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Dominik Burek

Publications and source records attributed to Dominik Burek.

8 recordsLinked to original sources
arXiv2026

Fixed-locus relations and purely non-symplectic automorphisms of order 6 on K3 surfaces

We study relations between invariants of the fixed loci of purely non-symplectic automorphisms of $K3$ surfaces. We obtain these relations by comparing the stringy Euler characteristic of higher dimensional Borcea--Voisin varieties with their Hodge numbers. For orders $4$ and $6$ we explain the geometric meaning of these relations and show how they follow from the Lefschetz and Riemann--Hurwitz formulas. For order $6$, the classifications of fixed loci for orders $2$ and $3$ give $817$ candidates. We use local conditions on discriminant forms, Hermitian trace forms and bounds for root-free lattices to reduce this number to $204$. All $142$ numerical types obtained from the $150$ deformation classes of Brandhorst and Hofmann satisfy these conditions. Also, we give explicit elliptic and projective models for each of the $20$ possible fixed loci of the generator.

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arXiv2026

Crepant resolutions of fiber products and Kummer fibrations of rational elliptic surfaces with non-reduced fibers

Let $S_1$ and $S_2$ be rational elliptic surfaces with section over an algebraically closed field of characteristic zero, and let $X=S_1\times_{\mathbb{P}^1}S_2$. We determine the local equations produced by pairs of Kodaira fibers, including non-reduced fibers, and construct crepant resolutions for the admissible local models. For every construction we record whether it is projective and compute the number of irreducible components and the Euler characteristic of the resolved central fiber. Non-existence is asserted only in the normal Gorenstein cases covered by the terminal factorial obstruction; the remaining entries are stated as open for the methods of this paper. Under an explicit global compatibility hypothesis, the local data give formulas for the Hodge numbers of a smooth projective resolved fiber product. We also study Kummer quotients by fiberwise involutions. Their fixed curves are analyzed through the monodromy-orbit description of normalized fiber products of multisections. The final section records what the same constructions imply after reduction in characteristics $5$ and $7$, without interpreting characteristic-$p$ Euler and Picard data as complex Hodge numbers.

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arXiv2024

Higher dimensional analogon of Borcea-Voisin Calabi-Yau manifolds, their Hodge numbers and $L$-functions

We construct a series of examples of Calabi-Yau manifolds in an arbitrary dimension and compute the main invariants. In particular, we give higher dimensional generalization of Borcea-Voisin Calabi-Yau threefolds. We give a method to compute a local zeta function using the Frobenius morphism for orbifold cohomology introduced by Rose. We compute Hodge numbers of the constructed examples using orbifold Chen-Ruan cohomology.

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arXiv2023

Zeta function of some Kummer Calabi-Yau 3-folds

We compute Hodge numbers and zeta function of a Kummer Calabi-Yau 3-folds introduced by M. Andreatta and J. Wiśniewski in arXiv:0804.4611 and investigated by M. Donten-Bury in arXiv:0812.3758.

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arXiv2019

Higher dimensional Calabi-Yau manifolds of Kummer type

Based on Cynk-Hulek method we construct complex Calabi-Yau varieties of arbitrary dimensions using elliptic curves with automorphism of order 6. Also we give formulas for Hodge numbers of varieties obtained from that construction. We shall generalize result of Katsura and Schütt to obtain arbitrarily dimensional Calabi-Yau manifolds which are Zariski in any characteristic $p\not\equiv 1\pmod{12}.$

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arXiv2018

A new upper bound for numbers with the Lehmer property and its application to repunit numbers

A composite positive integer $n$ has the Lehmer property if $ϕ(n)$ divides $n-1,$ where $ϕ$ is an Euler totient function. In this note we shall prove that if $n$ has the Lehmer property, then $n\leq 2^{2^{K}}-2^{2^{K-1}}$, where $K$ is the number of prime divisors of $n$. We apply this bound to repunit numbers and prove that there are at most finitely many numbers with the Lehmer property in the set $$ \left\{\frac{g^{n}-1}{g-1}\ \bigg|\ n,g\in\mathbb{N},\ ν_{2}(g)+ν_{2}(g+1)\leq L\ \right\}, $$ where $ν_{2}(g)$ denotes the highest power of $2$ that divides $g$, and $L\geq 1$ is a fixed real number.

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arXiv2017

Hodge Numbers of Generalised Borcea-Voisin Threefolds

We shall reproof formulas for the Hodge numbers of Calabi-Yau threefolds of Borcea-Voisin type constructed by A. Cattaneo and A. Garbagnati, using the orbifold cohomology formula and the orbifold Euler characteristic.

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