Termination of 4-fold Canonical Flips

We prove the termination of 4-fold canonical flips. This paper is a supplement of [KMM, Theorem 5-1-15] and [M, Main Theorem 2.1]. We prove that a sequence of 4-fold log flips for canonical pairs terminates after finitely many steps. Let us recall the definition of canonical flips, which is slightly different from the usual one (cf. [S, (2.11) Adjoint Diagram]). Definition 1 (Canonical flip). Let X be a normal projective variety and B an effective Q-divisor such that the pair (X,B) is canonical, that is, discrep(X,B) ≥ 0. Let φ : (X,B) −→ Z be a small contraction corresponding to a (KX + B)-negative extremal face. If there exists a normal projective variety X and a projective morphism φ : X −→ Z such that (1) φ is small; (2) KX+ +B + is φ-ample, where B is the strict transform of B, then we call φ the canonical flip or log flip of φ. We call the following diagram a flipping diagram: (X,B) 99K (X, B) φ ց ւ φ Z . The following is the main theorem of this paper. Theorem 2 (Termination of 4-fold canonical flips). Let X be a normal projective 4-fold and B an effective Q-divisor such that (X,B) is canonical. Consider a sequence of log flips starting from (X,B) = (X0, B0): (X0, B0) 99K (X1, B1) 99K (X2, B2) 99K · · · , where φi : Xi −→ Zi is a contraction and φi + : Xi + = Xi+1 −→ Zi is the log flip. Then this sequence terminates after finitely many steps. Date: 2003/1/15. 1991 Mathematics Subject Classification. Primary 14E30; Secondary 14J35, 14E05.