Projectively induced Kähler--Einstein surfaces
Abstract
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 (PP2,m gFS) and (PP1×PP1,m(gFS⊕ gFS)), m∈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.
Related
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BibTeX
@misc{loi2026projectively,
title = {{Projectively induced Kähler--Einstein surfaces}},
author = {Andrea Loi and Mirel Caibăr},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22832},
eprint = {2609.22832},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22832},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Projectively induced Kähler--Einstein surfaces
AU - Andrea Loi
AU - Mirel Caibăr
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.22832
UR - https://arxiv.org/abs/2609.22832
AB - 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 (PP^2,m g_FS) and (PP^1×PP^1,m(g_FS⊕ g_FS)), m∈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.
ER -