arXiv subjects
Publications and source records attributed to Dominik Burek.
We give new relations between geometric invariants of $K3$ surfaces with purely non-symplectic automorphisms of order 4 and 6. Our approach is based on a comparison of two methods of computation of formulas for the Euler characteristic of higher dimensional generalized Borcea-Vosin Calabi-Yau manifolds constructed in [Bur21].
We compute Hodge numbers and zeta function of a Kummer Calabi-Yau 3-folds introduced by M. Andreatta and J. Wi\'sniewski in arXiv:0804.4611 and investigated by M. Donten-Bury in arXiv:0812.3758.
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.
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 $p\not\equiv 1\pmod{12}.$
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.
We construct examples of modular rigid Calabi--Yau threefolds, which give a realization of some new weight 4 cusp forms.
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.