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**Rigidity of irreducible Hermitian symmetric spaces of the compact type under Kähler deformation.**
*(English)*
Zbl 0902.32014

The main result of the paper is an essential contribution to the problem of rigidity of complex structures of irreducible Hermitian symmetric spaces of the compact type.

The authors solve the problem under an additional Kähler assumption. They prove: Given an irreducible Hermitian symmetric space \(S\) of the compact type and a regular holomorphic family \(\pi: X\to\mathbb{E}\) of compact complex manifolds over the unit disk \(\mathbb{E}\) such that \(X_t: =\pi^{-1}(t)\) is biholomorphic to \(S\) for \(t\neq 0\) and such that the central fiber \(X_0\) is Kähler. Then \(X_0\) is biholomorphic to \(S\).

For \(\mathbb{P}^n\) and the hyperquadric \(Q^n\), \(n\geq 3\), the problem has been solved in general [cf. J.-M. Hwang, Math. Z. 221, 513-519 (1996; Zbl 0853.32031)]. Since these spaces have been characterized within the category of compact Kähler manifolds in terms of their first Chern classes, their rigidity under the Kähler assumption is well known. The main method of the proof is as follows: Let \(G\) be the identity component of the group of isometries of the irreducible Hermitian symmetric space \((S,h)\) of the compact type with rank \(S\geq 2\). By the embedding theorem of Harish-Chandra \(S\) can be viewed as an equivariant compactification of \(\mathbb{C}^n\) with respect to some \(n\)-dimensional abelian subgroup of \(G^\mathbb{C}\). Let \(K\) be the isotropy group of some \(s\in S\) in the compact group \(G\). The Harish-Chandra-embedding yields suitable holomorphic coordinates to show that the structure group of the holomorphic tangent bundle of \(S\) can be holomorphically reduced to the reductive group \(K^\mathbb{C}\), i.e. \(S\) admits an integrable holomorphic \(K^\mathbb{C}\)-structure. The authors apply an extension theorem of Ochiai, to see that if \(M\) is a simply connected compact complex manifold with an integrable holomorphic \(K^\mathbb{C}\)-structure, then \(M\) is biholomorphic to \(S\).

After this the main point of the proof is to show that the induced integrable \(K^\mathbb{C}\)-structures on the fibers \(X_t\), \(t\neq 0\), converge to an (integrable) holomorphic \(K^\mathbb{C}\)-structure on the central fiber \(X_0\). The holomorphic \(K^\mathbb{C}\)-structure on \(S\) is determined by an holomorphic \(K^\mathbb{C}\)-bundle of homogeneous cones on \(S\), where the fiber in \(s\in S\) is the homogeneous complex submanifold of the projectified complex tangent space consisting of all directions tangent to minimal rational curves through \(s\). The authors study the limit of these cone bundles on \(X_t\) for \(t\to 0\) and use deformation theory of rational curves to ensure that the limit is a non-degenerate \(K^\mathbb{C}\)-cone bundle on \(X_0\).

The authors solve the problem under an additional Kähler assumption. They prove: Given an irreducible Hermitian symmetric space \(S\) of the compact type and a regular holomorphic family \(\pi: X\to\mathbb{E}\) of compact complex manifolds over the unit disk \(\mathbb{E}\) such that \(X_t: =\pi^{-1}(t)\) is biholomorphic to \(S\) for \(t\neq 0\) and such that the central fiber \(X_0\) is Kähler. Then \(X_0\) is biholomorphic to \(S\).

For \(\mathbb{P}^n\) and the hyperquadric \(Q^n\), \(n\geq 3\), the problem has been solved in general [cf. J.-M. Hwang, Math. Z. 221, 513-519 (1996; Zbl 0853.32031)]. Since these spaces have been characterized within the category of compact Kähler manifolds in terms of their first Chern classes, their rigidity under the Kähler assumption is well known. The main method of the proof is as follows: Let \(G\) be the identity component of the group of isometries of the irreducible Hermitian symmetric space \((S,h)\) of the compact type with rank \(S\geq 2\). By the embedding theorem of Harish-Chandra \(S\) can be viewed as an equivariant compactification of \(\mathbb{C}^n\) with respect to some \(n\)-dimensional abelian subgroup of \(G^\mathbb{C}\). Let \(K\) be the isotropy group of some \(s\in S\) in the compact group \(G\). The Harish-Chandra-embedding yields suitable holomorphic coordinates to show that the structure group of the holomorphic tangent bundle of \(S\) can be holomorphically reduced to the reductive group \(K^\mathbb{C}\), i.e. \(S\) admits an integrable holomorphic \(K^\mathbb{C}\)-structure. The authors apply an extension theorem of Ochiai, to see that if \(M\) is a simply connected compact complex manifold with an integrable holomorphic \(K^\mathbb{C}\)-structure, then \(M\) is biholomorphic to \(S\).

After this the main point of the proof is to show that the induced integrable \(K^\mathbb{C}\)-structures on the fibers \(X_t\), \(t\neq 0\), converge to an (integrable) holomorphic \(K^\mathbb{C}\)-structure on the central fiber \(X_0\). The holomorphic \(K^\mathbb{C}\)-structure on \(S\) is determined by an holomorphic \(K^\mathbb{C}\)-bundle of homogeneous cones on \(S\), where the fiber in \(s\in S\) is the homogeneous complex submanifold of the projectified complex tangent space consisting of all directions tangent to minimal rational curves through \(s\). The authors study the limit of these cone bundles on \(X_t\) for \(t\to 0\) and use deformation theory of rational curves to ensure that the limit is a non-degenerate \(K^\mathbb{C}\)-cone bundle on \(X_0\).

Reviewer: E.Oeljeklaus (Bremen)

### MSC:

32M15 | Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) |

32J27 | Compact Kähler manifolds: generalizations, classification |