  In this paper, we consider the Schr\"odinger equation with a
mass-supercritical focusing nonlinearity, in the exterior of a smooth, compact,
convex obstacle of $\R^{d}$ with Dirichlet boundary conditions. We prove that
solutions with negative energy blow up in finite time. Assuming furthermore
that the nonlinearity is energy-subcritical, we also prove (under additional
symmetry conditions) blow-up with the same optimal ground-state criterion than
in the work of Holmer and Roudenko on $\R^{d}$. The classical proof of Glassey,
based on the concavity of the variance, fails in the exterior of an obstacle
because of the appearance of boundary terms with an unfavorable sign in the
second derivative of the variance. The main idea of our proof is to introduce a
new modified variance which is bounded from below and strictly concave for the
solutions that we consider.
