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Georges's

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.

2 deleted 1 characters in body

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.

1

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.