Log Canonical Singularities Research Articles

Non-rational centers of log canonical singularities

We show that if (X,B) is a log canonical pair with dimX⩾d+2, whose non-klt centers have dimension ⩾d, then X has depth ⩾d+2 at every closed point.

Normal del Pezzo surfaces of rank one with log canonical singularities

We study normal del Pezzo surfaces of rank one with only log canonical singularities. Under some additional assumptions, we classify such surfaces. Moreover, we prove that every normal del Pezzo surface of rank one with only rational log canonical singularities has at most five singular points.

A vanishing theorem and asymptotic regularity of powers of ideal sheaves

Let I be an ideal sheaf on P n . In the first part of this paper, we bound the asymptotic regularity of powers of I as s p ⩽ reg I p ⩽ s p + e , where e is a constant and s is the s-invariant of I . We also give the same upper bound for the asymptotic regularity of symbolic powers of I under some conditions. In the second part, by using multiplier ideal sheaves, we give a vanishing theorem of powers of I when it defines a local complete intersection subvariety with log canonical singularities.

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 semi-ample. Moreover, we consider semi-ampleness of the anti-log canonical divisor of any weak log Fano pair with log canonical singularities. We show semi-ampleness 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 semi-ampleness under such conditions. In particular we derive semi-ampleness of the anti-canonical divisors of log canonical weak Fano varieties whose lc centers are at most 1-dimensional. We also investigate the Kleiman–Mori cones of weak log Fano pairs with log canonical singularities.

A bound for the Castelnuovo–Mumford regularity of log canonical varieties

In this note, we give a bound for the Castelnuovo–Mumford regularity of a homogeneous ideal I in terms of the degrees of its generators. We assume that I defines a local complete intersection with log canonical singularities.

Log canonical singularities are Du Bois

A recurring difficulty in the Minimal Model Program is that while log terminal singularities are quite well behaved (for instance, they are rational), log canonical singularities are much more complicated; they need not even be Cohen-Macaulay. The aim of this paper is to prove that log canonical singularities are Du Bois. The concept of Du Bois singularities, introduced by Steenbrink, is a weakening of rationality. We also prove flatness of the cohomology sheaves of the relative dualizing complex of a projective family with Du Bois fibers. This implies that each connected component of the moduli space of stable log varieties parametrizes either only Cohen-Macaulay or only non-Cohen-Macaulay objects.

The canonical sheaf of Du Bois singularities

We prove that a Cohen–Macaulay normal variety X has Du Bois singularities if and only if π ∗ ω X ′ ( G ) ≃ ω X for a log resolution π : X ′ → X , where G is the reduced exceptional divisor of π. Many basic theorems about Du Bois singularities become transparent using this characterization (including the fact that Cohen–Macaulay log canonical singularities are Du Bois). We also give a straightforward and self-contained proof that (generalizations of) semi-log-canonical singularities are Du Bois, in the Cohen–Macaulay case. It also follows that the Kodaira vanishing theorem holds for semi-log-canonical varieties and that Cohen–Macaulay semi-log-canonical singularities are cohomologically insignificant in the sense of Dolgachev.

The links of 3-dimensional singularities defined by Brieskorn polynomials

In this paper, we completely determine the diffeomorphism types of the 5-dimensional links of 3-dimensional log-canonical singularities defined by Brieskorn polynomials. Moreover, we show that if k is an integer with 1 ≤ k < 611, then there is no link K defined by a Brieskorn polynomial in ℂ4 such that the order of H2(K) is 6k. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

A simple characterization of Du Bois singularities

We prove the following theorem characterizing Du Bois singular- ities. Suppose that Y is smooth and that X is a reduced closed subscheme. Let � : e Y ! Y be a log resolution of X in Y that is an isomorphism outside of X. If E is the reduced pre-image of X in e Y , then X has Du Bois singularities if and only if the natural map OX ! R��OE is a quasi-isomorphism. We also deduce Kollar's conjecture that log canonical singularities are Du Bois in the special case of a local complete intersection and prove other results related to adjunction. 1. Introduction and Background In this paper, we prove a simple new characterization of Du Bois singularities. Inspired by this characterization, we deduce several new theorems, including results related to adjunction, and progress towards a conjecture of Kollar. Du Bois singularities were initially defined by Steenbrink as a setting where certain aspects of Hodge theory for smooth varieties still hold; see (Ste81a), (Ste81b) and (DB81). They are defined by the cohomology of a complex which is difficult to understand since it requires the computation of resolutions of singularities for several varieties (specifically a simplicial or cubic hyperresolution is required; see (Del74), (GNPP88), or (Car85)). Our new characterization of Du Bois singularities requires only a single resolution. Our main result is:

Factoriality of nodal three-dimensional varieties and connectedness of the locus of log canonical singularities

Shokurov's vanishing theorem is used for the proof of the -factoriality of the following nodal threefolds: a complete intersection of hypersurfaces and in of degrees and , , such that is smooth and ; a double cover of a smooth hypersurface of degree branched over the surface cut on by a hypersurface of degree , provided that .

Classification of three-dimensional exceptional log canonical hypersurface singularities. II

We study three-dimensional exceptional canonical hypersurface singularities which do not satisfy the condition of well-formedness. The result obtained completes the classification of three-dimensional exceptional log canonical hypersurface singularities begun in [4].

Dimensions of jet schemes of log singularities

We characterize Kawamata log terminal singularities and log canonical singularities by dimensions of jet schemes. Our main result is Theorem 2.4.

Stable reductive varieties II: projective case

We construct a moduli space of stable projective pairs with a nontrivial action of a connected reductive group. These stable reductive pairs are higher-dimensional analogs of stable n-pointed curves and generalize to the non-commutative case a functorial compactification of moduli of abelian varieties, and of toric pairs. We prove that, confirming a prediction of log Minimal Program, stable reductive pairs have semi log canonical singularities.

Log–canonical forms and log canonical singularities

For a normal subvariety V of ℂn with a good 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\{0}. For this purpose we introduce sheaves of m–canonical and L2,m–canonical forms on normal complex spaces. For the case of affine varieties with good C*–action we give an explicit formula for these sheaves in terms of the grading of the dualizing sheaf and its tensor powers.

Classification of three-dimensional exceptional log canonical hypersurface singularities. I

We describe three-dimensional exceptional strictly log canonical hypersurface singularities and give a detailed classification of three-dimensional exceptional canonical hypersurface singularities under the condition of well-formedness.

F-regular and F-pure rings vs. log terminal and log canonical singularities

We investigate the relationship of F-regular (resp. F-pure) rings and log terminal (resp. log canonical) singularities. Also, we extend the notions of F-regularity and F-purity to “F-singularities of pairs." The notions of F-regular and F-pure rings in characteristic p &gt; 0 p &gt; 0 are characterized by a splitting of the Frobenius map, and define some classes of rings having “mild" singularities. On the other hand, there are notions of log terminal and log canonical singularities defined via resolution of singularities in characteristic zero. These are defined also for pairs of a normal variety and a Q \mathbb Q -divisor on it, and play important roles in birational algebraic geometry. As an analog of these singularities of pairs, we introduce the concept of “F-singularities of pairs," namely strong F-regularity, divisorial F-regularity and F-purity for a pair ( A , Δ ) (A,\Delta ) of a normal ring A A of characteristic p &gt; 0 p &gt; 0 and an effective Q \mathbb Q -divisor Δ \Delta on Spec ⁡ A \operatorname {Spec} A . The main theorem of this paper asserts that, if K A + Δ K_{A}+\Delta is Q \mathbb Q -Cartier, then the above three variants of F-singularities of pairs imply KLT, PLT and LC properties, respectively. We also prove some results for F-singularities of pairs which are analogous to singularities of pairs in characteristic zero.

The indices of log canonical singularities

Let ( P ∈ X , Δ) be a three-dimensional log canonical pair such that Δ has only standard coefficients and that P is a center of log canonical singularities for ( X , Δ). Then we get an effective bound of the indices of these pairs and actually determine all the possible indices. Furthermore, under certain assumptions including the log Minimal Model Program, an effective bound is also obtained in dimension n ≥ 4.

Sparseness of Exceptional Quotient Singularities

Definition 1.1 (see [3, 5.6]). Let (X P ) be a normal singularity and let D = ∑ diDi be an effective Q -divisor on X such that the pair (X, D) has log-canonical singularities only. A pair (X, D) is said to be exceptional if there is at most one divisor E of the function field K(X) with discrepancy a(E, D) = −1. A log-canonical singularity (X P ) is said to be exceptional if a pair (X, D) is exceptional whenever it is log-canonical.

Rational, Log Canonical, Du Bois Singularities II: Kodaira Vanishing and Small Deformations

Kollar's conjecture, that log canonical singularities are Du Bois, is proved in the case of Cohen–Macaulay 3-folds. This in turn is used to derive Kodaira vanishing for this class of varieties. Finally it is proved that small deformations of Du Bois singularities are again Du Bois.

Exceptional quotient singularities

A singularity is said to be exceptional (in the sense of V. Shokurov), if for any log canonical boundary, there is at most one exceptional divisor of discrepancy -1. This notion is important for the inductive treatment of log canonical singularities. The exceptional singularities of dimension 2 are known: they belong to types E 6 , E 7 , E 8 after Brieskorn. In our previous paper, it was proved that the quotient singularity defined by Klein's simple group in its 3-dimensional representation is exceptional. In the present paper, the classification of all the three-dimensional exceptional quotient singularities is obtained. The main lemma states that the quotient of the affine 3-space by a finite group is exceptional if an only if the group has no semiinvariants of degree 3 or less. It is also proved that for any positive ε, there are only finitely many ε-log terminal exceptional 3-dimensional quotient singularities.