Let $E$ be a hermitian holomorphic vector bundle over a complex manifold $X$. Then $\Theta(E)$, the curvature of $E$, is a section of $\bigwedge^{1,1}X\otimes\operatorname{End}(E)$. However, we have the following isomorphism: $$\bigwedge^{1,1}X\otimes\operatorname{End}(E) \cong \left(T^{1,0}X\otimes E\otimes\overline{T^{1,0}X\otimes E}\right)^*.$$ Under this isomorphism, $\Theta(E)$ corresponds to a hermitian form $\theta_E$ on $T^{1,0}X\otimes E$. This is used to define Griffiths and Nakano positivity of $E$.

Now suppose $E$ and $F$ are two hermitian holomorphic vector bundles over $X$. The curvature satisfies $\Theta(E\otimes F) = \Theta(E)\otimes\operatorname{id}_F + \operatorname{id}_E\otimes\Theta(F)$.

Is there a similar decomposition for $\theta_{E\otimes F}$ under the above isomorphism? If not, what about the case where $F$ is a line bundle?

I've been playing around with tensor products for a while now but I can't seem to get it straight.

Let $E$ be a hermitian holomorphic vector bundle over a complex manifold $X$. Then $\Theta(E)$, the curvature of $E$, is a section of $\bigwedge^{1,1}X\otimes\operatorname{End}(E)$. However, we have the following isomorphism: $$\bigwedge\nolimits^{\!1,1}X\otimes\operatorname{End}(E) \cong \left(T^{1,0}X\otimes E\otimes\overline{T^{1,0}X\otimes E}\right)^*.$$ Under this isomorphism, $\Theta(E)$ corresponds to a hermitian form $\theta_E$ on $T^{1,0}X\otimes E$. This is used to define Griffiths and Nakano positivity of $E$.

Now suppose $E$ and $F$ are two hermitian holomorphic vector bundles over $X$. The curvature satisfies $\Theta(E\otimes F) = \Theta(E)\otimes\operatorname{id}_F + \operatorname{id}_E\otimes\Theta(F)$.

Is there a similar decomposition for $\theta_{E\otimes F}$ under the above isomorphism? If not, what about the case where $F$ is a line bundle?

I've been playing around with tensor products for a while now but I can't seem to get it straight.