Some Finiteness Results for Calabi-Yau Threefolds

The moduli theory of Calabi–Yau threefolds is investigated,and using Griffiths' work on the period map, some finitenessresults are derived. In particular, a prediction of Morrison'scone conjecture is confirmed.     

Towards the classification of weak Fano threefolds with ρ = 2

In this paper, examples of type II Sarkisov links between smooth complex projective Fano threefolds with Picard number one are provided. To show examples of these links, we study smooth weak Fano threefolds X with Picard number two and with a divisorial extremal ray. We assume that the pluri-anticanonical morphism of X contracts only a finite number of curves. The numerical classification of these particular smooth weak Fano threefolds is completed and the geometric existence of some numerical cases is proven.     

Homological algebra in pre-Abelian categories

We construct derived functors in additive categories in which each morphism has a kernel, co-kernel, image, and coimage, but the image and coimage are not necessarily isomorphic. We prove that these derived functors possess the usual properties. The main difficulty is that the 3×3-lemma does not necessarily hold in the categories under consideration.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 94, pp. 131–141, 1979.     

Deformation-obstruction theory for complexes via Atiyah and Kodaira–Spencer classes

We give a universal approach to the deformation-obstruction theory of objects of the derived category of coherent sheaves over a smooth projective family. We recover and generalise the obstruction class of Lowen and Lieblich, and prove that it is a product of Atiyah and Kodaira–Spencer classes. This allows us to obtain deformation-invariant virtual cycles on moduli spaces of objects of the derived category on threefolds.     

Linear Systems on Fano Threefolds. I

Abstract

In this note, we classify all the polarized Fano threefold (X, H) with Bs|H|¬ = ?. As corollaries we obtained that (1) the very ample part of the conjecture of Fujita holds for smooth Fano threefolds and (2) global Seshadri constants of ample divisors on Fano threefolds are bounded from below by 1 except three types of polarized Fano threefolds.     

Examples of Calabi–Yau Threefolds as Covers of Almost-Fano Threefolds

We give a method for producing examples of Calabi–Yau threefolds as covers of degree d ≤ 8 of almost-Fano threefolds, computing explicitely their Euler– Poincaré characteristic. Such a method generalizes the well-knownclassical construction of Calabi–Yau threefolds as double covers of the projective space branched along octic surfaces.     

A categorical invariant for cubic threefolds

We prove a categorical version of the Torelli theorem for cubic threefolds. More precisely, we show that the non-trivial part of a semi-orthogonal decomposition of the derived category of a cubic threefold characterizes its isomorphism class.     

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.     

Calabi-Yau threefolds arising from fiber products of rational quasi-elliptic surfaces, II

We construct examples of supersingular Calabi-Yau threefolds in characteristic 2 making use of the method by Schoen. Unirational Calabi-Yau threefolds of five different topological types are obtained. There are two examples with the third Betti number zero among them, and they are counted as other examples of non-liftable Calabi-Yau threefolds in characteristic 2 after the one given by Schröer.     

Projective threefolds on which acts with 2-dimensional general orbits

The birational geometry of projective threefolds on which acts with 2-dimensional general orbits is studied from the viewpoint of the minimal model theory of projective threefolds. These threefolds are closely related to the minimal rational threefolds classified by Enriques, Fano and Umemura. The main results are (i) the -birational classification of such threefolds and (ii) the classification of relatively minimal models in the fixed point free cases.

     

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).     

Degenerations of cubic threefolds and matroids

We present a surprising connection between cubic threefolds and the well-known regular matroid by making use of intermediate Jacobians of cubic threefolds realized as Prym varieties. As a corollary we obtain a new proof of the nonrationality of generic cubic threefolds.

     

Hilbert Modular Threefolds of Arithmetic Genus One

In 1981, Weisser proved that there are exactly four Galois cubic number fields with Hilbert modular threefolds of arithmetic genus one. In this paper, we extend Weisser's work to cover all cubic number fields. Our main result is that there are exactly 33 fields with Hilbert modular threefolds of arithmetic genus one. These fields are enumerated explicitly.     

On an example of Aspinwall and Morrison

In this paper, a family of smooth multiply-connected Calabi-Yau threefolds is investigated. The family presents a counterexample to global Torelli as conjectured by Aspinwall and Morrison.

     

Ruled threefolds homeomorphic to ℙ2 × ℙ1 and their divisors

We study a class of ruled threefolds, namely, manifolds X with a projection p:X→ℙ², such that each fiber is isomorphic to ℙ¹, and which are homeomorphic to ℙ²×ℙ¹; and we characterize ample and very ample line bundles on such threefolds. This paper was written with the financial support of M.P.I. The author is a member of G.N.S.A.G.A. of the C.N.R.     

Moduli of reflexive sheaves on smooth projective 3-folds

We compute the expected dimension of the moduli space of torsion-free rank 2 sheaves at a point corresponding to a stable reflexive sheaf, and give conditions for the existence of a perfect tangent-obstruction complex on a class of smooth projective threefolds; this class includes Fano and Calabi-Yau threefolds. We also explore both local and global relationships between moduli spaces of reflexive rank 2 sheaves and the Hilbert scheme of curves.     

A lower bound on Seshadri constants of hyperplane bundles on threefolds

We give the lower bound on Seshadri constants for the case of very ample line bundles on threefolds. We consider the situation when the Seshadri constant is strictly less than 2 and give a version of Bauer’s theorem (Math Ann 313(3):547–583, 1999, Theorem 2.1) for singular surfaces so we can prove the same result for smooth threefolds.     

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.