A well-known Calabi’s rigidity theorem on holomorphic isometric immersions into the complex projective space is generalized to the case that the target is the complex Grassmann manifolds. Our strategy is to use the differential geometry of vector bundles and a generalization of do Carmo and Wallach theory developed in Nagatomo (Harmonic maps into Grassmann manifolds. arXiv:mathDG/1408.1504). We introduce the associated maps with holomorphic maps to obtain a general rigidity theorem (Theorem 5.6). As applications, several rigidity results on Einstein–Hermitian holomorphic maps are exhibited and we also give an interpretation of the existence of a Kähler structure with an \(S^1\)-action on the moduli spaces of holomorphic isometric embeddings of a compact Kähler manifold into complex quadrics. Theorem 5.6 also implies classification theorems for equivariant holomorphic maps.

将众所周知的关于全同等距浸入复杂投影空间中的卡拉比的刚性定理推广到目标是复杂的格拉斯曼流形的情况。我们的策略是使用矢量束的微分几何以及对Nagatomo中开发的do Carmo和Wallach理论的推广（将谐波映射成Grassmann流形。arXiv：mathDG / 1408.1504）。我们将相关图和全纯图一起引入以获得一般刚度定理（定理5.6）。作为应用，展示了爱因斯坦-埃尔米特全纯图上的几个刚度结果，并且我们还给出了具有\（S ^ 1 \）的Kähler结构的存在的解释。紧Kähler流形到复杂二次曲面的全同等距嵌入的模空间上的作用。定理5.6还隐含了等变全纯映射的分类定理。