Finiteness Theorems for Holomorphic Maps

Abstract

This Part is devoted to a direction of complex analysis which has its roots in the theorem of Liouville (Liouville (1844) for doubly periodic functions; Cauchy (1844) in the contemporary formulation) and Picard (1879) on the nonexistence of nonconstant holomorphic functions f: ℂ→D = {z∈ ℂ: |z| < 1} and f: ℂ→ℂ\{0, 1}. The first results on the finiteness of sets of holomorphic maps were obtained in the second half of the past century within the framework of Riemann surfaces, which subject then began to take shape. Schwarz (1879) and Poincaré (1885) proved the finiteness of the group Aut R g of the automorphism of compact Riemann surfaces of genus g > 1. Hurwitz (1893) completed this result by establishing the explicit bound # Aut R g ≦84(g — 1); we owe to him several other remarkable results on maps of Riemann surfaces (some of these will be set forth in § 2 and § 3 of Chap. 1). De Franchis (1913) and Severi (1926) proved the finiteness of the set Hol*(R g1, R g2) of nonconstant holomorphic maps of compact Riemann surfaces R g1→R g2 in the hypothesis g 2> 1. Moreover, they established that, for a fixed Riemann surface R g1, the number of all pairs (f, R g2), where R g2 is a compact Riemann surface of genus g 2>1 and f∈Hol*(R g1, R g2), is finite (and admits estimates depending only on g 1).KeywordsRiemann SurfaceAlgebraic VarietyBraid GroupCompact Riemann SurfaceChern NumberThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.