scholarly journals On the log canonical ring with Kodaira dimension two

2020 ◽  
pp. 2050121
Author(s):  
Haidong Liu
Keyword(s):  
Kodaira Dimension ◽  
Finitely Generated ◽  
Log Canonical ◽  
Canonical Pair ◽  
Canonical Ring

We prove that the log canonical ring of a projective log canonical pair with Kodaira dimension two is finitely generated.

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.

Derived Category and Canonical Bundle --II

2007 ◽  
Author(s):  
D. Huybrechts
Keyword(s):  
Hochschild Cohomology ◽  
Derived Categories ◽  
Kodaira Dimension ◽  
Derived Category ◽  
Canonical Bundle ◽  
Derived Equivalence ◽  
One Step ◽  
Canonical Ring ◽  
Special Case

Based on the work of Orlov, Kawamata, and others, this chapter shows that the (numerical) Kodaira dimension and the canonical ring are preserved under derived equivalence. The same techniques can be used to derive the invariance of Hochschild cohomology under derived equivalence. Going one step further, it is shown that the nefness of the canonical bundle is detected by the derived category. The chapter also studies the relation between derived and birational (or rather K-) equivalence. The special case of a central conjecture predicts that two birational Calabi-Yau varieties have equivalent derived categories.

Semi-Normal Log Centres and Deformations of Pairs

2013 ◽  
Vol 57 (1) ◽  
pp. 191-199 ◽  
Author(s):  
János Kollár
Keyword(s):  
Log Canonical ◽  
Canonical Pair

AbstractWe show that some of the properties of log canonical centres of a log canonical pair also hold for certain subvarieties that are close to being a log canonical centre. As a consequence, we obtain that, in working with deformations of pairs where all the coefficients of the boundary divisor are bigger than ½, embedded points never appear on the boundary divisor.

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.

THE BOGOMOLOV–MIYAOKA–YAU INEQUALITY FOR LOG CANONICAL SURFACES

2001 ◽  
Vol 64 (2) ◽  
pp. 327-343 ◽  
Author(s):  
ADRIAN LANGER
Keyword(s):  
Vector Bundles ◽  
Universal Cover ◽  
Kodaira Dimension ◽  
Projective Surface ◽  
Chern Classes ◽  
Singular Surfaces ◽  
Smooth Projective Surface ◽  
Log Canonical ◽  
Surface Singularities ◽  
Negative Kodaira Dimension

Let X be a smooth projective surface of non-negative Kodaira dimension. Bogomolov [1, Theorem 5] proved that c21 [les ] 4c2. This was improved to c21 [les ] 3c2 by Miyaoka [12, Theorem 4] and Yau [19, Theorem 4]. Equality c21 [les ] 3c2 is attained, for example, if the universal cover of X is a ball (if κ(X) = 2 then this is the only possibility). Further generalizations of inequalities for Chern classes for some singular surfaces with (fractional) boundary were obtained by Sakai [16, Theorem 7.6], Miyaoka [13, Theorem 1.1], Kobayashi [6, Theorem 2; 7, Theorem 12], Wahl [18, Main Theorem] and Megyesi [10, Theorem 10.14; 11, Theorem 0.1].In [8] we introduced Chern classes of reflexive sheaves, using Wahl's local Chern classes of vector bundles on resolutions of surface singularities. Here we apply them to obtain the following generalization of the Bogomolov–Miyaoka–Yau inequality.

scholarly journals Divisorial Models of Normal Varieties

2017 ◽  
Vol 60 (4) ◽  
pp. 1053-1064 ◽  
Author(s):  
Stefano Urbinati
Keyword(s):  
Toric Varieties ◽  
Finitely Generated ◽  
Normal Varieties ◽  
Canonical Variety ◽  
Canonical Ring

AbstractWe prove that the canonical ring of a canonical variety in the sense of de Fernex and Hacon is finitely generated. We prove that canonical varieties are Kawamata log terminal (klt) if and only if is finitely generated. We introduce a notion of nefness for non-ℚ-Gorenstein varieties and study some of its properties. We then focus on these properties for non-ℚ-Gorenstein toric varieties.

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 The Dual Complex of a Semi-log Canonical Surface

2021 ◽  
Author(s):  
Morgan V Brown
Keyword(s):  
Minimal Model ◽  
Stable Curves ◽  
Log Canonical ◽  
Canonical Pair ◽  
Higher Dimensional ◽  
Canonical Surface ◽  
Dual Complex ◽  
Moduli Theory

Abstract Semi-log canonical varieties are a higher-dimensional analogue of stable curves. They are the varieties appearing as the boundary $\Delta $ of a log canonical pair $(X,\Delta )$ and also appear as limits of canonically polarized varieties in moduli theory. For certain three-fold pairs $(X,\Delta ),$ we show how to compute the PL homeomorphism type of the dual complex of a dlt minimal model directly from the normalization data of $\Delta $.

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