International Congress of Mathematicians

686 A. EremenkoTheorem 7 Let Y be a projective variety, and q divisors Dj of degrees dj in Ysatisfy the intersection condition of Theorem 6. Let f : G —¥ Y be a holomorphicmap whose image is not contained in UjDj. ThenQj(q - 2m)T(r, /) < £ ~^N(r, Dj, f) + o(T(r, /)),j=iwhen r —t oo avoiding a set of finite logarithmic measure.3This theorem is stated in [13] only for the case Y = P", m = n but the sameproof applies to the more general statement. When m = n = 1 we obtain a roughform of the Second Alain Theorem of Nevanlinna; with worse error term, and moreimportantly, without the ramification term. A corollary from Theorem 7 is thedefect relation:2_,ô(Dj,f) < 2m, where ô(Dj,f) = 1 — lim sup 'J ' . (3.4)r-s-oo djl (r, f)The key result of potential theory used in the proof of Theorem 7 is of independentinterest [11]:Theorem 8 Suppose that a finite set of subharmonic functions {WJ} in a region inthe plane has the property that the pointwise minima w, A Wj are subharmonic forevery pair. Then the pointwise minimum of all these functions is subharmonicThis is derived in turn from the following:Theorem 9 Let Gi,G2,Gz be three pair-wise disjoint regions, and pi,p2,Pz theirharmonic measures. Then there exist Borei sets Ej c dGj such that Pj(Ej) =1, j = 1,2,3, and Ei^E 2^E z = 0.For regions in R 2 (the only case needed for theorems 7 and 8) this is easyto prove: just take Ej to be the set of accessible points from Gj and notice thatat most two points can be accessible from all three regions [11]. It is interestingthat Theorem 9 holds for regions in R" for all n, but the proof of this (based onadvanced stochastic analysis rather then potential theory) is very hard [30].We notice that the number 2 in Picard's Theorem 1, as well as in Theorem7, thus admits an interpretation which seems to be completely different from thecommon one: with our approach it has nothing to do with the Euler characteristicof the sphere or its canonical class, but comes from Theorem 9. Recently, Siu[29] gave a proof of a result similar to Theorem 7 (with Y = P", m = n) usingdifferent arguments which are inspired by "Vojta's analogy" between Nevanlinnatheory and Diophantine approximation. However Siu's proof gives a weaker estimateem « 2.718m instead of 2m in (3.4), and his assumptions on the intersection ofdivisors are stronger than those in Theorem 7.The constant 2m in (3.4) is best possible. Aloreover, one can give a rathercomplete characterization of extremal holomorphic curves of finite lower order. Werecall that the lower order of a holomorphic curve isA = l iminf^A>i/).r-s-oo logr