Lemma 11.26.223EB10AE428C

Proof. The trace of $\alpha _{h,v}$ with respect to $\omega $ is $h(K_{\omega ,h}v,v)\geq 0$. A nonzero Hermitian form with nonnegative trace has a positive eigenvalue, since if all its eigenvalues were nonpositive, their nonnegative sum would force all of them to vanish. This gives the required tangent direction. Conversely, a tangent vector with positive evaluation shows that $\alpha _{h,v}\ne 0$. $\square$