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Although intersection homology lacks a ring structure, certain expressions (called uniform) in the intersection homology of an irreducible projective variety $X$ always give the same value, when computed via the decomposition theorem on any…

We prove that smooth projective varieties with equivalent derived categories have isogenous (and sometimes isomorphic) Picard varieties. In particular their irregularity and number of independent vector fields are the same. This is turn…

We show that smooth projective horospherical varieties with nef tangent bundles are rational homogeneous spaces.

Fix integers $a\geq 1$, $b$ and $c$. We prove that for certain projective varieties $V\subset{\bold P}^r$ (e.g. certain possibly singular complete intersections), there are only finitely many components of the Hilbert scheme parametrizing…

We investigate the structure of smooth holomorphic foliations with numerically flat tangent bundles on compact K\"ahler manifolds. Extending earlier results on non-uniruled projective manifolds by the second and fourth authors, we show that…

Let f: X -> Y be a smooth family of canonically polarized complex varieties over a smooth base. Generalizing the classical Shafarevich hyperbolicity conjecture, Viehweg conjectured that Y is necessarily of log general type if the family has…

In this paper, we show that if the tangent bundle of a smooth projective variety is strictly nef, then it is isomorphic to a projective space; if a projective variety $X^n$ $(n>4)$ has strictly nef $\Lambda^2 TX$, then it is isomorphic to…

We give a negative answer to a question posed by Severi in 1951, whether the Abelian Varieties are the only projective manifolds with trivial Chern classes. By Yau' s celebrated result, compact K\"ahler manifolds with trivial Chern classes…

We study homogeneous Lorentzian manifolds $M = G/L$ of a connected reductive Lie group $G$ modulo a connected reductive subgroup $L$, under the assumption that $M$ is (almost) $G$-effective and the isotropy representation is totally…

We describe varieties of minimal rational tangents on the wonderful symmetric varieties by marked Dynkin diagrams. An irreducible component of a variety of minimal rational tangents is a rational homogeneous space, and hence, we have a…

We study geometric properties of certain obstructed equisingular families of projective hypersurfaces with emphasis on smoothness, reducibility, being reduced, and having expected dimension. In the case of minimal obstructness, we give a…

We give an introduction to the theory of varieties of minimal rational tangents, emphasizing its aspect as a fusion of algebraic geometry and differential geometry, more specifically, a fusion of Mori geometry of minimal rational curves and…

We classify all smooth projective horospherical varieties of Picard group $\mathbb{Z}^2$ and we give a first description of their geometry via the Log Minimal Model Program.

We discuss algebraic vector bundles on smooth k-schemes X contractible from the standpoint of A^1-homotopy theory; when k = C, the smooth manifolds X(C) are contractible as topological spaces. The integral algebraic K-theory and integral…

Lasell and Ramachandran show that the existence of rational curves of positive self-intersection on a smooth projective surface $X$ implies that all the finite dimensional linear representations of the fundamental group $\pi_1(X)$ are…

Let $(M,\omega)$ be an almost symplectic manifold ($\omega$ is a non degenerate, not closed, 2-form). We say that a vector field $X$ of $M$ is locally Hamiltonian if $L_X\omega=0,d(i(X)\omega)=0$, and it is Hamiltonian if, furthermore, the…

In this paper, it is proved that a connected 3-dimensional Riemannian manifold or a closed connected semi-Riemannian manifold $M^n$($n>1$) admitting a projective vector field with a non-linearizable singularity is projectively flat.

We continue the study, begun by the second author in math.AG/0701889, of secant defective manifolds having "simple entry loci". We prove that such manifolds are rational and describe them in terms of tangential projections. Using also our…

We review the theory of quaternionic Kahler and hyperkahler structures. Then we consider the tangent bundle of a Riemannian manifold M with a metric connection D (with torsion) and with its well estabilished canonical complex structure.…

We describe the equivariant cohomology ring of rationally smooth projective embeddings of reductive groups. These embeddings are the projectivizations of reductive monoids. Our main result describes their equivariant cohomology in terms of…