Primitive Contractions of Calabi–Yau Threefolds I

We construct examples of primitive contractions of Calabi–Yau threefolds with exceptional locus being ?¹ × ?¹, ?², and smooth del Pezzo surfaces of degrees ≤ 5. We describe the images of these primitive contractions and find their smoothing families. In particular, we give a method to compute the Hodge numbers of a generic fiber of the smoothing familly of each Calabi–Yau threefold with one isolated singularity obtained after a primitive contraction of type II. As an application, we get examples of natural conifold transitions between some families of Calabi–Yau threefolds.     

Higher dimensional Calabi–Yau manifolds of Kummer type

Based on Cynk–Hulek method from [7] we construct complex Calabi–Yau varieties of arbitrary dimensions using elliptic curves with an automorphism of order 6. Also we give formulas for Hodge numbers of varieties obtained from that construction. We shall generalize the result of [11] to obtain arbitrarily dimensional Calabi–Yau manifolds which are Zariski in any characteristic .     

A bound for the orders of the torsion groups of surfaces with c 1 2 = 2 - 1

We shall give a bound for the orders of the torsion groups of minimal algebraic surfaces of general type whose first Chern numbers are twice the Euler characteristics of the structure sheaves minus 1, where the torsion group of a surface is the torsion part of the Picard group. Namely, we shall show that the order is at most 3 if the Euler characteristic is 2, that the order is at most 2 if the Euler characteristic is greater than or equal to 3, and that the order is 1 if the Euler characteristic is greater than or equal to 7.     

New construction of complex manifold via conifold transition

For a generic anti-canonical hypersurface in each smooth toric Fano 4-fold with rank 2 Picard group, we prove there exist three isolated rational curves in it. Moreover, for all these 4-folds except one, the contractions of generic anti-canonical hypersurfaces along the three rational curves can be deformed to smooth threefolds which is diffeomorphic to connected sums of S3 ×S3 . In this manner, we obtain complex structures with trivial canonical bundles on some connected sums of S3 × S3 . This construction is an analogue of that made by Friedman [On threefolds with trivial canonical bundle. In: Complex Geometry and Lie Theory, volume 53 of Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 1991, 103-134], Lu and Tian [Complex structures on connected sums of S3 × S3 . In: Manifolds and Geometry, Pisa, 1993, 284-293] who used only quintics in P4 .     

Calabi–Yau Coalgebras

We present a method for constructing the minimal injective resolution of a simple comodule of a path coalgebra of quivers with relations. Dual to the Calabi–Yau condition of algebras, we introduce the concept of a Calabi–Yau coalgebra, and then describe the Calabi–Yau coalgebras of low global dimensions.     

Deformations of Quadratic Algebras with Antisymmetric Relations

A finitely generated quadratic algebra with antisymmetric generating relations is called a weakly symmetric algebra. The automorphism group and Calabi–Yau property of a Poincaré–Birkhoff–Witt (PBW)-deformation of a weakly symmetric algebra are discussed. It is shown that the Calabi–Yau property of a PBW-deformation of a weakly symmetric algebra is equivalent to that of the corresponding augmented PBW-deformation under some mild conditions.     

Calabi–Yau Threefolds and Moduli of Abelian Surfaces I

We describe birational models and decide the rationality/unirationality of moduli spaces A _d (and A _d ^lev ) of (1, d)-polarized Abelian surfaces (with canonical level structure, respectively) for small values of d. The projective lines identified in the rational/unirational moduli spaces correspond to pencils of Abelian surfaces traced on nodal threefolds living naturally in the corresponding ambient projective spaces, and whose small resolutions are new Calabi–Yau threefolds with Euler characteristic zero.     

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 holds. In this case, period integrals of X are expected to be expressible in terms of the special values and . 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 toward this interpretation based on a case of double octics.     

Rigid Curves in Complete Intersection Calabi–Yau Threefolds

Working over the complex numbers, we study curves lying in a complete intersection K3 surface contained in a (nodal) complete intersection Calabi–Yau threefold. Under certain generality assumptions, we show that the linear system of curves in the surface is a connected componend of the the Hilbert scheme of the threefold. In the case of genus one, we deduce the existence of infinitesimally rigid embeddings of elliptic curves of arbitrary degree in the general complete intersection Calabi–Yau threefold.     

A fake projective plane with an order 7 automorphism

A fake projective plane is a compact complex surface (a compact complex manifold of dimension 2) with the same Betti numbers as the complex projective plane, but not isomorphic to the complex projective plane. As was shown by Mumford, there exists at least one such surface.In this paper we prove the existence of a fake projective plane which is birational to a cyclic cover of degree 7 of a Dolgachev surface.     

Galois closure and Lagrangian varieties

Let X be a complex projective variety and consider the morphism