Let $X$ be a complex manifold, let $(E,h_E)$ be a Hermitian holomorphic vector bundle on $X$, and let $q:E\to Q$ be a holomorphic quotient vector bundle. Let $S=\ker q$, and equip $Q$ with the quotient Hermitian metric $h_Q$ determined by the $h_E$-orthogonal smooth splitting $E=S\oplus S^\perp$. Use the convention that Chern curvature is tested as the Hermitian form $h(i\Theta(\xi,\bar\xi)\cdot,\cdot)$, and that for the second fundamental form $B\in A^{1,0}(X,\operatorname{Hom}(S,S^\perp))$ its adjoint contribution satisfies

\begin{align*} h_E\bigl(i(B\wedge B^*)(\xi,\bar\xi)v,v\bigr)=\|B^*_{\bar\xi}v\|_{h_E}^2 \end{align*}

for $x\in X$, $\xi\in T_x^{1,0}X$, and $v\in S_x^\perp$. If $(E,h_E)$ is Griffiths semipositive, then $(Q,h_Q)$ is Griffiths semipositive.

More precisely, identify $Q$ smoothly and isometrically with $S^\perp\subset E$ by the restriction $q|_{S^\perp}:S^\perp\to Q$. Let $B\in A^{1,0}(X,\operatorname{Hom}(S,S^\perp))$ be the second fundamental form of the holomorphic subbundle $S\subset E$, and let $B^*\in A^{0,1}(X,\operatorname{Hom}(S^\perp,S))$ be its Hermitian adjoint, so that $B^*_{\bar\xi}:S_x^\perp\to S_x$ is the adjoint of $B_\xi:S_x\to S_x^\perp$. For every $x\in X$, every $\xi\in T_x^{1,0}X$, and every $u\in Q_x$, let $\tilde u\in S_x^\perp\subset E_x$ be the unique lift satisfying $q_x(\tilde u)=u$, and set

\begin{align*} A_\xi u:=B^*_{\bar\xi}\tilde u\in S_x. \end{align*}

Then the Chern curvatures satisfy

\begin{align*} h_Q\bigl(i\Theta(Q,h_Q)(\xi,\bar\xi)u,u\bigr)=h_E\bigl(i\Theta(E,h_E)(\xi,\bar\xi)\tilde u,\tilde u\bigr)+\|A_\xi u\|_{h_E}^2. \end{align*}

In particular, if $(E,h_E)$ is Griffiths positive, $\xi\ne0$, and $u\ne0$, then the left-hand side is positive.