Georges's answer is very nice so I will just add some comments.

In the curve case the assumption "unramified" is not necessary; in fact, every finite cover of a Riemann surface is still a Riemann surface (this is essentially Riemann Existence Theorem).

In higher dimension the situation is more complicate and "unramified" is definitely necessary. In fact, Donaldson and Auroux proved that every real, symplectic 4-manifold $(X, \omega)$ can be realized as a finite branched cover of $\mathbb{CP}^2$, and moreover such a cover $f \colon X \to \mathbb{CP}^2$ is completely determined, up to symplectomorphisms, by

• the branch curve $D \subset \mathbb{CP}^2$ and
• the monodromy representation $\theta \colon \pi_1(\mathbb{CP}^2 -D) \to S_{N}$, where $N:= \deg f$.

Finally, $X$ is complex - projective if and only if $D$ is isotopic to a complex curve.

Georges' answer is very nice so I will just add some comments.

In the curve case the assumption "unramified" is not necessary; in fact, every finite cover of a Riemann surface is still a Riemann surface (this is essentially Riemann Existence Theorem).

In higher dimension the situation is more complicate and "unramified" is definitely necessary. In fact, Donaldson and Auroux proved that every real, symplectic 4-manifold $(X, \omega)$ can be realized as a finite branched cover of $\mathbb{CP}^2$, and moreover such a cover $f \colon X \to \mathbb{CP}^2$ is completely determined, up to symplectomorphisms, by

• the branch curve $D \subset \mathbb{CP}^2$ and
• the monodromy representation $\theta \colon \pi_1(\mathbb{CP}^2 -D) \to S_{N}$, where $N:= \deg f$.

Finally, $X$ is complex - projective if and only if $D$ is isotopic to a complex curve.

Georges's answer is very nice so I will just add some comments.

In the curve case the assumption "unramified" is not necessary; in fact, every finite cover of a Riemann surface is still a Riemann surface (this is essentially Riemann Existence Theorem).

In higher dimension the situation is more complicate and "unramified" is definitely necessary. In fact, Donaldson and Auroux proved that every real, symplectic 4-manifold $(X, \omega)$ can be realized as a finite, branched cover of $\mathbb{CP}^2$, and moreover such a cover $f \colon X \to \mathbb{CP}^2$ is completely determined, up to symplectomorphisms, by

• the branch curve $D \subset \mathbb{CP}^2$ and
• the monodromy representation $\theta \colon \pi_1(\mathbb{CP}^2 -D) \to S_{N}$, where $N:= \deg f$.

Finally, $X$ is complex - projective if and only if $D$ is isotopic to a complex curve.

Georges's answer is very nice so I will just add some comments.

In the curve case the assumption "unramified" is not necessary; in fact, every finite cover of a Riemann surface is still a Riemann surface (this is essentially Riemann Existence Theorem).

In higher dimension the situation is more complicate and "unramified" is definitely necessary. In fact, Donaldson and Auroux proved that every real, symplectic 4-manifold $(X, \omega)$ can be realized as a finite branched cover of $\mathbb{CP}^2$, and moreover such a cover $f \colon X \to \mathbb{CP}^2$ is completely determined, up to symplectomorphisms, by

• the branch curve $D \subset \mathbb{CP}^2$ and
• the monodromy representation $\theta \colon \pi_1(\mathbb{CP}^2 -D) \to S_{N}$, where $N:= \deg f$.

Finally, $X$ is complex - projective if and only if $D$ is isotopic to a complex curve.