Polarized pairs, log minimal models, and Zariski decompositions

Abstract

We continue our study of the relation between log minimal models and various types of Zariski decompositions. Let (X, B) be a projective log canonical pair. We will show that (X, B) has a log minimal model if either K X + B birationally has a Nakayama-Zariski decomposition with nef positive part, or that K X + B is big and birationally it has a Fujita or CKM Zariski decomposition. Along the way we introduce polarized pairs (X, B + P ) where (X, B) is a usual projective pair and P is nef, and study the birational geometry of such pairs.

Non-vanishing theorem for generalized log canonical pairs with a polarization

“…Hence, a weaker version of the conjecture, called weak non-vanishing conjecture (cf. [BH,Question 3.5]), is expected to hold. In addition to it, in general, the class of generalized lc pairs are strictly larger than the class of lc pairs.…”

“…It is important to consider the situation of Theorem 1.3 because the situation is deeply concerned with the existence of flips for (K X + B + M X )-flipping contractions. Property of being log abundant for R-divisors is much stronger than effectivity up to R-linear equivalence (Subsection 2.2), therefore Theorem 1.3 gives a partial answer to [BH,Question 3.5] by Birkar and Hu. In the case of log pairs, the existence of log MMP with scaling of an ample divisor preserving property of being log abundant implies the existence of a log minimal model ( [H3,Corollary 1.2]). We believe that Theorem 1.3 and the arguments in [HH,Section 5] (or [H3]) with the aid of [HanLi] by Han-Li will give a proof of the existence of a minimal model in the setting of Theorem 1.3.…”

Log canonical pairs with good augmented base loci

“…The corollary implies the following result which was conjectured in [6] and first proved in [4,Theorem 1.6]: both papers put the extra assumption that K X + B + P birationally has a CKM-Zariski decomposition. Recall that a lc polarized pair (X, B + P ) consists of a lc pair (X, B) together with a nef divisor P .…”

Log canonical pairs over varieties with maximal Albanese dimension

“…Remark 3. 7 We point out that a crucial step in the argument in [5] relies on the extension theorem from [16] which was obtained via an analytic method. So, both Theorem 1.1 and [5] do not have a pure algebraic proof at this point.…”