Local moduli of holomorphic bundles

We study moduli of holomorphic vector bundles on non-compact varieties. We discuss filtrability and algebraicity of bundles and calculate dimensions of local moduli. As particularly interesting examples, we describe numerical invariants of bundles on some local Calabi-Yau threefolds.     

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.     

Generalised Kummer constructions and Weil restrictions

We use a generalised Kummer construction to realise all but one known weight four newforms with complex multiplication and rational Fourier coefficients in Calabi-Yau threefolds defined over Q. The Calabi-Yau manifolds are smooth models of quotients of the Weil restrictions of elliptic curves with CM of class number three.     

Small resolutions and non-liftable Calabi-Yau threefolds

We use properties of small resolutions of the ordinary double point in dimension three to construct smooth non-liftable Calabi-Yau threefolds. In particular, we construct a smooth projective Calabi-Yau threefold over \mathbbF₃{\mathbb{F}_3} that does not lift to characteristic zero and a smooth projective Calabi-Yau threefold over \mathbbF₅{\mathbb{F}_5} having an obstructed deformation. We also construct many examples of smooth Calabi-Yau algebraic spaces over \mathbbF_p{\mathbb{F}_p} that do not lift to algebraic spaces in characteristic zero.     

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.     

Note on a Conjecture of Gopakumar-Vafa

We rephrase the Gopakumar-Vafa conjecture on genus zero Gromov-Witten invariants of Calabi-Yau threefolds in terms of the virtual degree of the moduli of pure dimension one stable sheaves and investigate the conjecture for K3 fibred local Calabi-Yau threefolds.     

On the existence of almost Fano threefolds with del Pezzo fibrations

By Jahnke–Peternell–Radloff and Takeuchi, almost Fano threefolds with del Pezzo fibrations were classified. Among them, there exist 10 classes such that the existence of members of these was not proved. In this paper, we construct such examples belonging to each of 10 classes.     

New families of Calabi-Yau threefolds without maximal unipotent monodromy

The aim of this paper is to construct families of Calabi-Yau threefolds without boundary points with maximal unipotent monodromy and to describe the variation of their Hodge structures. In particular five families are constructed. In all these cases the variation of the Hodge structures of the Calabi-Yau threefolds is basically the variation of the Hodge structures of a family of curves. This allows us to write explicitly the Picard-Fuchs equation for the one-dimensional families. These Calabi-Yau threefolds are desingularizations of quotients of the product of a (fixed) elliptic curve and a K3 surface admitting an automorphisms of order 4 (with some particular properties). We show that these K3 surfaces admit an isotrivial elliptic fibration.     

Stability and Fourier–Mukai transforms on elliptic fibrations

We systematically develop Bridgeland's [7] and Bridgeland–Maciocia's [10] techniques for studying elliptic fibrations, and identify criteria that ensure 2-term complexes are mapped to torsion-free sheaves under a Fourier–Mukai transform. As an application, we construct an open immersion from a moduli of stable complexes to a moduli of Gieseker stable sheaves on elliptic threefolds. As another application, we give various 1–1 correspondences between fibrewise semistable torsion-free sheaves and codimension-1 sheaves on Weierstrass surfaces.     

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.     

On some moduli spaces of stable vector bundles on cubic and quartic threefolds

We study certain moduli spaces of stable vector bundles of rank 2 on cubic and quartic threefolds. In many cases under consideration, it turns out that the moduli space is complete and irreducible and a general member has vanishing intermediate cohomology. In one case, all except one component of the moduli space has such vector bundles.     

Prime Fano threefolds and integrable systems

For a general K3 surface S of genus g, with 2 ≤ g ≤ 10, we prove that the intermediate Jacobians of the family of prime Fano threefolds of genus g containing S as a hyperplane section, form generically an algebraic completely integrable Hamiltonian system. The first author is partially supported by grant MI1503/2005 of the Bulgarian Foundation for Scientific Research.     

On the quasi-projectivity of compactifiable strongly pseudoconvex manifolds

We construct new examples of non-Kahlerian 1-convex threefolds X with exceptional set≅P₁ (resp. ≅F₂). Also the structure of Pic(X) will be studied. On the other hand, we shall investigate the quasi-projective structure of certain Kahlerian compactifiable 1-convex manifolds; particular attention will be given to 3-fold cases through concrete examples.     

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.

     

Non-annulation effective et positivité locale des fibrés en droites amples adjoints

We show how to use effective non-vanishing to prove that Seshadri constants of some ample divisors are bigger than 1 on smooth threefolds whose anticanonical bundle is nef or on Fano varieties of small coindice. We prove the effective non-vanishing conjecture of Ionescu–Kawamata in dimension 3 in the case of line bundles of “high” volume.     

Some remarks on varieties with projectively isomorphic hyperplane sections

We study here the projective varieties with the property that there exists a projective isomorphism between two of their generic hyperplane sections. The case of surfaces had already been studied by Fubini and Fano in the 1920s. The latter gave the list of all (possibly signular) surfaces with projectively isomorphic hyperplane sections. The proof, however, was essentially wrong. By means of a different approach, we are able to supply a proof of Fano's claims. Moreover, we show some general properties of varieties with projectively isomorphic hyperplane sections: they have uniruled hyperplane sections and are related to varieties with small dual varieties. In particular we are able to conclude that threefolds with projectively isomorphic hyperplane sections either have rational sections or are P²-bundles over a curve.A member of GNSAGA of CNR.     

Remarks on BCOV invariants and degenerations of Calabi-Yau manifolds

For a one parameter family of Calabi-Yau threefolds, Green et al.(2009) expressed the total singularities in terms of the degrees of Hodge bundles and Euler number of the general fiber. In this paper,we show that the total singularities can be expressed by the sum of asymptotic values of BCOV(BershadskyCecotti-Ooguri-Vafa) invariants, studied by Fang et al.(2008). On the other hand, by using Yau's Schwarz lemma, we prove Arakelov type inequalities and Euler number bound for Calabi-Yau family over a compact Riemann surface.     

Modularity of a Certain Calabi-Yau Threefold

 The Langlands program predicts that certain Calabi-Yau threefolds are modular in the sense that their L-series correspond to the Mellin transforms of weight 4 newforms. Here we prove that the L-function of the threefold given by is , the unique normalized eigenform in .     

Modularity of a Certain Calabi-Yau Threefold

 The Langlands program predicts that certain Calabi-Yau threefolds are modular in the sense that their L-series correspond to the Mellin transforms of weight 4 newforms. Here we prove that the L-function of the threefold given by is , the unique normalized eigenform in . (Received 21 May 1999; in revised form 27 July 1999)     

Borcea–Voisin Calabi–Yau threefolds and invertible potentials

We prove that the Borcea–Voisin mirror pairs of Calabi–Yau threefolds admit projective birational models that satisfy the Berglund–Hübsch–Chiodo–Ruan transposition rule. This shows that the two mirror constructions provide the same mirror pairs, as soon as both can be defined.