J. Wahl characterized projective spaces in characteristic zero by cohomological condition of tangent bundles; in addition, he remarked that a counter-example in characteristic two is constructed from odd-dimensional hyperquadrics $Q_{2n-1}$ ($n > 1$).

This is caused by existence of a common point which every embedded tangent space to the quadric contains.

In general, a projective variety in $P^N$ is said to be strange if it admits such a common point.

A non-linear smooth projective curve is strange if and only if it is a conic in characteristic two (E. Lluis, P. Samuel).

S. Kleiman and R. Piene showed that a non-linear smooth hypersurface in $P^N$ is strange if and only if it is a quadric of odd-dimension in characteristic two.

In this talk, we investigate complete intersection varieties, and prove that, a non-linear smooth complete intersection variety in $P^N$ is strange if and only if it is a quadric in $P^N$ of odd dimension in characteristic two; these conditions are also equivalent to non-vanishing of $0$-cohomology of $(-1)$-twist of the tangent bundle. (The details of our results are stated in arXiv:1304.1634v1.)

We will present a new approach to Barth-Larsen theorem about the extendability of algebraic cycles of projective subvarieties of small codimension. We will use merely geometric tools, which will allow to generalize the result to other ambient spaces. We will also discuss the relation of this theory with the famous Hartshorne's conjecture.

A Fano manifold $X$ with nef tangent bundle is of flag-type if it has the same type of elementary contractions as a complete flag manifold. In this talk, we present a method to associate a Dynkin diagram $\mathcal{D}(X)$ with any such $X$, based on the numerical properties of its contractions. We then show that $\mathcal{D}(X)$ is the Dynkin diagram of a semisimple Lie group. Furthermore, we prove that a flag-type manifold of classical type is a complete flag manifold.

This is joint work with R. Mu .A Nqoz, G. Occhetta, L. E. Sol Na Conde and J. Wisniewski.

Several results on the structures of embedded manifolds swept out by high dimensional linear spaces, quadric hypersurfaces, or cubic hypersurfaces, have been provided. In this talk, I will discuss the structures of manifolds swept out by high dimensional hypersurfaces with arbitrary degree on the assumption that the famous Hartshorne conjecture about complete intersections is true.