Jarosław A. Wiśniewski

Projective contact manifolds

Abstract.The present work is concerned with the study of complex projective manifolds X which carry a complex contact structure. In the first part of the paper we show that if KX is not nef, then either X is Fano and b2(X)=1, or X is of the form ℙ(TY), where Y is a projective manifold. In the second part of the paper we consider contact manifolds where KX is nef.

Small contractions of symplectic 4-folds

We classify small contractions of (holomorphically) symplectic 4-folds. AMS MSC: 14E30, 14J35.

Variétés complexes dont l'éclatée en un point est de Fano

We classify complex projective manifolds X for which there exists a point a such that the blow-up of X at a is Fano. As a consequence, we get that, in dimension greater or equal than three, the quadric is the only complex manifold X for which there exists two distinct points a and b such that the blow-up of X with center {a,b} is Fano. To cite this article: L. Bonavero et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 463–468.

On Fano manifolds, which are P^K-bundles over P^2

In our earlier paper [8] we discussed Fano manifolds X that are of the form X = ℙ( ) with a rank-2 vector bundle on a surface S . Here we study a more general situation of Fano manifolds, ruled over the complex projective plane P 2 as P r-1 -bundles, i.e., being of the form ℙ( ) with -a bundle of rank r ≥ 3 on P 2 .

A survey on the Campana-Peternell Conjecture

In 1991 Campana and Peternell proposed, as a natural algebro-geometric extension of Moris characterization of the projective space, the problem of classifying the complex projective Fano manifolds whose tangent bundle is nef, conjecturing that the only varieties satisfying these properties are rational homogeneous. In this paper we review some background material related to this problem, with special attention to the partial results recently obtained by the authors.

4-dimensional symplectic contractions

Local symplectic contractions are resolutions of singularities which admit symplectic forms. Four dimensional symplectic contractions are (relative) Mori Dream Spaces. In particular, any two such resolutions of a given singularity are connected by a sequence of Mukai flops. We discuss the cone of movable divisors on such a resolution; its faces are determined by curves whose loci are divisors, we call them essential curves. The movable cone is divided into nef chambers which are related to different resolutions; this subdivision is determined by classes of 1-cycles. We also study schemes parametrizing minimal essential curves and show that they are resolutions, possibly non-minimal, of surface Du Val singularities. Some examples, with an exhaustive description, are provided.

On

We provide a construction of 81 symplectic resolutions of a 4-dimensional quotient singularity obtained by an action of a group of order 32. The existence of such resolutions is known by a result of Bellamy and Schedler. Our explicit construction is obtained via GIT quotient of the spectrum of a ring graded in the Picard group generated by the divisors associated to the conjugacy classes of symplectic reflections of the group in question. As the result we infer the geometric structure of these resolutions and their flops. Moreover, we represent the group in question as a group of automorphisms of an abelian 4-fold so that the resulting quotient has singularities with symplectic resolutions. This yields a new Kummer-type symplectic 4-fold.

81

A projective manifold X is called Fano if its anticanonical divisor − K x is ample. Fano manifolds form a very distinguished class: in each dimension there is only a finite number of deformation classes of them and they are classified in dimension ≤ 3, the case dim X = 3 due to Fano, Roth, Iskovskih and Shokurov. In dimension ≥ 4 not much is known about Fano manifolds in general. However, due to results of Mori, Kawamata and Shokurov, Fano manifolds with Picard number ρ(X) bigger than 1 admit special morphisms, called Fano-Mori contractions, which can be used to study the structure of such Fano’s. The case ρ(X) = 1 seems to be harder to approach, see [IP] for an overview on Fano varieties.

symplectic resolutions of a

Internalins comprise a class of Listeria monocytogenes proteins responsible for activation of signalling pathways leading to phagocytic uptake of the bacterium by the host cell. In this paper, a possible role of Lmo0171—a new member of the internalin family was investigated. Disruption of the lmo0171 gene resulted in important cell morphology alterations along with a decrease in the ability to invade three eukaryotic cell lines, that is Int407, Hep-2 and HeLa and diminished adhesion efficiency to int407, thereby suggesting bifunctionality of the newly characterised Lmo0171 internalin.