an:05962774
Zbl 1226.14021
Birkar, Caucher
On existence of log minimal models. II
EN
J. Reine Angew. Math. 658, 99-113 (2011).
00287230
2011
j
14E30
minimal models; nonvanishing conjecture
Let \((X,B)\) be a projective log canonical pair (e.g. \(X\) is smooth and \(B=\sum b_iB_i\) is a sum of smooth codimension one subvarieties meeting transversely with \(0\leq b_i\leq 1\)). \((X,B)\) is pseudo-effective if there is a sequence of effective divisors \(M_i\geq 0\) such that \(K_X+B= \lim_{i\to \infty} M_i\). According to the Minimal Model Conjecture, if \((X,B)\) is pseudo-effective, then it has a minimal model \(\phi :X\dasharrow Z\) (in particular \(K_Z+\phi _* B\) is nef so that \((K_Z+\phi _* B)\cdot C\geq 0\) for any curve \(C\subset X\)) and if \((X,B)\) is not pseudo-effective, then it has a Mori fiber space (in particular there is a birational map \(\phi :X\dasharrow Z\) and a morphism \(Z\to W\) such that \(-(K_Z+\phi _* B)\) is ample over \(W\)). The Weak Nonvanishing Conjecture says that any pseudo-effective log canonical pair \((X,B)\) is effective so that \(K_X+B\equiv M\geq 0\).
In this paper, the author shows the important result that the Weak Nonvanishing Conjecture implies the Minimal Model Conjecture and that if \((X,B)\) is a \(\mathbb Q\)-factorial dlt pair, then the birational map \(\phi :X\dasharrow Z\) to the minimal model (or Mori fiber space) is given by a finite sequence of divisorial contractions and flips.
For part I, cf. [Compos. Math. 146, No. 4, 919--928 (2010; Zbl 1197.14011)].
Christopher Hacon (Salt Lake City)
Zbl 1197.14011