Second fundamental form of a holomorphic subbundle ($A\in {\mathcal{A}}^{1,0}(\mathrm{Hom}(S,Q))$)

For a holomorphic subbundle $S\subset E$ and quotient $Q=E/S$, the second fundamental form is $A(s)=(I-{\pi }_S)D_{E}s$. The Leibniz rule gives $A(fs)=fA(s)$, and holomorphicity of the subbundle makes its $(0,1)$ part vanish. Thus $A\in {\mathcal{A}}^{1,0}(\mathrm{Hom}(S,Q))$: the form factor is a cotangent bundle, not a tangent bundle.

In an adapted smooth unitary frame with input-index connection convention, the Chern connection matrix has blocks $({\theta }_F,B;-{\overline{B}}^t,{\theta }_Q)$, where the second fundamental form of a holomorphic subbundle $B$ has type $(1,0)$. The Cartan curvature equation with input indices gives the displayed formula. If the ambient curvature is zero, ${\Theta }_F(X,\overline{X})=-B(X)B(X{)}^{†}$ is negative semidefinite. Dualizing the determinant line bundle therefore gives semipositive curvature.