Vanishing theorems for ample vector bundles

We prove a general vanishing theorem for the cohomology of products of symmetric and skew-symmetric powers of an ample vector bundle on a smooth complex projective variety. Special cases include an extension of classical theorems of Griffiths and Le Potier to the whole Dolbeault cohomology, and an answer to a problem raised by Demailly. An application to degeneracy loci is given.Comment: 12 pages, LaTeX2

On the variety of four dimensional lie algebras

Lie algebras of dimension $n$ are defined by their structure constants , which can be seen as sets of $N = n^2 (n -- 1)/2$ scalars (if we take into account the skew-symmetry condition) to which the Jacobi identity imposes certain quadratic conditions. Up to rescaling, we can consider such a set as a point in the projective space $P^{N--1}$. Suppose $n =4$, hence $N = 24$. Take a random subspace of dimension $12$ in $P^{23}$ , over the complex numbers. We prove that this subspace will contain exactly $1033$ points giving the structure constants of some four dimensional Lie algebras. Among those, $660$ will be isomorphic to $gl\_2$ , $195$ will be the sum of two copies of the Lie algebra of one dimensional affine transformations, $121$ will have an abelian, three-dimensional derived algebra, and $57$ will have for derived algebra the three dimensional Heisenberg algebra. This answers a question of Kirillov and Neretin.Comment: To appear in Journal of Lie Theor

On linear spaces of skew-symmetric matrices of constant rank

We describe the space of projective planes of complex skew-symmetric matrices of order six and constant rank four. We prove that it has four connected components, all of dimension 26 and homogeneous under the action of PGL_6.Comment: 12 page