Trace of Chern curvature ($\mathrm{Tr}{F}_h=\bar{\partial }\partial \log \det h$)

In a holomorphic local frame, a Hermitian metric on a holomorphic vector bundle has matrix $h$. The trace of its Chern connection curvature is $\bar{\partial }\partial \log \det h$. A frame change multiplies $\det h$ by the squared modulus of a holomorphic unit, whose logarithm has vanishing mixed derivative. For two metrics, the quotient of their determinants is a global positive function, and the trace curvatures differ by the exterior derivative of its logarithmic $\partial$-derivative.