On existence of log minimal models

Abstract

In this paper, we prove that the log minimal model program in dimension $d-1$ implies the existence of log minimal models for effective lc pairs (eg of nonnegative Kodaira dimension) in dimension $d$. In fact, we prove that the same conclusion follows from a weaker assumption, namely, the log minimal model program with scaling in dimension $d-1$. This enables us to prove that effective lc pairs in dimension five have log minimal models. We also give new proofs of the existence of log minimal models for effective lc pairs in dimension four and the Shokurov reduction theorem. Other applications appear in a paper of Birkar-Paun.

References in corpus (4)

• Termination of (many) 4-dimensional log flips
• Finite generation of the log canonical ring in dimension four
• Minimal models, flips and finite generation : a tribute to V.V. SHOKUROV and Y.-T. SIU
• Mld's vs thresholds and flips

Cited by in corpus (31)

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• Fundamental theorems for the log minimal model program
• On a connectedness principle of Shokurov-Kollár type
• Boundedness of the base varieties of certain fibrations
• Log canonical pairs with good augmented base loci
• Existence of log canonical flips and a special LMMP
• Minimal models, flips and finite generation : a tribute to V.V. SHOKUROV and Y.-T. SIU
• On the termination of flips for log canonical generalized pairs
• Finite generation of a canonical ring
• On the $4$-dimensional minimal model program for Kähler varieties
• Notes on the log minimal model program
• Divisorial algebras and modules on schemes
• Iitaka conjecture $C_{n,m}$ in dimension six
• Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs
• Supplement to the paper "On existence of log minimal models II"
• Higher Dimensional Elliptic Fibrations and Zariski Decompositions
• Log Iitaka conjecture for abundant log canonical fibrations
• Lectures on birational geometry
• Remarks on the abundance conjecture
• Introduction to the Minimal Model Program and the existence of flips
• Families of canonically polarized manifolds over log Fano varieties
• Log minimal models according to Shokurov
• Semi-stable minimal model program for varieties with trivial canonical divisor
• A remark on the abundance conjecture
• On termination of log flips in dimension four
• On existence of log minimal models and weak Zariski decompositions
• Some non-vanishing results on log canonical pairs of dimension 4