射影诱导的Kähler--Einstein曲面
Projectively induced Kähler--Einstein surfaces
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- The Ohio State University（俄亥俄州立大学）
- Università di Cagliari（卡利亚里大学）
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中文总结 AI 辅助
本文分类了由全纯等距浸入复射影空间诱导度量的紧致Kähler--Einstein曲面，证明仅有射影平面和双曲面的乘积，且均为齐次。
中文摘要 AI 辅助
我们分类了其度量由到有限维复射影空间的全纯等距浸入诱导的紧致Kähler--Einstein曲面。不施加对称性假设，也不限制余维数。我们证明这样的曲面仅有 \\[ (\PP^2,m g_{\FS}) \quad\text{和}\quad \bigl(\PP^1\times\PP^1,m(g_{\FS}\oplus g_{\FS})\bigr), \qquad m\in\mathbb Z_{>0}, \\] 分别由Veronese和Segre--Veronese嵌入实现。主要的新工具是一个与余维数无关的排除整个Fano指数为一分支的方法，结合了公共反典范根构造与Gram--Gauss秩估计，并在五次情形中使用等变曲率论证。因此，每个连通的紧致Kähler--Einstein曲面，若其度量由到有限维复射影空间的全纯等距浸入诱导，则必为齐次的。
英文摘要
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 \[ (\PP^2,m g_{\FS}) \quad\text{and}\quad \bigl(\PP^1\times\PP^1,m(g_{\FS}\oplus g_{\FS})\bigr), \qquad m\in\mathbb Z_{>0}, \] 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.
