Sławomir Cynk

Higher-Dimensional Modular Calabi-Yau Manifolds

We construct several examples of higher-dimensional Calabi?Yau manifolds and prove their modularity.

Geometry and Arithmetic of Certain Double Octic Calabi-Yau Manifolds

We study Calabi?Yau manifolds constructed as double coverings of mathbb{P}3 branched along an octic surface. We give a list of 87 examples corresponding to arrangements of eight planes defined over mathbb{Q}. The Hodge numbers are computed for all examples. There are 10 rigid Calabi?Yau manifolds and 14 families with h1,2 = 1. The modularity conjecture is verified for all the rigid examples.

Hodge numbers of nodal double octics

Let B be a surface of even degree d in P3 with nodes as the only singular points. In [1] Clemens proved that the topology of the small resolution of the double covering of P3 branched along B depends not only on the number of nodes but also on the so-called defect, a non-negative integer describing their configuration. The aim of this note is to give an elementary proof of the Clemens. result and to present some geometric interpretation of defect.

Double covers and Calabi-Yau varieties

Introduction. In the present paper we study examples of double coverings of the projective space P branched over an octic surface. A double covering of P branched over a smooth octic is a Calabi-Yau threefold. If the octic is singular then so is the double covering and we study its resolution of singularities. In this paper we restrict our considerations to the case of octics with only non-isolated singularities of a special type, namely looking locally like plane arrangements. Our research was inspired by a paper of Persson [5] where K3 surfaces arising as double covers of P branched over curves of degree six are studied. In this note we also adopt some methods introduced in [3] by Hunt in studying Fermat covers of P branched over plane arrangements. The main results of this note are Theorem 2.1 and Theorem 3.5 which can be formulated together as follows Theorem. Let S ⊂ P be an octic arrangement with no q-fold curve for q ≥ 4 and no p-fold point for p ≥ 6. Then the double covering of P branched along S has a non-singular model Y which is a Calabi-Yau threefold. Moreover if S contains no triple elliptic curves and l3 triple lines then the Euler characteristic e(Y ) of Y is given as follows

MODULAR CALABI–YAU THREEFOLDS OF LEVEL EIGHT

In the studies on the modularity conjecture for rigid Calabi–Yau threefolds several examples with the unique level 8 cusp form were constructed. According to the Tate conjecture, correspondences inducing isomorphisms on the middle cohomologies should exist between these varieties. In the paper, we construct several examples of such correspondences. In the constructions elliptic fibrations play a crucial role. In fact we show that all but three examples are in some sense built upon two modular curves from the Beauville list.

Non-liftable Calabi–Yau spaces

We construct many new non-liftable three-dimensional Calabi–Yau spaces in positive characteristic. The technique relies on lifting a nodal model to a smooth rigid Calabi–Yau space over some number field as introduced by one of us jointily with D. van Straten.

NON-FACTORIAL NODAL COMPLETE INTERSECTION THREEFOLDS

We give a bound on the minimal number of singularities of a nodal projective complete intersection threefold which contains a smooth complete intersection surface that is not a Cartier divisor.

Euler characteristic of a complete intersection

In this paper we study the behaviour of the degree of the Fulton–Johnson class of a complete intersection under a blow–up with a smooth center under the assumption that the strict transform is again a complete intersection. Our formula is a generalization of the genus formula for singular curves in smooth surfaces.

The geometry and arithmetic of a Calabi-Yau siegel threefold

In this paper we treat in details a modular variety

Calabi–Yau Conifold Expansions

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