Jaroslaw Buczynski

Title: Webs of Legendrian projective spaces

Abstract:
We show that there are no nontrivial webs of Legendrian projective spaces on projective 
contact manifolds. This answers positively a question of S. Kebekus and J.-M. Hwang. 
As a consequence, if the variety of minimal rational tangents (VMRT) of a contact manifold 
is linear (i.e. it is a union of linear subspaces), then the contact manifold must be either a 
projective space or a projectivisation of a tangent bundle. As another consequence, there is 
no degree 1 embedding of an n-dimensional projective space into a 2n+1 dimensional 
contact Fano manifold with second Betti number 1. An analogous result for webs of 
Lagrangian tori in hyperkähler manifolds was conjectured by A. Beauville and shown by 
J.-M. Hwang, R. Weiss, D. Greb, Ch. Lehn, S. Rollenske, D. Matsushita.