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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 (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.

Does there exist a flat plane cubic?

More generally,

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?

If such a curve existed, and it had total Milnor number $\delta$, its curvature would be $(3-d)/2 + \delta/d$.

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A general result of Hulin 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 and Chern. Hence the answer to your second question is also no.

See also this related question.

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  • $\begingroup$ Interesting, thanks! I had completely missed that other question (and so did my Google searches). $\endgroup$ – Marco Golla Jan 3 '15 at 11:28
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Actually, the rigidity result is local, and hence is even stronger: A very old result of Calabi implies that any connected (local) piece of a holomorphic curve in $\mathbb{CP}^n$ with constant Gaussian curvature in the induced metric from the Fubini-Study metric is a piece of a rational curve that is a rational normal curve in the smallest linear projective space that contains it. In fact, it must be an orbit of a representation of $\mathrm{SU}(2)$ into $\mathrm{SU}(n{+}1)$.

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