Finiteness Theorems for Holomorphic Maps

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 g₂> 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 g₂>1 and f∈Hol_*(R _g1, R _g2), is finite (and admits estimates depending only on g₁).