Define $Pic^0(X)$ ($X$ complete complex algebraic variety) as the kernel of the first Chern class $c_1:Pic(X)\to H^2(X,\mathbb Z)$. From the long exact sequence arising from the short exponential sequence, $$H^0(X,\mathbb Z)\to H^0(X,\mathcal O)\to H^0(X,\mathcal O^*)\to H^1(X,\mathbb Z)\to H^1(X,\mathcal O)\to H^1(X,\mathcal O^*)\to H^2(X,\mathbb Z)$$
we get that $Pic^0(X)$ is isomorphic to $H^1(X,\mathcal O)$ quotiented by the image of $H^1(X,\mathbb Z)$ in $H^1(X,\mathcal O)$. Griffiths-Harris writes $$Pic^0(X)\cong \frac{H^1(X,\mathcal O)}{H^1(X,\mathbb Z)}$$ so I understand that the map $H^1(X,\mathbb Z)\to H^1(X,\mathcal O)$ is injective. Why is this?
I have a simple reason for that, i.e. that a complex analysis theorem whose name I have forgotten assures us that $H^0(X,\mathcal O)\to H^0(X,\mathcal O^*)$ via the exponential map is a surjection. So the following map in the long exact sequence is $0$, i.e. the one we are investigation is an injection. Is this the correct reason?
Thank you in advance.