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Sławomir Cynk

Publications and source records attributed to Sławomir Cynk.

6 recordsLinked to original sources
arXiv2026

Crepant resolutions of four-dimensional finite quotients

We give counterexamples to the conjecture of Dais-Henk-Ziegler on projective crepant resolutions of complete-intersection quotient singularities. For finite $G\subset\operatorname{SL}_4(\mathbb{Z})$, we determine whether $\mathbb{C}^4/G$ admits a projective crepant resolution, with the representations considered up to conjugacy in $\operatorname{SL}_4(\mathbb{C})$. Ten explicit criteria decide all cases. The examples also show that the Molien series and the age polynomial do not determine existence. We also discuss questions of Ito-Reid on maximal cyclic subgroups and the generation of cohomology in degree two.

math.AG↗
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.

math.AG↗
arXiv2026

Classification of double octic Calabi-Yau threefolds defined by an arrangement of eight planes II

We present a complete classification of all arrangements of eight planes in projective threespace that give rise to double octic Calabi-Yau threefolds. Building on earlier work, we determine all 455 combinatorial types and describe the projective automorphism groups associated with each arrangement. These automorphisms allow us to identify geometrically distinguished subfamilies and elements, and to construct a rich collection of elliptic and K3 fibrations arising naturally from the singularities of the arrangements. As applications, we exhibit two one-parameter families of Calabi-Yau threefolds that each possess three pairwise non-equivalent maximally unipotent monodromy points-phenomena that are rare in known examples. We further construct two Calabi-Yau threefolds whose third cohomology decomposes, via complex multiplication, into complementary two-dimensional Hodge structures of types (1,0,1,0) and (0,1,0,1). This decomposition yields new instances of modularity not previously observed in the Calabi-Yau setting.

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arXiv2025

Elliptic and K3 fibrations on double octic Calabi-Yau threefolds

We study fibrations by elliptic curves and K3 surfaces of double octic Calabi-Yau threefolds determined by singular lines and points of multiplicity at least 4 of the defining octic arrangement. As a consequence we conclude that every double octic admits many elliptic and K3 fibrations. We apply descriptions of some K3 fibrations to construct a birational map between elements of two families of double octics.

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arXiv2022

Periods of singular double octic Calabi-Yau threefolds and modular forms

By the modularity theorem every rigid Calabi-Yau threefold $X$ has associated modular form $f$ such that the equality of $L$-functions $L(X,s)=L(f,s)$ holds. In this case period integrals of $X$ are expected to be expressible in terms of the special values $L(f,1)$ and $L(f,2)$. We propose a similar interpretation of period integrals of a nodal model of $X$. It is given in terms of certain variants of a Mellin transform of $f$. We provide numerical evidence towards this interpretation based on a case of double octics.

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