complex geometry - How can I see the projection $pi:H^1(X,mathcal{O}_X)rightarrow Pic^{0}(X)$ in terms of holomorphic structures on $Xtimesmathbb{C}$?

Hi, as the title says I'm looking for a way to see the projection $pi:H^1(X,mathcal{O}_X)rightarrow operatorname{Pic}^{0}(X)$ in terms of holomorphic structures on $Xtimesmathbb{C}$. ($X$ is a compact complex manifold and $operatorname{Pic}^0(X)$ is the kernel of $c_1:H^1(X,mathcal{O}_X^{ * })rightarrow H^2(X,mathbb{Z})$ in the long exact sequence induced by the exponential sequence). Since
$$H^1(X,mathcal{O}_X) simeq H_{overline{partial}}^{0,1}$$
by the Dolbeault isomorphism, I take
$[c]in H^1(X,mathcal{O}_X)$,
so I take the corresponding $[gamma]in H_{overline{partial}}^{0,1}$ and a representative $gamma$ of the class $[gamma]$. My wrong thought was that $pi([c])=(Xtimesmathbb{C},overline{partial}+gamma)$, i.e. to associate to $[c]$ the trivial bundle with the "holomorphic structure" $overline{partial}+gamma$, but it is not a holomorphic structure unless $gammawedgegamma=0$! So how can I see explicitly (if it is possible) the map $pi$ in terms of holomorphic structures on the trivial line bundle?

Thank you in advance.