Let $X$ be a complex manifold, and let $E \to X$ be a holomorphic Hermitian vector bundle of positive rank $r \in \mathbb{N}$. Let $\nabla^E$ be the Chern connection on $E$, and let $\Theta_E \in A^{1,1}(X; \operatorname{End} E)$ denote its curvature. Equip the determinant line bundle $\det E = \Lambda^r E$ with the Hermitian metric induced from $E$, and let $\nabla^{\det E}$ be its Chern connection with curvature $\Theta_{\det E} \in A^{1,1}(X; \operatorname{End}(\det E))$. Let $\operatorname{tr}: A^{1,1}(X; \operatorname{End} E) \to A^{1,1}(X)$ be the scalar trace map obtained by taking the fiberwise trace on $\operatorname{End} E$, and identify $A^{1,1}(X; \operatorname{End}(\det E))$ with $A^{1,1}(X)$ by letting an endomorphism of the line $\det E_x$ act as multiplication by a scalar. Then

\begin{align*} \Theta_{\det E} = \operatorname{tr}(\Theta_E). \end{align*}