Log Canonical Singularities Research Articles

Minimal compactifications of the affine plane with nef canonical divisors

We shall consider minimal analytic compactifications of the affine plane with singularities. In previous work, Kojima and Takahashi proved that any minimal analytic compactification of the affine plane, which has at worse log canonical singularities, is a numerical del Pezzo surface (i.e., a normal complete algebraic surface with the numerically ample anti-canonical divisor) and has only rational singularities. In this article, we show that any minimal analytic compactification of the affine plane with the nef canonical divisor has an irrational singularity.

Log canonical foliation singularities on surfaces

AbstractWe give a classification of the dual graphs of the exceptional divisors on the minimal resolutions of log canonical foliation singularities on surfaces. As 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 special log canonical foliation singularities.

Semi-ampleness of NQC generalized log canonical pairs

We establish a Kollár-type gluing theory for NQC generalized log canonical pairs and use it to prove semi-ampleness results of NQC generalized pairs. As consequences, we prove the existence of flips for any NQC generalized log canonical pair, and show that NQC generalized log canonical singularities are Du Bois.

Deformations of Log Terminal and Semi Log Canonical Singularities

Abstract In this paper, we prove that klt singularities are invariant under deformations if the generic fiber is $\mathbb {Q}$ -Gorenstein. We also obtain a similar result for slc singularities. These are generalizations of results of Esnault-Viehweg [Math. Ann.271 (1985), 439–449] and S. Ishii [Math. Ann.275 (1986), 139–148; Singularities (Iowa City, IA, 1986) Contemporary Mathematics, vol. 90 (American Mathematical Society, Providence, RI, 1989), 135–145].

Compactifications of the affine plane over nonclosed fields

In this article, we give the classification of normal del Pezzo surfaces of rank one with at most log canonical singularities containing the affine plane defined over an algebraically non-closed field of characteristic zero. As an application, we have a criterion for del Pezzo fibrations with canonical singularities whose generic fibers are not smooth to contain vertical $\mathbb{A}^2$-cylinders.

The Kähler–Ricci Flow With Log Canonical Singularities

Abstract We establish the existence of the Kähler–Ricci flow on projective varieties with log canonical singularities. This generalizes some of the existence results of Song–Tian [36] in case of projective varieties with klt singularities. We also prove that the normalized Kähler–Ricci flow will converge to the Kähler–Einstein metric with negative Ricci curvature on semi-log canonical models in the sense of currents. Finally, we also construct Kähler–Ricci flow solutions performing divisorial contractions and flips with log canonical singularities.

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.

The violation of the Lipman–Zariski conjecture in positive characteristic

We study the failure of the Lipman–Zariski conjecture in positive characteristic. For rational double points, the conjecture holds true except for a short finite list of exceptions. For log canonical surface singularities, the conjecture continues to hold with the same list of exceptions under an additional tameness hypothesis. In particular, among rational double points in characteristic p≥7 Lipman's counterexample is the only one, and the conjecture holds for all tame F-pure normal surface singularities.

Some remarks on finiteness of extremal rays of divisorial type

Let $X$ be a normal $\mathbf{Q}$-factorial projective variety with at most log canonical singularities. We shall give a sufficient condition for the existence of at most finitely many $K_{X}$-negative extremal rays $R(\subset \overline{\mathrm{NE}}(X))$ of divisorial type. As an application, we show that for a nonisomorphic surjective endomorphism $f\colon X\to X$ of a normal projective $\mathbf{Q}$-factorial terminal 3-fold $X$ with $\kappa(X) > 0$, a suitable power $f^{k}\ (k > 0)$ of $f$ descends to a nonisomorphic surjective endomorphism $g\colon X_{\textit{min}}\to X_{\textit{min}}$ of a minimal model $X_{\textit{min}}$ of $X$.

Local and global applications of the Minimal Model Program for co-rank 1 foliations on threefolds

We provide several applications of the minimal model program to the local and global study of co-rank 1 foliations on threefolds. Locally, we prove a singular variant of Malgrange’s theorem, a classification of terminal foliation singularities and the existence of separatrices for log canonical singularities. Globally, we prove termination of flips, a connectedness theorem on log canonical centres, a non-vanishing theorem and some hyperbolicity properties of foliations.

On minimal log discrepancies and kollár components

AbstractIn this article, we prove a local implication of boundedness of Fano varieties. More precisely, we prove that$d$-dimensional$a$-log canonical singularities with standard coefficients, which admit an$\epsilon$-plt blow-up, have minimal log discrepancies belonging to a finite set which only depends on$d,\,a$and$\epsilon$. This result gives a natural geometric stratification of the possible mld's in a fixed dimension by finite sets. As an application, we prove the ascending chain condition for minimal log discrepancies of exceptional singularities. We also introduce an invariant for klt singularities related to the total discrepancy of Kollár components.

Blowups with log canonical singularities

We show that the minimum weight of a weighted blow-up of $\mathbf A^d$ with $\varepsilon$-log canonical singularities is bounded by a constant depending only on $\varepsilon $ and $d$. This was conjectured by Birkar. Using the recent classification of $4$-dimensional empty simplices by Iglesias-Vali\~no and Santos, we work out an explicit bound for blowups of $\mathbf A^4$ with terminal singularities: the smallest weight is always at most $32$, and at most $6$ in all but finitely many cases.

On the construction of a complete Kähler-Einstein metric with negative scalar curvature near an isolated log-canonical singularity

In this short note we are concerned with the Kähler-Einstein metrics near cone type log canonical singularities. By two different approaches, we construct a Kähler-Einstein metric with negative scalar curvature in a neighborhood of the cone over a Calabi-Yau manifold which is complete towards the vertex. This provides a local model for the further study of the global Kähler-Einstein metrics on singular varieties. In the first approach, we show that the singularity is uniformized by a complex ball and hence the induced metric from the Bergman metric of the ball is the desired one. In the second approach, we obtain a Kähler-Einstein metric with the desired properties by Calabi Ansatz. At last, we show that the obtained metrics are the same.

Singularities of Non-ℚ-Gorenstein Varieties Admitting a Polarized Endomorphism

Abstract In this paper, we introduce a new generalization of log canonical singularities for non-$\mathbb{Q}$-Gorenstein varieties. We prove that if a normal complex projective variety has a non-invertible polarized endomorphism, then it has log canonical singularities in our sense. As a corollary, we give an affirmative answer to a conjecture of Broustet and Höring [ 6].

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.

Singularities of linear systems and boundedness of Fano varieties

We study log canonical thresholds (also called global log canonicalthreshold or $\alpha$-invariant) of $\mathbb{R}$-linear systems. We prove existence of positive lower bounds in different settings, in particular, proving a conjecture of Ambro. We then show that the Borisov-Alexeev-Borisov conjecture holds; that is, given a natural number $d$ and a positive real number $\epsilon$, the set of Fano varieties of dimension $d$ with $\epsilon$-log canonical singularities forms a bounded family. This implies that birational automorphism groups of rationally connected varieties are Jordan which, in particular, answers a question of Serre. Next we show that if the log canonical threshold of the anti-canonical system of a Fano variety is at most one, then it is computed by some divisor, answering a question of Tian in this case.

On the stability of extensions of tangent sheaves on Kähler–Einstein Fano/Calabi–Yau pairs

Let S be a smooth projective variety and $$\Delta $$ a simple normal crossing $${\mathbb {Q}}$$ -divisor with coefficients in (0, 1]. For any ample $${\mathbb {Q}}$$ -line bundle L over S, we denote by $$\mathscr {E}(L)$$ the extension sheaf of the orbifold tangent sheaf $$T_S(-\log (\Delta ))$$ by the structure sheaf $$\mathcal {O}_S$$ with the extension class $$c_1(L)$$ . We prove the following two results: These results generalize Tian’s result where $$-K_S$$ is ample and $$\Delta =\emptyset $$ . We give two applications of these results. The first is to study a question by Borbon–Spotti about the relationship between local Euler numbers and normalized volumes of log canonical surface singularities. We prove that the two invariants differ only by a factor 4 when the log canonical pair is an orbifold cone over a marked Riemann surface. In particular we complete the computation of Langer’s local Euler numbers for any line arrangements in $${\mathbb {C}}^2$$ . The second application is to derive Miyaoka–Yau-type inequalities on K-semistable log-smooth Fano pairs and Calabi–Yau pairs, which generalize some Chern-number inequalities proved by Song–Wang.

Extracting non-canonical places

Let (X,B) be a log canonical pair and V be a finite set of divisorial valuations with log discrepancy in [0,1). We prove that there exists a projective birational morphism π:Y→X so that the exceptional locus is divisorial, the exceptional divisors are Q-Cartier, and they correspond to elements of V. We study how two such models are related. As applications, we apply the main theorem to the study of deformations of log canonical singularities. Furthermore, we prove the existence of rationalization for log canonical singularities.

Berkovich log discrepancies in positive characteristic

We introduce and study a log discrepancy function on the space of semivaluations centered on an integral noetherian scheme of positive characteristic. Our definition shares many properties with the analogue in characteristic zero; we prove that if log resolutions exist, then our definition agrees with previous approaches to log discrepancies of semivaluations that these resolutions. We then apply this log discrepancy to a variety of topics in singularity theory over fields of positive characteristic. Strong F-regularity and sharp $F$-purity of Cartier subalgebras are detected using positivity and non-negativity of log discrepancies of semivaluations, just as Kawamata log terminal and log canonical singularities are defined using divisorial log discrepancies, making precise a long-standing heuristic. We prove, in positive characteristic, several theorems of Jonsson and Mustata in characteristic zero regarding log canonical thresholds of graded sequences of ideals. Along the way, we give a valuation-theoretic proof that asymptotic multiplier ideals are coherent on strongly F-regular schemes.

On exchange spectra of valued quivers and cluster algebras

We introduce and study a log discrepancy function on the space of semivaluations centered on an integral noetherian scheme of positive characteristic. Our definition shares many properties with the analogue in characteristic zero; we prove that if log resolutions exist, then our definition agrees with previous approaches to log discrepancies of semivaluations that these resolutions. We then apply this log discrepancy to a variety of topics in singularity theory over fields of positive characteristic. Strong F-regularity and sharp $F$-purity of Cartier subalgebras are detected using positivity and non-negativity of log discrepancies of semivaluations, just as Kawamata log terminal and log canonical singularities are defined using divisorial log discrepancies, making precise a long-standing heuristic. We prove, in positive characteristic, several theorems of Jonsson and Mustata in characteristic zero regarding log canonical thresholds of graded sequences of ideals. Along the way, we give a valuation-theoretic proof that asymptotic multiplier ideals are coherent on strongly F-regular schemes.