I am reading Huybrecht's introduction to complex geometry, and be stuck. First he introduces the exponential sheaf sequence, so that we have : $H^1(X,Z)\to H^1(X,\mathcal{O}_X)\to H^1(X,\mathcal{O}^*_X)\to H^2(X,Z)$.
Then he says if $X$ is compact, then the map $H^1(X,Z)\to H^1(X,\mathcal{O}_X)$ is injective. Why we have this?
Since $Pic(X)$ is isomorphic to $H^1(X,\mathcal{O}^*_X$), he says discrete part of $Pic(X)$ can be measured by it image in $H^2(X,Z)$ and continuous part comes from vector space $H^1(X,\mathcal{O}_X)$. Why $H^1(X,\mathcal{O}_X)$ is a vector space and what does he mean by discrete and continuous part?
I apologize if the question is very fundamental since I am totally new to complex geometry.
It's important to be aware of the whole long exact sequence. In general, you have an exact \[ A \xrightarrow{u} B \to C \xrightarrow{w} D \] then $w$ is injective iff $u$ is surjective. So in your situation we can look at the map $H^0(X, \mathscr{O}) \to H^0(X, \mathscr{O}^*)$. Since $X$ is compact — and let me just assume connected — by the maximum principle this map on $H^0$ is just the old exponential $\mathbf C \to \mathbf C^*$.
For the second: the point is that each $\mathscr{O}(U)$ is a vector space, so all the $H^p(X, \mathscr{O})$ are vector spaces too. Maybe this is easiest to believe from the Čech point of view. I usually think of a vector space as a continuous thing — it's our local model for a manifold, after all. On the other hand, the $H^i(X,\mathbf{Z})$ are finitely generated abelian groups.