scholarly journals On minimal model theory for log abundant lc pairs

2020 ◽  
Vol 2020 (767) ◽  
pp. 109-159
Author(s):  
Kenta Hashizume ◽  
Zheng-Yu Hu
Keyword(s):  
Model Theory ◽  
Minimal Model ◽  
Canonical Divisor ◽  
Detailed Proof ◽  
Log Canonical

AbstractUnder the assumption of the minimal model theory for projective klt pairs of dimension n, we establish the minimal model theory for lc pairs {(X/Z,\Delta)} such that the log canonical divisor is relatively log abundant and its restriction to any lc center has relative numerical dimension at most n. We also give another detailed proof of results by the second author, and study termination of log MMP with scaling.

scholarly journals MINIMAL MODEL THEORY FOR LOG SURFACES IN FUJIKI’S CLASS

2020 ◽  
pp. 1-27
Author(s):  
OSAMU FUJINO
Keyword(s):  
Model Theory ◽  
Minimal Model ◽  
Log Canonical

We establish the minimal model theory for $\mathbb{Q}$ -factorial log surfaces and log canonical surfaces in Fujiki’s class ${\mathcal{C}}$ .

scholarly journals Polarized pairs, log minimal models, and Zariski decompositions

2014 ◽  
Vol 215 ◽  
pp. 203-224 ◽  
Author(s):  
Caucher Birkar ◽  
Zhengyu Hu
Keyword(s):  
Minimal Model ◽  
Birational Geometry ◽  
Minimal Models ◽  
Positive Part ◽  
Log Canonical ◽  
Canonical Pair ◽  
The Way

AbstractWe 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 KX + B birationally has a Nakayama–Zariski decomposition with nef positive part, or if KX +B is big and birationally has a Fujita–Zariski or Cutkosky–Kawamata–Moriwaki–Zariski decomposition. Along the way we introduce polarized pairs (X,B +P), where (X,B) is a usual projective pair and where P is nef, and we study the birational geometry of such pairs.

scholarly journals Reduction maps and minimal model theory

2012 ◽  
Vol 149 (2) ◽  
pp. 295-308 ◽  
Author(s):  
Yoshinori Gongyo ◽  
Brian Lehmann
Keyword(s):  
Model Theory ◽  
Minimal Model ◽  
Model Program ◽  
Minimal Model Program ◽  
Birational Model ◽  
Terminal Pair

AbstractWe use reduction maps to study the minimal model program. Our main result is that the existence of a good minimal model for a Kawamata log terminal pair (X,Δ) can be detected on a birational model of the base of the (KX+Δ)-trivial reduction map. We then interpret the main conjectures of the minimal model program as a natural statement about the existence of curves on X.

scholarly journals Morphisms of projective varieties from the viewpoint of minimal model theory

2003 ◽  
Vol 413 ◽  
pp. 1-72 ◽  
Author(s):  
Marco Andreatta ◽  
Massimiliano Mella
Keyword(s):  
Model Theory ◽  
Minimal Model ◽  
Projective Varieties

scholarly journals Polarized pairs, log minimal models, and Zariski decompositions

2014 ◽  
Vol 215 ◽  
pp. 203-224
Author(s):  
Caucher Birkar ◽  
Zhengyu Hu
Keyword(s):  
Minimal Model ◽  
Birational Geometry ◽  
Minimal Models ◽  
Positive Part ◽  
Log Canonical ◽  
Canonical Pair ◽  
The Way

AbstractWe 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 eitherKX+Bbirationally has a Nakayama–Zariski decomposition with nef positive part, or ifKX+Bis big and birationally has a Fujita–Zariski or Cutkosky–Kawamata–Moriwaki–Zariski decomposition. Along the way we introduce polarized pairs (X,B+P), where (X,B) is a usual projective pair and wherePis nef, and we study the birational geometry of such pairs.

scholarly journals Log canonical pairs with good augmented base loci

2014 ◽  
Vol 150 (4) ◽  
pp. 579-592 ◽  
Author(s):  
Caucher Birkar ◽  
Zhengyu Hu
Keyword(s):  
Minimal Model ◽  
Surjective Morphism ◽  
Normal Variety ◽  
Base Locus ◽  
Log Canonical ◽  
Canonical Pair ◽  
Interesting Special Case ◽  
Base Loci ◽  
Special Case

AbstractLet $(X,B)$ be a projective log canonical pair such that $B$ is a $\mathbb{Q}$-divisor, and that there is a surjective morphism $f: X\to Z$ onto a normal variety $Z$ satisfying $K_X+B\sim _{\mathbb{Q}} f^*M$ for some big $\mathbb{Q}$-divisor $M$, and the augmented base locus ${\mathbf{B}}_+(M)$ does not contain the image of any log canonical centre of $(X,B)$. We will show that $(X,B)$ has a good log minimal model. An interesting special case is when $f$ is the identity morphism.

scholarly journals On numerically effective log canonical divisors

2002 ◽  
Vol 30 (9) ◽  
pp. 521-531 ◽  
Author(s):  
Shigetaka Fukuda
Keyword(s):  
Complex Numbers ◽  
Canonical Divisor ◽  
Log Canonical ◽  
Iitaka Dimension

Let(X,Δ)be a4-dimensional log variety which is proper over the field of complex numbers and with only divisorial log terminal singularities. The log canonical divisorKX+Δis semiample, if it is numerically effective (NEF) and the Iitaka dimensionκ(X,KX+Δ)is strictly positive. For the proof, we use Fujino's abundance theorem for semi-log canonical threefolds.

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