maximal rank) and moreover, a discrete subgroup of the Lie group Cn . G has for its fundamental domain the 2n-dimensional real closed “parallelopiped” given by    2n  F = (tj wj ) ; 0 ≤ tj ≤ 1 .   j=1 F is clearly compact and connected.

Then we verify that the inverse image of U (w) under φ splits into the disjoint union of translates of U (w) by various elements of G, and further that each of these translates is mapped homeomorphically by φ onto U (w). In this way φ : W −→ W/G becomes a topological covering with deck transformation group G. 3 Remark. We note that the statement of the above theorem and its proof can be easily adapted to any of the following categories: 1. Complex manifolds and holomorphic maps (as done above); 2.

G G G G is is is is discrete; Kleinian and acts properly discontinuously at each point of ∆; Kleinian and acts properly discontinuously at some point of ∆; Kleinian. Proof. (1) ⇒ (2): Suppose that G does not act properly discontinuously at some point z0 of ∆. This holds if and only if there exists an infinite sequence of distinct points {zn } in the G-orbit of z0 tending to z0 . For each n, pick gn ∈ G such that gn (zn ) = z0 . Thus {gn } is an infinite sequence of distinct points of G. We will show that G has an accumulation point, contradicting its discreteness.

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