The Mahler measure of a Calabi–Yau threefold and special $$$$-values

The aim of this paper is to prove a Mahler measure formula of a four-variable Laurent polynomial whose zero locus defines a Calabi–Yau threefold. We show that its Mahler measure is a rational linear combination of a special \(L\)-value of the normalized newform in \(S_4(\Gamma _0(8))\) and a Riemann zeta value. This is equivalent to a new formula for a \(_6F_5\)-hypergeometric series evaluated at 1.     

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.     

Moduli of Polarized Calabi–Yau Pairs

We prove that the irreducible components of the moduli space of polarized Calabi–Yau pairs are projective.     

The construction of numerically Calabi–Yau orders on projective surfaces

In this paper, we construct a vast collection of maximal numerically Calabi–Yau orders utilising a noncommutative analogue of the well-known commutative cyclic covering trick. Such orders play an integral role in the Mori program for orders on projective surfaces and although we know a substantial amount about them, there are relatively few known examples.     

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.     

Global Smoothing of Calabi–Yau Threefolds II

The moduli spaces of Calabi–Yau threefolds are conjectured to be connected by the combination of birational contraction maps and flat deformations. In this context, it is important to calculate dim Def(X) from dim Def(~X) in terms of certain geometric information of f, when we are given a birational morphism f:~XX from a smooth Calabi–Yau threefold ~X to a singular Calabi–Yau threefold X. A typical case of this problem is a conjecture of Morrison-Seiberg which originally came from physics. In this paper we give a mathematical proof to this conjecture. Moreover, by using output of this conjecture, we prove that certain Calabi–Yau threefolds with nonisolated singularities have flat deformations to smooth Calabi–Yau threefolds. We shall use invariants of singularities closely related to Du Bois's work to calculate dim Def(X) from dim Def(~X).     

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.     

Symplectic Calabi–Yau 6-manifolds

In this article, we construct simply connected symplectic Calabi–Yau 6-manifolds by applying Gompf's symplectic fiber sum operation along T⁴${\mathbb{T}}^4$. Using our method, we also construct symplectic non-Kähler Calabi–Yau 6-manifolds with fundamental group Z$\mathbb{Z}$. This paper also produces the first examples of simply connected and non-simply connected symplectic Calabi–Yau 6-manifolds with fundamental groups _Zp×_Zq${\mathbb{Z}}_p\times {\mathbb{Z}}_q$, and Z×_Zq$\mathbb{Z}\times {\mathbb{Z}}_q$ for any p≥1$p\ge 1$ and q≥2$q\ge 2$via co-isotropic Luttinger surgery.     

Zeta functions of alternate mirror Calabi–Yau families

We prove that if two Calabi–Yau invertible pencils have the same dual weights, then they share a common factor in their zeta functions. By using Dwork cohomology, we demonstrate that this common factor is related to a hypergeometric Picard–Fuchs differential equation. The factor in the zeta function is defined over the rationals and has degree at least the order of the Picard–Fuchs equation. As an application, we relate several pencils of K3 surfaces to the Dwork pencil, obtaining new cases of arithmetic mirror symmetry.     

Calabi–Yau Manifolds with Affine Structures

Mathematical Notes -     

Calabi–Yau threefolds with non-Gorenstein involutions

The concept of non-Gorenstein involutions on Calabi–Yau threefolds is a higher dimensional generalization of non-symplectic involutions on K3 surfaces. We present some elementary facts about Calabi–Yau threefolds with non-Gorenstein involutions. We give a classification of the Calabi–Yau threefolds of Picard rank one with non-Gorenstein involutions, whose fixed locus is not zero-dimensional.     

Cleft extensions of Koszul twisted Calabi–Yau algebras

Let H be a twisted Calabi–Yau (CY) Hopf algebra and σ a 2-cocycle on H. Let A be an N-Koszul twisted CY algebra such that A is a graded H^σ- module algebra. We show that the cleft extension A#_σH is also a twisted CY algebra. This result has two consequences. Firstly, the smash product of an N-Koszul twisted CY algebra with a twisted CY Hopf algebra is still a twisted CY algebra. Secondly, the cleft objects of a twisted CY Hopf algebra are all twisted CY algebras. As an application of this property, we determine which cleft objects of U(D, λ), a class of pointed Hopf algebras introduced by Andruskiewitsch and Schneider, are Calabi–Yau algebras.     

Rigid realizations of modular forms in Calabi–Yau threefolds

We construct examples of modular rigid Calabi–Yau threefolds, which give a realization of some new weight 4 cusp forms.     

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.     

Poincaré–Birkhoff–Witt Deformation of Koszul Calabi–Yau Algebras

The Calabi–Yau property of the Poincaré–Birkhoff–Witt deformation of a Koszul Calabi–Yau algebra is characterized. Berger and Taillefer (J Noncommut Geom 1:241–270, 2007, Theorem 3.6) proved that the Poincaré–Birkhoff–Witt deformation of a Calabi–Yau algebra of dimension 3 is Calabi–Yau under some conditions. The main result in this paper generalizes their result to higher dimensional Koszul Calabi–Yau algebras. As corollaries, the necessary and sufficient condition obtained by He et al. (J Algebra 324:1921–1939, 2010) for the universal enveloping algebra, respectively, Sridharan enveloping algebra, of a finite-dimensional Lie algebra to be Calabi–Yau, is derived.     

Calabi–Yau Algebras Viewed as Deformations of Poisson Algebras

From any algebra A defined by a single non-degenerate homogeneous quadratic relation f, we prove that the quadratic algebra B defined by the potential w?=?fz is 3-Calabi–Yau. The algebra B can be viewed as a 3-Calabi–Yau completion of Keller. The algebras A and B are both Koszul. The classification of the algebras B in three generators, i.e., when A has two generators, leads to three types of algebras. The second type (the most interesting one) is viewed as a deformation of a Poisson algebra S whose Poisson bracket is non-diagonalizable quadratic. Although the potential of S has non-isolated singularities, the homology of S is computed. Next the Hochschild homology of B is obtained.