Projectively induced Kähler--Einstein surfaces

We classify compact Kähler–Einstein surfaces whose metric is induced by a holomorphic isometric immersion into a finite-dimensional complex projective space. No symmetry assumption and no bound on the codimension are imposed. We prove that the only such surfaces are $$({\text{PP}}^2,m{g}_{\text{FS}})\text{and}({\text{PP}}^1\times {\text{PP}}^1,m(g_{\text{FS}}\oplus g_{\text{FS}})),m\in {\mathbb{Z}}_{>0},$$ realized respectively by the Veronese and Segre–Veronese embeddings. The main new ingredient is a codimension-independent exclusion of the entire Fano-index-one branch, combining a common anticanonical root construction with Gram–Gauss rank estimates and, in degree five, an equivariant curvature argument. Consequently, every connected compact Kähler–Einstein surface whose metric is induced by a holomorphic isometric immersion into a finite-dimensional complex projective space is homogeneous.