Deciding if surfaces in $\mathbb{C}P^2$ are varieties

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Let $S$ be a smooth submanifold of $\mathbb{C}P^2$ given in some very explicit way (for example given combinatorially in some triangulation of $\mathbb{C}P^2$). Is it decidable whether or not $S$ is (smoothly, say) isotopic to some smooth complex projective curve $C$?