Abstract
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.
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Nagatomo, Y. Holomorphic maps into Grassmann manifolds (harmonic maps into Grassmann manifolds III). Ann Glob Anal Geom 60, 33–63 (2021). https://doi.org/10.1007/s10455-021-09765-6
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DOI: https://doi.org/10.1007/s10455-021-09765-6