Publications Mathématiques de l'Institut des Hautes Études Scientifiques

If X is a smooth affine variety of dimension d over an algebraically closed field k, and if (d−1)!∈k × then any stably trivial vector bundle of rank (d−1) over X is trivial. The hypothesis that X is smooth can be weakened to X is normal if d≥4.

If $U:[0,+\infty [\times M$ is a uniformly continuous viscosity solution of the evolution Hamilton-Jacobi equation $$ \partial _{t}U+ H(x,\partial _{x}U)=0, $$ where $M$ is a not necessarily compact manifold, and $H$ is a Tonelli Hamiltonian, we prove the set $\Sigma (U)$ , of points in $]0,+\infty [\times M$ where $U$ is not differentiable, is locally contractible. Moreover, we study the homotopy type of $\Sigma (U)$ . We also give an application to the singularities of the distance function to a closed subset of a complete Riemannian manifold.