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 Fano manifolds of Picard number one

K{\"u}chle classified the Fano fourfolds that can be obtained as zero loci of global sections of homogeneous vector bundles on Grassmannians. Surprisingly, his classification exhibits two families of fourfolds with the same discrete invariants. Kuznetsov asked whether these two types of fourfolds are deformation equivalent. We show that the answer is positive in a very strong sense, since the two families are in fact the same! This phenomenon happens in higher dimension as well

Configurations of lines and models of Lie algebras

The automorphism groups of the 27 lines on the smooth cubic surface or the 28 bitangents to the general quartic plane curve are well-known to be closely related to the Weyl groups of $E\_6$ and $E\_7$. We show how classical subconfigurations of lines, such as double-sixes, triple systems or Steiner sets, are easily constructed from certain models of the exceptional Lie algebras. For ${\mathfrak e}\_7$ and ${\mathfrak e}\_8$ we are lead to beautiful models graded over the octonions, which display these algebras as plane projective geometries of subalgebras. We also interpret the group of the bitangents as a group of transformations of the triangles in the Fano plane, and show how this allows to realize the isomorphism $PSL(3,F\_2)\simeq PSL(2,F\_7)$ in terms of harmonic cubes.Comment: 31 page

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 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