Why does $H^1(X,\mathbb{Z})$ span a full lattice in $H^1(X,O_X)$ if $X$ is kahler?

Question

We have the exponential short exact sequence for compact complex manifolds. Why does the image of $H^1(X,\mathbb{Z})$ span $H^1(X,O_X)$ over $\mathbb{R}$ if $X$ is kahler? The map is injective, I was wondering why the case can't happen: (for example) that the whole lattice is sent to a real line in $H^1(X,O_X)$ with generators sent to $\mathbb{Q}$ linearly independent irrationals?