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Sándor J Kovács

Publications and source records attributed to Sándor J Kovács.

At least 19 recordsLinked to original sources
arXiv2024

Rational singularities

A resolution-free definition of rational singularities is introduced, and it is proved that for a variety admitting a resolution of singularities, so in particular in characteristic zero, this is equivalent to the usual definition. It is also demonstrated that pseudo-rational singularities are rational. The main theorem, a Kempf-type criterion for rational singularities, is new even in characteristic zero contrasting an earlier result of Cutkosky that a similar result with slightly different assumptions could not hold. Several applications are included. In particular, an Elkik-type theorem, proving that Cohen-Macaulay dlt singularities are rational in arbitrary characteristic.

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arXiv2024

KSB stability is automatic in codimension 3

KSB stability holds at codimension 1 points trivially, and it is quite well understood at codimension 2 points, since we have a complete classification of 2-dimensional slc singularities. We show that it is automatic in codimension 3.

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arXiv2022

Du Bois property of log centers

In this note we generalize the results of [KK10] and [KK20] by showing that if a closed subset V of X is "close enough" to being a union of log canonical centers, then it is Du Bois.

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arXiv2022

Arakelov inequalities in higher dimensions

We develop a Hodge theoretic invariant for families of projective manifolds that measures the potential failure of an Arakelov-type inequality in higher dimensions, one that naturally generalizes the classical Arakelov inequality over regular quasi-projective curves. We show that for families of manifolds with ample canonical bundle this invariant is uniformly bounded. As a consequence we establish that such families over a base of arbitrary dimension verify the aforementioned Arakelov inequality, answering a question of Viehweg.

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arXiv2019

Frobenius split anticanonical divisors

In this note I extend two theorems of Sommese regarding abelian varieties to arbitrary characteristic; that an abelian variety cannot be an ample divisor in a smooth projective variety and that a cone over an abelian variety of dimension at least two is not smoothable.

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arXiv2018

Deformations of log canonical and F-pure singularities

We introduce a lifting property for local cohomology, which leads to a unified treatment of the dualizing complex for flat morphisms with semi-log-canonical, Du Bois or F-pure fibers. As a consequence we obtain that, in all 3 cases, the cohomology sheaves of the relative dualizing complex are flat and commute with base change. We also derive several consequences for deformations of semi-log-canonical, Du Bois and F-pure singularities.

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arXiv2018

Deformations of log canonical singularities

We prove that the cohomology sheaves of the relative dualizing complex of a flat family of varieties with semi-log-canonical or Du Bois singularities are flat and commute with base change. This is a local version of our earlier similar result where the family was assumed to be projective. We also derive several consequences for deformations of semi-log-canonical and of Du Bois singularities.

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arXiv2017

The minimal model program for b-log canonical divisors and applications

We discuss the minimal model program for b-log varieties, which is a pair of a variety and a b-divisor, as a natural generalization of the minimal model program for ordinary log varieties. We show that the main theorems of the log MMP work in the setting of the b-log MMP. If we assume that the log MMP terminates, then so does the b- log MMP. Furthermore, the b-log MMP includes both the log MMP and the equivariant MMP as special cases. There are various interesting b-log varieties arising from different objects, including the Brauer pairs, or "non-commutative algebraic varieties which are finite over their centres". The case of toric Brauer pairs is discussed in further detail.

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arXiv2016

On the boundedness of SLC surfaces of general type

The purpose of this note is to give a new proof of Alexeev's boundedness result for stable surfaces which is independent of the base field and to highlight some important consequences of this result.

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arXiv2016

Inversion of adjunction for rational and Du Bois pairs

We prove several results about the behavior Du Bois singularities and Du Bois pairs in families. Some of these generalize existing statements about Du Bois singularities to the pair setting while others are new even in the non-pair setting. We also prove a new inversion of adjunction result for Du Bois and rational pairs. In the non-pair setting this asserts that if a family over a smooth base has a special fiber $X_0$ with Du Bois singularities and the general fiber has rational singularities, then the total space has rational singularities near $X_0$.

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arXiv2014

Potentially Du Bois spaces

We investigate properties of potentially Du Bois singularities, that is, those that occur on the underlying space of a Du Bois pair. We show that a normal variety $X$ with potentially Du Bois singularities and Cartier canonical divisor $K_X$ is necessarily log canonical, and hence Du Bois. As an immediate corollary, we obtain the Lipman-Zariski conjecture for varieties with potentially Du Bois singularities. We also show that for a normal surface singularity, the notions of Du Bois and potentially Du Bois singularities coincide. In contrast, we give an example showing that in dimension at least three, a normal potentially Du Bois singularity $x \in X$ need not be Du Bois even if one assumes the canonical divisor $K_X$ to be $\mathbb{Q}$-Cartier.

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arXiv2014

An optimal extension theorem for 1-forms and the Lipman-Zariski conjecture

Let $X$ be a normal variety. Assume that for some reduced divisor $D \subset X$, logarithmic 1-forms defined on the snc locus of $(X, D)$ extend to a log resolution $\tilde X \to X$ as logarithmic differential forms. We prove that then the Lipman-Zariski conjecture holds for $X$. This result applies in particular if $X$ has log canonical singularities. Furthermore, we give an example of a 2-form defined on the smooth locus of a three-dimensional log canonical pair $(X, \emptyset)$ which acquires a logarithmic pole along an exceptional divisor of discrepancy zero, thereby improving on a similar example of Greb, Kebekus, Kovács and Peternell.

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arXiv2014

On positivity and base loci of vector bundles

The aim of this note is to shed some light on the relationships among some notions of positivity for vector bundles that arose in recent decades. Our purpose is to study several of the positivity notions studied for vector bundles with some notions of asymptotic base loci that can be defined on the variety itself, rather than on the projectivization of the given vector bundle. We relate some of the different notions conjectured to be equivalent with the help of these base loci, and we show that these can help simplify the various relationships between the positivity properties present in the literature. In particular, we define augmented and restricted base loci $\mathbb{B}_+ (E)$ and $\mathbb{B}_- (E)$ of a vector bundle $E$ on the variety $X$, as generalizations of the corresponding notions studied extensively for line bundles. As it turns out, the asymptotic base loci defined here behave well with respect to the natural map induced by the projectivization of the vector bundle $E$.

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arXiv2013

The cone of curves of K3 surfaces revisted

The main theorem of "S. J. Kovács: The cone of curves of a K3 surface, Math. Ann. 300 (1994), no. 4, 681-691" is proved in arbitrary characteristic. The proof is essentially the same as in the original paper where it was stated only over the complex numbers.

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arXiv2013

Steenbrink vanishing extended

The notion of DB index, a measure of how far a singularity of a pair is from being Du Bois, is introduced and used to generalize vanishing theorems of Steenbrink and others with simpler and more natural proofs than the originals.

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