A nodal projective curve in $\mathbb{CP}^2$ inherits a Kähler metric from the Fubini-Study metric, and hence a Riemannian metric. In particular, with respect to this metric, a line has constant Gaussian curvature 1; according to [Vitter](http://www.iumj.indiana.edu/docs/23067/23067.asp) (page 826), the Fermat conic $\{x^2+y^2+z^2=0\}$ has constant curvature 1/2, but no other Fermat curve has constant curvature. In fact, Vitter explicitly computes the Gaussian curvature of the zero set of a given homogeneous polynomial.



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> Does there exist a flat plane cubic?
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More generally,



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> Fix a degree $d>2$: does there exist a smooth, degree-$d$ plane curve with constant Gaussian curvature? Does there exist an irreducible degree-$d$ nodal curve with constant Gaussian curvature?
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If such a curve existed, and it had total Milnor number $\delta$, its curvature would be $(3-d)/2 + \delta/d$.



Projective curves of constant curvature
A general result of [Hulin](https://doi.org/10.1007/BF02921947) shows that if $M$ is a compact complex manifold with a holomorphic embedding $f:M\to\mathbb{CP}^n$ such that $f^\*g\_{FS}$ is Einstein, then the Einstein constant is positive. Hence the answer to your first question is **no.**


Also, if furthermore $\mathrm{dim}\_{\mathbb{C}} M=n-1,$ i.e. $f(M)$ is a complex hypersurface, then $f(M)$ is either a hyperplane or a hyperquadric, by a theorem of [Smyth](https://www.jstor.org/stable/1970441) and [Chern](https://doi.org/10.4310/jdg/1214427878). Hence the answer to your second question is also **no.**


See also this related question: [Geometry of complex elliptic curves](https://mathoverflow.net/questions/68952/)


