= Solution
A <holomorphic line bundle> is a complex line bundle with holomorphic transition functions, and a <holomorphic section> is one whose coefficient in every holomorphic local frame is holomorphic. Given a <Hermitian metric on a holomorphic vector bundle> $h$, its <Chern connection> is the connection $\nabla$ satisfying
$$
\nabla^{0,1}=\bar\partial_L,
\qquad
d\,h(s,t)=h(\nabla s,t)+h(s,\nabla t).
$$
Let $e$ be a nonvanishing <holomorphic local frame>, put $H=h(e,e)>0$, and write $\nabla e=Ae$. The first condition forces $A^{0,1}=0$, while metric compatibility forces
$$
A=H^{-1}\partial H=\partial\log H.
$$
This determines $\nabla$ uniquely and also constructs it. If $e'=ge$ for a nowhere-zero holomorphic function $g$, then $A'=A+g^{-1}\partial g$, exactly the <connection one-form> transformation law, so the local constructions glue.
For a line bundle, $A\wedge A=0$, and the <curvature form of a connection> is
$$
F(\nabla)=dA=\bar\partial\partial\log H=-\partial\bar\partial\log H.
$$
It has type $(1,1)$. Under $e'=ge$, the extra term $g^{-1}\partial g$ is closed, so the curvature is unchanged and therefore global. This is the <local formula for the Chern connection on a line bundle>.
Any other Hermitian metric has the form $\widehat h=e^u h$ for a global smooth real function $u$. Its local squared norm is $\widehat H=e^uH$, whence
$$
F(\widehat\nabla)-F(\nabla)=\bar\partial\partial u.
$$
Connections $\nabla_1$ and $\nabla_2$ induce the <tensor product connection>
$$
(\nabla_1\otimes\nabla_2)(s_1\otimes s_2)
=\nabla_1s_1\otimes s_2+s_1\otimes\nabla_2s_2.
$$
Its connection form in a product frame is $A_1+A_2$, so the <curvature of a tensor product connection> is $F_1+F_2$. For Chern connections, equip $L_1\otimes L_2$ with the product metric
$$
h(s_1\otimes s_2,t_1\otimes t_2)=h_1(s_1,t_1)h_2(s_2,t_2).
$$
The tensor product connection has the correct $(0,1)$ part and preserves this metric, so uniqueness identifies it with the Chern connection of $h$.
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