Remarks on special kinds of the relative log minimal model program (Q2326783)

Let \(\pi:X\to Z\) be a projective morphism between normal quasi-projective varities and let \(\Delta\ge 0\) be a \(\mathbb{R}\)-divisor on \(X\) such that the pair \((X,\Delta)\) is log canonical (lc for short), then it is a conjecture in the (log) Minimal Model Program (MMP for short) that \((X,\Delta)\) admits a (good) minimal model or a Mori Fibre space over \(Z\). Though the question in full generality outreaches the current research, important progress has been made in the case that \((X,\Delta)\) is klt by [\textit{C. Birkar} et al., J. Am. Math. Soc. 23, No. 2, 405--468 (2010; Zbl 1210.14019)] (except the goodness of the minimal models). For the lc case, the problem is expected to be much more hard; yet it is known in the following two special cases: \begin{itemize} \item if there is \(A\ge 0\) \(\mathbb{R}\)-Cartier on \(X\) such that \(K_X+\Delta+A\sim_{\mathbb{R}}0\) over \(Z\). \item if there is an open subset \(U\subset Z\) such that \((\pi^{-1}(U),\Delta|_{\pi^{-1}(U)}) \) has a good minimal model over \(U\) and any lc center of \((X,\Delta)\) intersects \(\pi^{-1}[U)\). \end{itemize} When \(\Delta\) is a \(\mathbb{Q}\)-divisor (and \(A\) is \(\mathbb{Q}\)-Cartier), these results are obtained by \textit{C. Birkar} [Publ. Math., Inst. Hautes Étud. Sci. 115, 325--368 (2012; Zbl 1256.14012)] and \textit{C. D. Hacon} and \textit{C. Xu} [Invent. Math. 192, No. 1, 161--195 (2013; Zbl 1282.14027)], respectively. The article under review generalizes them to the general case that \(\Delta\) is an \(\mathbb{R}\)-divisor (and \(A\) is \(\mathbb{R}\)-Cartier); and the first result is further generalized (Theorem 4.1). Moreover, from the second result one can prove the existence of lc closures (Corollary 1.3), and from the first result one can deduce the existence of lc flips (Theorem 1.4). Indeed, let \((X,\Delta)\) be a lc pair and \(\phi:X\to W\) be a flipping contraction, then the lc flip of \(\phi\) is a log canonical model of \((X,\Delta)\) over \(W\), whose existence can be deduced from the first result above by taking \(0\le A\sim_{\mathbb{R},W}-(K_X+\Delta)\). The article under review is not a simple generalization of the ideas of Birkar [loc. cit., Zbl 1256.14012] and Hacon and Xu [loc. cit., Zbl 1282.14027] to the \(\mathbb{R}\)-boundary case. In fact the key ingredients of the two works mentioned above depend on the finite generation of the log canonical rings for klt pairs, which cannot be discussed for lc pairs with \(\mathbb{R}\)-boundary divisors. This article take a different approach and prove the \(\mathbb{R}\)-boundary version of the two results in a uniform way, as I summarize in the sequel. First, by blowing up the vertical lc centers and by Szabó's resolution lemma [\textit{E. Szabó}, J. Math. Sci., Tokyo 1, No. 3, 631--639 (1994; Zbl 0835.14001)] one can assume that \(X\) and \(Z\) are projective varieties. Second, by the standard reduction argument (Theorem 4.7.1 in [\textit{O. Fujino}, Foundations of the minimal model program. Tokyo: Mathematical Society of Japan (2017; Zbl 1386.14072)]) and by the related results on abundance conjecture for \(\mathbb{Q}\)-boundary case, one can see that the goodness of the minimal model follows as long as it exists. The remaining problem is to prove the existence of a minimal model, to this end one first chooses carefully a dlt model \((Y,\Gamma)\) of pair \((X,\Delta)\) satisfying the condition in Lemma 2.14, then runs a \((K_Y+\Gamma)\)-MMP (with scaling), and uses the argument of the special termination [\textit{O. Fujino}, Oxf. Lect. Ser. Math. Appl. 35, 63--75 (2007; Zbl 1286.14025)] and induction to prove its termination, this MMP then gives rise to a minimal model of \((X,\Delta)\) (in virtue of Lemma 2.15). The main technical part of the present article is to find a dlt model as above: first, we take the relative Iitaka fibration of \(X\dashrightarrow V\), and by Lemma 3.1 (one needs the weak semistable reduction [\textit{D. Abramovich} and \textit{K. Karu}, Invent. Math. 139, No. 2, 241--273 (2000; Zbl 0958.14006)] in its proof) one can assume that \(K_X+\Delta\sim_{V,\mathbb{R}}0\); second, we replace \((X,\Delta)\) by a dlt blow-up as in [\textit{K. Hashizume}, Ann. Inst. Fourier 68, No. 5, 2069--2107 (2018; Zbl 1423.14111)], and if no log canonical center of \((X,\Delta)\) dominate \(V\) this has been done in Proposition 3.3; in general, one first slightly diminishes some coefficients in the boundary to get rid of the vertical lc centers, then by running different kinds of MMPs and by constructing an auxiliary sequence of pairs, one finds the desired dlt model.