= Trace of Chern curvature
{title2=$\operatorname{Tr}F_h=\bar\partial\partial\log\det h$}
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.
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