Differential geometry of complex projective plane conics (Q6171996)

Let \(\mathbb{P}^2\) be the complex projective plane and \(h\) its Fubini-Study metric. Fix \(\mathcal{C}\subset \mathbb{P}^2\) a smooth curve; then \(\mathcal{C}\) with the restricted metric becomes a Riemannian surface. The paper studies the geometric invariants of \(\mathcal{C}\) and performs an useful comparison with its algebraic invariants. A large part of this paper concerns the case of complex conics for which the Gaussian curvature is computed. Finally, the deformations of smooth conics into a pair of lines are discussed and the classification of cubics up to Hermitian transformations is obtained.