Semi-ampleness of the numerically effective part of Zariski decomposition
Atsushi Moriwaki · Kyoto journal of mathematics · 1986
The minimal model conjecture is the central problem in the classification theory o f algebraic varieties.Though big progress was m ade in this conjecture by the effort of Kawamata [K an m any difficulities yet remain to be overcome.Let f: X--> Y be the contraction with respect to an extremal ray which is isomorphic in codimension one.The existence of a flip off : X-->Y is one of key steps to prove the minimal model conjecture (for the definition of a flip, see § 4 ) .As will be showed in §4, it is equivalent to saying that the pluricanonical ring R = 7 ,=0 H °(X, (),(nd(K x + f* (r A )))) is finitely generated, where A is an am ple divisor on Y , r is a sufficiently large integer an d d is a n integer such that d K , is a Cartier divisor.The m ain aim o f this article is to show that the existence of suitable decomposition o f K ,± f* (r A ) is enough for R to be finitely generated.Indeed we have the following theorem which is, in some sense, a generalization of the theorem in [Ka3, Theorem 6.1] and (A.5) in [F4]. Theorem O. L et (X , 4 ) b e a norm al com plete v ariety w ith only log-terminalsingularities and D a nef and good Q-Cartier divisor on X .A ssume that (K1 + 4 )+ D admits a Zariski decomposition with rational coefficients (K1 ± 4 )+ D = P ± N , where P is the nef part of this decom position.If there are two positive rational num bers a, b such that v(X, D)_v(X, Dd-aP) and x(X, D+bP) 0, then P is semi-maple.(See § 1 , for the definitions of the terminologies used in the theorem.)Applying the theorem to the case where D is the pullback of rA by some birational morphism, we see that if K ,H -f*(rA ) admits a Zariski decomposition in the sense o f § 4 by changing X , then there exists a flip of f : X Y .It seems to the author that the theory of generalized Zariski decomposition m ust play a key role to establish the minimal model conjecture.§ 1 is preliminaries to prepare our notation and to recall some o f known results.§ 2 is devoted to prove the Theorem O. Our proof of the theorem is just