A note on log canonical thresholds

Note on quasi-polarized canonical Calabi-Yau threefolds

Let $(X,L)$ be a quasi-polarized canonical Calabi-Yau threefold. In this note, we show that $\vert mL\vert$ is basepoint free for $m\geq 4$. Moreover, if the morphism $Φ_{\vert 4L\vert}$ is not birational onto its image and $h^0(X,L)\geq 2$, then $L^3=1$. As an application, if $Y$ is a $n$-dimensional Fano manifold such that $-K_Y=(n-3)H$ for some ample divisor $H$, then $\vert mH\vert$ is basepoint free for $m\geq 4$ and if the morphism $Φ_{\vert 4H\vert}$ is not birational onto its image, then $Y$ is either a weighted hypersurface of degree $10$ in the weighted projective space $\mathbb{P}(1…

Kodaira vanishing theorem for log-canonical and semi-log-canonical pairs

We prove the Kodaira vanishing theorem for log-canonical and semi-log-canonical pairs. We also give a relative vanishing theorem of Reid--Fukuda type for semi-log-canonical pairs.

Abundance theorem for numerically trivial log canonical divisors of semi-log canonical pairs

We prove the abundance theorem for numerically trivial log canonical divisors of log canonical pairs and semi-log canonical pairs.

IMPANGA lecture notes on log canonical thresholds

These are lecture notes from the IMPANGA 2010 Summer school. They give an introduction to log canonical thresholds, covering some basic properties, examples, and some recent results and open questions.

Note on asymptotic behaviour of the canonical ring

The theory of the Oukounkov body is a useful tool for studying the asymptotic behaviour of the canonical ring of a line bundle over a projective manifold. In this note, combined with the algebraic reduction, we study the asymptotic behaviour of the canonical ring of a line bundle over an arbitrary compact normal irreducible complex space.

A relative spannedness for log canonical pairs and quasi-log canonical pairs

We establish a relative spannedness for log canonical pairs, which is a generalization of the basepoint-freeness for varieties with log-terminal singularities by Andreatta--Wiśniewski. Moreover, we establish a generalization for quasi-log canonical pairs.

Log Iitaka conjecture for abundant log canonical fibrations

We prove that the log Iitaka conjecture holds for log canonical fibrations when log canonical divisor of a sufficiently general fiber is abundant.

Restrictions of log canonical algebras of general type

We introduce a diophantine property of a log canonical algebra, and use it to describe the restriction of a log canonical algebra of general type to a log canonical center of codimension one.

Simple connectedness of Fano log pairs with semi-log canonical singularities

We show that any union of slc strata of a Fano log pair with semi-log canonical singularities is simply connected. In particular, Fano log pairs with semi-log canonical singularities are simply connected, which confirms a conjecture of the first author.

Log-canonical forms and log canonical singularities

For a normal subvariety $V$ of ${\bf C}^n$ with a good ${\bf C}^*$-action we give a simple characterization for when it has only log canonical, log terminal or rational singularities. Moreover we are able to give formulas for the plurigenera of isolated singular points of such varieties and of the logarithmic Kodaira dimension of $V\backslash \{0\}$. For this purpose we introduce sheaves of $m$-canonical and $L^{2,m}$-canonical forms on normal complex spaces. For the case of affine varieties with good ${\bf C}^*$-action we give an explicit formula for these sheaves in terms of the grading of t…

ACC for log canonical thresholds

We show that log canonical thresholds satisfy the ACC

New examples of terminal and log canonical singularities

We give new examples of terminal and log canonical singularities.

Non-vanishing theorem for log canonical pairs

We obtain a correct generalization of Shokurov's non-vanishing theorem for log canonical pairs. It implies the base point free theorem for log canonical pairs. We also prove the rationality theorem for log canonical pairs. As a corollary, we obtain the cone theorem for log canonical pairs. We do not need Ambro's theory of quasi-log varieties.

Discreteness of log discrepancies over log canonical triples on a fixed pair

For a fixed pair and fixed exponents, we prove the discreteness of log discrepancies over all log canonical triples formed by attaching a product of ideals with given exponents.

ACC for log canonical threshold polytopes

We show that the log canonical threshold polytopes of varieties with log canonical singularities satisfy the ascending chain condition.

On log canonical rings

We discuss the relationship among various conjectures in the minimal model theory including the finite generation conjecture of the log canonical rings and the abundance conjecture. In particular, we show that the finite generation conjecture of the log canonical rings for log canonical pairs can be reduced to that of the log canonical rings for purely log terminal pairs of log general type.

On weak Fano varieties with log canonical singularities

We prove that the anti-canonical divisors of weak Fano 3-folds with log canonical singularities are semiample. Moreover, we consider semiampleness of the anti-log canonical divisor of any weak log Fano pair with log canonical singularities. We show semiampleness dose not hold in general by constructing several examples. Based on those examples, we propose sufficient conditions which seem to be the best possible and we prove semiampleness under such conditions. In particular we derive semiampleness of the anti-canonical divisors of log canonical weak Fano 4-folds whose lc centers are at most 1-…

Log canonical thresholds and coregularity

We prove the ascending chain condition for log canonical thresholds of bounded coregularity.

On log canonical rational singularities

We prove that the class of log canonical rational singularities is closed under the basic operations of the minimal model program. We also give some supplementary results on the minimal model program for log canonical surfaces.

On log canonical inversion of adjunction

We prove a result on the inversion of adjunction for log canonical pairs that generalizes Kawakita's result to log canonical centers of arbitrary codimension.

Log canonical thresholds on Gorenstein canonical del Pezzo surfaces

We classify all the effective anticanonical divisors on weak del Pezzo surfaces. Through this classification we obtain the smallest number among the log canonical thresholds of effective anticanonical divisors on a given Gorenstein canonical del Pezzo surface.

The Zariski-Lipman conjecture for log canonical spaces

In this paper we give an elementary proof of the Zariski-Lipman conjecture for log canonical spaces.

Subadjunction for quasi-log canonical pairs and its applications

We establish a kind of subadjunction formula for quasi-log canonical pairs. As an application, we prove that a connected projective quasi-log canonical pair whose quasi-log canonical class is anti-ample is simply connected and rationally chain connected. We also supplement the cone theorem for quasi-log canonical pairs. More precisely, we prove that every negative extremal ray is spanned by a rational curve. Finally, we treat the notion of Mori hyperbolicity for quasi-log canonical pairs.

Minimal models of semi-log-canonical pairs

We compare the minimal model of a log canonical pair with the minimal model of its reduced boundary. These results are then used to study the existence of the minimal model of a semi-log-canonical pair using its normalization.

Sources of log canonical centers

We define the "source" and the "spring" of a log canonical center and use them to solve several problems in higher-codimension adjunction. The main application is to the construction of semi log canonical pairs. Version 2: References updated. Version 3: Many small changes.

Kollár-type effective freeness for quasi-log canonical pairs

We prove Kollár-type effective basepoint-free theorems for quasi-log canonical pairs.

Accumulation point theorem for generalized log canonical thresholds

In this paper we show that the set of accumulation points of generalized log canonical thresholds for certain DCC sets comes from the set of generalized log canonical thresholds of dimension $1$ less of the same DCC sets.

ACC for generalized log canonical thresholds for complex analytic spaces

We show that generalized log canonical thresholds for complex analytic spaces satisfy the ACC and we characterize the accumulation points.

Fujita-type freeness for quasi-log canonical threefolds

In this paper, we show that Fujita's basepoint-freeness conjecture for projective quasi-log canonical singularities holds true in dimension three. Immediately, we prove Fujita-type basepoint-freeness for projective semi-log canonical threefolds.

Log canonical thresholds of smooth Fano threefolds. With an appendix by Jean-Pierre Demailly

We compute global log canonical thresholds of some smooth Fano threefolds.

The Angehrn-Siu Type Effective Freeness For Quasi-Log Canonical Pairs

We prove the Angehrn-Siu Type effective freeness and effective point separation for quasi-log canonical pairs. As a natural consequence, we obtain that these two results hold for semi-log canonical pairs. One of the main ingredients of our proof is the inversion of adjunction for quasi-log canonical pairs, which is established in this paper.

Introduction to the log minimal model program for log canonical pairs

We describe the foundation of the log minimal model program for log canonical pairs according to Ambro's idea. We generalize Kollár's vanishing and torsion-free theorems for embedded simple normal crossing pairs. Then we prove the cone and contraction theorems for quasi-log varieties, especially, for log canonical pairs.

Canonical and log canonical thresholds of Fano complete intersections

It is proved that the global log canonical threshold of a Zariski general Fano complete intersection of index 1 and codimension $k$ in ${\mathbb P}^{M+k}$ is equal to one, if $M\geqslant 2k+3$ and the maximum of the degrees of defining equations is at least 8. This is an essential improvements of the previous results about log canonical thresholds of Fano complete intersections. As a corollary we obtain the existence of K\" ahler-Einstein metrics on generic Fano complete intersections described above.

Existence of canonical models for Kawamata log terminal pairs

We prove that a Kawamata log terminal pair has the canonical model.

Potential log canonical centers

Given an ambient variety $X$ and a fixed subvariety $Z$ we give sufficient conditions for the existence of a boundary $Δ$ such that $Z$ is a log canonical center for the pair $(X, Δ)$. We also show that under some additional hypotheses $Δ$ can be chosen such that $(X, Δ)$ has log canonical singularities.

Boundedness of $ε$-log Canonical Complements on Surfaces

We prove a conjecture due to V.V. Shokurov on the boundedness of $ε$-log canonical complements on surfaces. As an application we give a new proof to the boundedness of weak log Fano surfaces.

Complements on log canonical Fano varieties

In this paper, we generalise the theory of complements to log canonical log fano varieties and prove boundedness of complements for them in dimension less than or equal to 3. We also prove some boundedness results for the canonical index of sdlt log Calabi-Yau varieties in dimension 2.

The global indices of log Calabi-Yau varieties --A supplement to Fujino's paper: The indices of log canonical singularities--

This paper gives the all possible global indices of log Calabi-Yau 3-folds with standard coefficients on the boundaries and having lc, non-klt singularities. This follows easily from the discussion in the paper: The indices of log canonical singularities by Fujino (AG/9909035).

Del Pezzo foliations with log canonical singularities

In this paper, we classify del Pezzo foliations on projective manifolds of rank at least 3 and with log canonical singularities in the sense of McQuillan.

On normalization of quasi-log canonical pairs

The normalization of a quasi-log canonical pair is a quasi-log canonical pair.

对数典范阈值的无穷小结构

Infinitesimal structure of log canonical thresholds

本文证明固定维数下对数典范阈值序列的聚点行为具有标准化结构，并进一步探讨了极小对数差异、典范阈值及K-半稳定阈值的类似性质。

Log-canonical models of singular pairs and its applications

We prove the existence of log canonical modifications for a log pair. As an application, together with Kollär's gluing theory, we remove the assumption in the first named author's work [Odaka11], which shows that K-semistable polarized varieties can only have semi-log-canonical singularities.

On foliations with semi-positive anti-canonical bundle

In this note, we describe the structure of regular foliations with semi-positive anti-canonical bundle on smooth projective varieties.

On Log Canonical Models of the Moduli Space of Stable Pointed Curves

We study the log canonical models of the moduli space MBar_{0,n} of pointed stable genus zero curves with respect to the standard log canonical divisors K+aD, where D denotes the boundary. In particular we show that, as a formal consequence of a conjecture by Fulton regarding the ample cone of MBar_{0,n}, these log canonical models are equal to certain of Hassett's weighted pointed curve spaces.

Log canonical foliation singularities on surfaces

We give a classification of the dual graphs of the exceptional divisors on the minimal resolutions of log canonical foliation singularities on surfaces. For an application, we show the set of foliated minimal log discrepancies for foliated surface triples satisfies the ascending chain condition and a Grauert-Riemenschneider type vanishing theorem for foliated surfaces with good log canonical foliation singularities.

Log pluricanonical representations and abundance conjecture

We prove the finiteness of log pluricanonical representations for projective log canonical pairs with semi-ample log canonical divisor. As a corollary, we obtain that the log canonical divisor of a projective semi log canonical pair is semi-ample if and only if so is the log canonical divisor of its normalization. We also treat many other applications.

ACC for log canonical thresholds and termination of log flips

We prove that the ascending chain condition (ACC) for log canonical (lc) thresholds in dimension $d$ and Special Termination in dimension $d$ imply the termination of any sequence of log flips starting with a $d$-dimensional lc pair of nonnegative Kodaira dimension. In particular, in characteristic zero, the latter termination in dimension 4 follows from Alexeev-Borisov's conjecture in dimension 3.

The locus of log canonical singularities

The LCS locus is an essential ingredient in the proof of fundamental results of Log Minimal Model Program, such as nonvanishing and base point freeness theorems. We prove in this paper that the LCS locus of a log canonical variety has seminormal singularities.

Fundamental theorems for semi log canonical pairs

We prove that every quasi-projective semi log canonical pair has a quasi-log structure with several good properties. It implies that various vanishing theorems, torsion-free theorem, and the cone and contraction theorem hold for semi log canonical pairs.

关于特征 $>3$ 下三维对数典范丰度的一个注记

Note on the 3-dimensional log canonical abundance in characteristic $>3$

本文在特征 $>3$ 的代数闭域上证明了射影对数典范三维偶对的非消没性及丰度的若干特殊情形，并应用于对数典范环的有限生成及丰度成立的充分条件。

Abundance theorem for semi log canonical surfaces in positive characteristic

We prove the abundance theorem for semi log canonical surfaces in positive characteristic.

Log canonical thresholds of quasi-ordinary hypersurface singularities

The log canonical thresholds of irreducible quasi-ordinary hypersurface singularities are computed, using an explicit list of pole candidates for the motivic zeta function found by the last two authors.

Higher direct images of log canonical divisors and positivity theorems

In this paper, we investigate higher direct images of log canonical divisors. After we reformulate Kollár's torsion-free theorem, we treat the relationship between higher direct images of log canonical divisors and the canonical extensions of Hodge filtration of gradedly polarized variations of mixed Hodge structures. As a corollary, we obtain a logarithmic version of Fujita-Kawamata's semi-positivity theorem. By this semi-positivity theorem, we generalize Kawamata's positivity theorem and apply it to the study of a log canonical bundle formula. The final section is an appendix, which is a res…

The Dual Complex of a semi-log canonical Surface

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