PomeCorollary 11.18.4.18D33491ABFD If $K_{\omega ,h}$ is positive definite for a Hermitian form $\omega $ on $X$, then $U_ E$ is $(n+r-1)$-convex. This holds, in particular, when $\omega $ is Gauduchon, $E$ is $\omega $-slope-stable, and $\deg _\omega E>0$.
Corollary 11.18.418D33491ABFD
Proof. For $v\ne 0$, the trace of $\alpha _{h,v}$ with respect to $\omega $ is $h(K_{\omega ,h}v,v)>0$, and a Hermitian form with positive trace has a positive eigenvalue. Apply theorem 11.18.2 with $s=1$.
Under the Gauduchon and stability hypotheses, the classical existence theorem for Hermitian–Einstein metrics on stable bundles gives a metric $h$ with $K_{\omega ,h}=\lambda \operatorname {Id}_ E$, where $\lambda $ is a constant. Taking the trace and integrating the contraction identity gives
\[ r\lambda \int _ X\omega ^ n =n\int _ X\operatorname {tr}(\mathrm iF_ h)\wedge \omega ^{n-1} =2\pi n\deg _\omega E>0. \]
Here $\operatorname {tr}F_ h=\bar\partial \partial \log \det M$, so $\mathrm i\operatorname {tr}F_ h/(2\pi )$ represents $c_1^{\mathrm{BC}}(E)$. Thus $\lambda >0$, and the first assertion applies. $\square$
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