Reduction maps and minimal model theory

Abstract

We use reduction maps to study the minimal model program. Our main result is that the existence of a good minimal model for a klt pair $(X,\Delta )$ can be detected on the base of the $(K_X+\Delta )$-trivial reduction map. Thus we show that the main conjectures of the minimal model program can be interpreted as a natural statement on the existence of curves on $X$.

19 pages, v2:major revised

References in corpus (7)

• Uniruledness of stable base loci of adjoint linear systems with and without Mori Theory
• Extension theorems, Non-vanishing and the existence of good minimal models
• Abundance conjecture
• Numerical trivial foliations, Iitaka Fibrations and the numerical dimension
• Tsuji's Numerically Trivial Fibrations and Abundance
• On Eckl's pseudo-effective reduction map
• Asymptotic base loci on singular varieties

Cited by in corpus (24)

• On Generalised Abundance, I
• Abundance for varieties with many differential forms
• On Generalised Abundance, II
• Invariance of Plurigenera and boundedness for Generalized Pairs
• Boundedness of elliptic Calabi-Yau varieties with a rational section
• Birational boundedness of low dimensional elliptic Calabi-Yau varieties with a section
• A note on the abundance conjecture
• Log canonical pairs with good augmented base loci
• Versions of injectivity and extension theorems
• Corrigendum: On subadditivity of the logarithmic Kodaira dimension
• A travel guide to the canonical bundle formula
• On minimal model theory for log abundant lc pairs
• Log canonical pairs with boundaries containing ample divisors
• Finiteness of log abundant log canonical pairs in log minimal model program with scaling
• Higher Dimensional Elliptic Fibrations and Zariski Decompositions
• Direct images of relative pluricanonical bundles
• Log Iitaka conjecture for abundant log canonical fibrations
• Pseudo-effective classes and pushforwards
• Minimal model theory for relatively trivial log canonical pairs
• On Nonvanishing for uniruled log canonical pairs
• Numerical analogues of the Kodaira dimension and the Abundance Conjecture
• Some non-vanishing results on log canonical pairs of dimension 4
• Counter-examples to the minimal model theory in characteristic two
• Remarks on log Calabi--Yau structure of varieties admitting polarized endomorphism