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Victor Przyjalkowski

Publications and source records attributed to Victor Przyjalkowski.

At least 19 recordsLinked to original sources
arXiv2026

Maximality of the Hodge level for smooth weighted complete intersections of general type

We prove that for every smooth well formed weighted complete intersection of general type of dimension $n>0$, the Hodge number $h^{0,n}(X)$ is positive; in other words, its Hodge level is maximal. We also obtain an explicit lower bound for the geometric genus $p_g(X)$. This implies that the only weighted complete intersection of general type that is not a numerical intersection with a linear cone with $p_g(X)=1$ is $X_{6,6}\subset\mathbb{P}(1,2,2,3,3)$.

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arXiv2026

Landau-Ginzburg models for Fano threefolds of Picard rank one and exceptional collections

We study fibers with isolated singularities of Landau-Ginzburg models for Fano threefolds of Picard rank one. We compare the data we get with maximal known lengths of exceptional collections in derived categories of coherent sheaves on the Fano threefolds, verify some predictions of Homological Mirror Symmetry, and present some expectations about exceptional collections for Fano threefolds.

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arXiv2025

Singularities of Landau-Ginzburg models for complete intersections and derived categories

Mirror symmetry predicts that bounded derived category of a smooth Fano variety is equivalent to Fukaya-Seidel category of its Landau-Ginzburg model. It is expected that fibers of Landau-Ginzburg model with ordinary double points correspond to an exceptional collection of a Fano variety. We verify this expectation on a numerical level for Fano complete intersections and Calabi-Yau compactifications of their toric Landau-Ginzburg models of Givental's type.

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arXiv2025

Fibers of Landau-Ginzburg models and rationality

In this article, we study how the rationality of a Fano threefold is reflected in its standard mirror Landau-Ginzburg model and its deformations. The main result is that a Fano threefold is rational if and only if the monodromy around every reducible fiber of its generic mirror Landau-Ginzburg model is unipotent.

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arXiv2025

G-coregularity of del Pezzo surfaces

We introduce and study the notion of $G$-coregularity of algebraic varieties endowed with an action of a finite group $G$. We compute $G$-coregularity of smooth del Pezzo surfaces of degree at least 6, and give a characterization of groups that can act on conic bundles with $G$-coregularity 0. We describe the relations between the notions of $G$-coregularity, $G$-log-canonical thresholds, $G$-rigidity, and exceptional quotient singularities.

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arXiv2025

Modularity of Landau-Ginzburg models

For each Fano threefold, we construct a family of Landau-Ginzburg models which satisfy many expectations coming from different aspects of mirror symmetry; they are log Calabi-Yau varieties with proper potential maps; they admit open algebraic torus charts on which the potential function $w$ restricts to a Laurent polynomial satisfying a deformation of the Minkowski ansatz; the general fibres of $w$ are Dolgachev-Nikulin dual to the anticanonical hypersurfaces in $X$. To do this, we study the deformation theory of Landau-Ginzburg models in arbitrary dimension, following the third-named author, Kontsevich, and Pantev, specializing to the case of Landau-Ginzburg models obtained from Laurent polynomials. Our proof of Dolgachev-Nikulin mirror symmetry is by detailed case-by-case analysis, refining work of Cheltsov and the fifth-named author.

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arXiv2023

Coregularity of smooth Fano threefolds

We study the coregularity of smooth Fano threefolds. We prove that for 100 out of 105 families of smooth Fano threefolds, a general member in the family has coregularity 0; moreover, for 92 families out of these 100, any member in the family has coregularity 0; for the remaining 5 families, we obtain some partial results. In particular, we show that there exist families of smooth Fano threefolds whose general elements have positive coregularity.

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arXiv2023

Fibers over infinity of Landau-Ginzburg models

We conjecture that the number of components of the fiber over infinity of Landau--Ginzburg model for a smooth Fano variety $X$ equals the dimension of the anticanonical system of $X$. We verify this conjecture for log Calabi--Yau compactifications of toric Landau--Ginzburg models for smooth Fano threefolds, complete intersections, and some toric varieties.

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arXiv2023

On Hodge level of weighted complete intersections of general type

We show that smooth varieties of general type which are well formed weighted complete intersections of Cartier divisors have maximal Hodge level, that is, their the rightmost middle Hodge numbers do not vanish. We show that this does not hold in the quasi-smooth case.

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arXiv2023

Well formedness vs weak well formedness

In the literature there are two definitions of well formed varieties in weighted projective spaces. According to the first one, well formed variety is the one whose intersection with the singular locus of the ambient weighted projective space has codimension at least two, while, according to the second one, well formed variety is the one who does not contain in codimension one a singular stratum of the ambient weighted projective space. We show that these two definitions indeed differ, and show that they coincide for quasi-smooth weighted complete intersections of dimension at least 3.

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arXiv2021

Laurent polynomials in Mirror Symmetry: why and how?

We survey the approach to mirror symmetry via Laurent polynomials, outlining some of the main conjectures, problems, and questions related to the subject. We discuss: how to construct Landau--Ginzburg models for Fano varieties; how to apply them to classification problems; and how to compute invariants of Fano varieties via Landau--Ginzburg models.

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arXiv2021

On singular log Calabi-Yau compactifications of Landau-Ginzburg models

We consider the procedure that constructs log Calabi-Yau compactifications of weak Landau-Ginzburg models of Fano varieties. We apply it for del Pezzo surfaces and coverings of projective spaces of index one. For the coverings of degree greater then 2 the log Calabi-Yau compactification is singular; moreover, no smooth projective log Calabi-Yau compactification exists. We also prove in the cases under consideration the conjecture saying that the number of components of the fiber over infinity %and of finite fibers is equal to the dimension of an anticanonical system of the Fano variety.

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arXiv2020

On automorphisms of quasi-smooth weighted complete intersections

We show that every reductive subgroup of the automorphism group of a quasi-smooth well formed weighted complete intersection is a restriction of a subgroup in the automorphism group in the ambient weighted projective space. Also, we provide examples demonstrating that an automorphism group of a quasi-smooth well formed Fano weighted complete intersection may be infinite and even non-reductive.

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arXiv2020

Bounds for smooth Fano weighted complete intersections

We prove that if a smooth variety with non-positive canonical class can be embedded into a weighted projective space of dimension $n$ as a well formed complete intersection and it is not an intersection with a linear cone therein, then the weights of the weighted projective space do not exceed $n+1$. Based on this bound we classify all smooth Fano complete intersections of dimensions $4$ and $5$, and compute their invariants.

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arXiv2020

Hodge level for weighted complete intersections

We give lower bounds for Hodge numbers of smooth well formed Fano weighted complete intersections. In particular, we compute their Hodge level, that is, the maximal distance between non-trivial Hodge numbers in the same row of the Hodge diamond. This allows us to classify varieties whose Hodge numbers are like that of a projective space, of a curve, or of a Calabi--Yau variety of low dimension.

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