A REMARK ON THE LOG MMP (PRIVATE NOTE)

Abstract. In this short note, we treat the log MMP without the assumption that the variety is Q-factorial. This short note is an answer to Takagi’s question. We will work over C throughout this note. For simplicity, we treat only klt pairs and Q-divisors in this note. Theorem 1. Assume that the log MMP holds for Q-factorial klt paris in dimension n. Then the following modified version of the log MMP works for (not necessarily Q-factorial) klt pairs in dimension n. Proof and explanation of the log MMP without Q-factoriality. We start with a projective morphism f: X − → Y, where X0: = X is a (not necessarily Q-factorial) normal variety, and a Q-divisor D0: = D on X such that (X,D) is klt. The aim is to set up a recursive procedure which creates intermediate fi: Xi − → Y and Di. After finitely many steps, we obtain a finial objects f ̃ : X ̃ − → Y and D̃. Assume that we already constructed fi: Xi − → Y and Di with the following properties: (i) fi is projective, (ii) Di is a Q-divisor on Xi, (iii) (Xi, Di) is klt. If KXi + Di is fi-nef, then we set X ̃: = Xi and D ̃: = Di. Assume that KXi + Di is not fi-nef. Then we can take a (KXi + Di)-negative extremal ray R (or, more generally, a (KXi + Di)-negative extremal face F) of NE(Xi/Y). Thus we have a contraction morphism ϕR: Xi − → Wi over Y. If dimWi < dimXi, then we set X ̃: = Xi and D ̃: = Di and stop the process. If ϕR is birational, then we put Xi+1:= ProjWi m≥0 ϕR∗OXi(m(KXi + Di)), Di+1: = the proper transform of ϕR∗Di on Xi+1 and repeat this process. We note that (Xi+1, Di+1) is the log canonical model of (Xi, Di) over Wi. If KWi + ϕR∗Di is Q-Cartier, then Xi+1 ' Wi. So, this process coincides with the usual one if the varieties Xi are Q-factorial. It is not difficult to see tha