Calabi–Yau varieties with semi-stable fibre structures

Motivated by the Strominger–Yau–Zaslow conjecture, we study Calabi–Yau varieties with semi-stable fibre structures. We use Hodge theory to study the higher direct images of wedge products of relative cotangent sheaves of certain semi-stable families over higher dimensional quasi-projective bases, and obtain some results on positivity. We then apply these results to study non-isotrivial Calabi–Yau varieties fibred by semi-stable Abelian varieties (or hyperkähler varieties).     

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.     

Calabi–Yau coverings over some singular varieties and new Calabi–Yau 3-folds with Picard number one

This note is a report on the observation that some singular varieties admit Calabi–Yau coverings. As an application, we construct 18 new Calabi–Yau 3-folds with Picard number one that have some interesting properties.     

Moduli of Polarized Calabi–Yau Pairs

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

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.     

Calabi–Yau threefolds with small Hodge numbers associated with a one-parameter family of polynomials

We construct several quintic Calabi–Yau threefolds over the rationals with small Hodge numbers, by using certain members of a family of polynomial solutions of a second order linear partial differential equation.     

A gradient estimate in the Calabi–Yau theorem

We prove a C ¹-estimate for the complex Monge–Ampère equation on a compact Kähler manifold directly from the C ⁰-estimate, without using a C ²-estimate. This was earlier done only under additional assumption of non-negative bisectional curvature.     

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.     

On the unipotence of autoequivalences of toric complete intersection Calabi–Yau categories

Consider the derived category of coherent sheaves, D ^b (X), on a compact Calabi–Yau complete intersection X in a toric variety. The scope of this work is to establish the (quasi-)unipotence of a class of elements in the group of autoequivalences, Aut(D ^b (X)). This is achieved by associating singularity categories, modelled by matrix factorizations, to the toric data. Each of these triangulated categories is equivalent to the derived category of coherent sheaves on X. The idea is then that, although the singularity categories share the group of autoequivalences, on each category there are elements in Aut(D ^b (X)), whose (quasi-)unipotence relations are easier to see than on the other categories.     

Calabi–Yau Manifolds with Affine Structures

Mathematical Notes -     

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.