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On compact splitting complex submanifolds of quotients of bounded symmetric domains. (English) Zbl 1375.53091

Let \(\Omega\) be an irreducible bounded symmetric domain, \(\Gamma\) be a torsion-free discrete subgroup of \(\text{Aut} ( \Omega)\) and \(X = \Omega / \Gamma\). Denote by \(g_{\Omega}\) the canonical Kähler-Einstein metric on \(\Omega\), with respect to which the minimal discs on \(\Omega\) have Gaussian curvature \(-2\) and put \(g\) for the Kähler-Einstein metric on \(X\), induced by \(g_{\Omega}\). If the identical inclusion \((S, g | _S) \hookrightarrow (X, g)\) is a totally geodesic embedding then \(S\) is a splitting submanifold of \(X\), i.e., there is a surjective holomorphic bundle map \(T(X) | _S \rightarrow T(S)\) of the holomorphic tangent bundles of \(X\) and \(S\), which restricts to the identity of \(T(S)\). The article characterizes the totally geodesic submanifolds \((S, g | _S) \hookrightarrow (X,g)\) among the splitting complex submanifolds of \(X\).
In order to formulate some of the results, let us consider the Borel embedding \(\Omega \hookrightarrow Z\) of \(\Omega\) in its compact dual Hermitian symmetric space \(Z\). A complex submanifold \(S \subset X\) is characteristic if for any point \(x \in S\) and any holomorphic tangent vector \(v \in T_x (S)\) there is a minimal rational curve on \(Z\) through \(x\), tangent to \(v\) at \(x\). The authors show that any compact splitting characteristic complex submanifold \(S \subset X\) is totally geodesic with respect to the Kähler-Einstein metric \(g\) and the normal bundle \(\mathcal{N} _{S/X} = T(S) ^{\perp}\) of \(S\) in \(X\) is the \(g\)-orthogonal complement of the holomorphic sub-bundle \(T(S)\) of \(T(X) | _S\). The argument views \(T(S)\) as a quotient bundle of \(T(X) | _S\) and makes use of the monotonicity of the curvatures of the Hermitian holomorphic sub-bundles of the Hermitian holomorphic vector bundles.
For an arbitrary complex manifold \(M\) let \(T^r _s (M) := T(M) ^{\otimes r} \otimes T^* (M) ^{\otimes s}\) be the holomorphic bundle of \((r,s)\)-tensors. The contraction of the curvature tensor \(R_{\xi \overline{\eta} \mu \overline{\nu}}\) of \(X\) by the Kähler-Einstein metric \(g\) is a parallel and, therefore, a holomorphic tensor of type \((2,2)\). It can be viewed as a holomorphic endomorphism \(R_{\tau} : T^1 _1 (X) \rightarrow T^1 _1 (X)\) or \(R_{\sigma} : S^2 T(X) \rightarrow S^2 T(X)\). Since \(R_{\sigma}\) is parallel and holomorphic, the symmetric square \(S^2 T(X)\) of \(T(X)\) has a parallel and holomorphic decomposition into eigen-sub-bundles of \(R_{\sigma}\). Calabi-Vesentini and Borel have established that for an arbitrary irreducible bounded symmetric domain \(\Omega\) of \(\mathrm{rank}( \Omega) \geq 2\) the endomorphism \(R_{\sigma} : S^2 T(X) \rightarrow S^2 T(X)\) has two non-zero eigenvalues and there is a parallel holomorphic direct sum decomposition \(S^2 T(X) = A \oplus B\) into eigen-sub-bundles. For an arbitrary point \(x\) of a Kähler manifold \(M\) with curvature tensor \(R_{\xi \overline{\eta} \mu \overline{\nu}}\), let \(\mathcal{Z}_x\) be the collection of pairs \((U,V)\) of subspaces \(U,V \subset T_x (M)\) of the holomorphic tangent space \(T_x (M)\) to \(M\) at \(x\) with \(R_{u \overline{u} v \overline{v}} =0\) for \(\forall u \in U\), \(\forall v \in V\) and with \(U \cap V = \{ 0 \}\). Then the maximum dimension \(\rho _x (M)\) of \(U + V\) for \((U, V) \in \mathcal{Z}_x\) is called the degree of the strong non-degeneracy of the bisectional curvature of \(M\) at \(x\). In particular, if \(X = \Omega / \Gamma\) is a torsion-free discrete quotient of an irreducible bounded symmetric domain \(\Omega\) of \(\mathrm{rank}(\Omega ) \geq 2\) then \(\rho = \rho _x (X)\) is constant on \(X\) and called the degree of the strong non-degeneracy of the bisectional curvature on \(X\).
The article under review shows that if \(S \subset X\) is a compact splitting complex submanifold with \(\text{rank } ( S^2 T(S)) > \max ( \text{rank} (A), \text{rank} (B))\) and \(\dim S > \rho\) for the eigen-bundle decomposition \(S^2 T(X) = A \oplus B\) of \(R_{\sigma}\) and the degree \(\rho\) of the strong non-degeneracy of the bisectional curvature of \(X\) then \((S, g | _S) \hookrightarrow (X, g)\) is a totally geodesically embedded Hermitian locally symmetric space of \(\text{rank } (S) \geq 2\). Another result asserts that if \(K\) is the isotropy group of the origin of \(\Omega = G/K\) and \(S \subset X\) is a compact splitting complex submanifold with \(\text{rank} ( S^2 T(S)) > \min ( \text{rank} (A), \text{rank} (B))\) and \(\dim (S) ^2 > \max ( \dim _{\mathbb R} K, \rho ^2)\) then \((S, g | _S) \hookrightarrow (X, g)\) is a totally geodesically embedded Hermitian locally symmetric space of rank \(\text{rank} (S) \geq 2\). After showing that any compact complex submanifold \(S \subset X\) of \(\dim (S) > \rho\) is de Rham irreducible with respect to the Kähler-Einstein metric \(g\), the article studies the kernels of the endomorphisms of \(S^2 T(S)\), induced by the projections of \(S^2 T(X)\) onto the eigen-sub-bundles \(A, B \subset S^2 T(X)\) of \(R_{\sigma}\), as well as the endomorphism of \(T^1 _1 (S)\), induced by \(R_{\tau} : T^1 _1 (X) \rightarrow T^1 _1 (X)\).
For an irreducible bounded symmetric domain \(\text{D} ^{I} _{2,p} = \text{SU} (2,p) / \text{S} ( \text{U} _2 \times \text{U} _p)\) of type \(I\) with \(p \geq 2\), any compact splitting complex submanifold \(S \subset \text{D} ^{I} _{2,p} / \Gamma\) of \(\dim (S) \geq p+1\) is shown to be a totally geodesically embedded Hermitian locally symmetric space of \(\mathrm{rank}(S) \geq 2\). Moreover, if \(p \geq 3\) then the universal cover of \(S\) is \(\text{D} ^{I} _{2,q}\) for some \(q \leq p\). The authors conjecture that the compact splitting complex submanifolds \(S \subset \text{D}^{I} _{2,p} / \Gamma\) of \(\dim (S) \geq p \geq 2\) are totally geodesic. If \(\text{D} ^{IV}_n = \text{SO} _o (2,n) / \text{SO} (2) \times \text{SO} (n)\) is an irreducible bounded symmetric domain of type \(IV\) with \(n \geq 3\) then the compact splitting complex submanifolds \(S \subset \text{D} ^{IV} _n / \Gamma\) of \(\dim (S) > \frac{n}{\sqrt{2}}\) are proved to be totally geodesically embedded Hermitian locally symmetric spaces, whose universal covers are biholomorphic to \(\text{D} ^{IV} _m\) for some \(m \leq n\). The authors raise a conjecture that \(\dim (S) \geq 2\) suffices for a compact splitting complex submanifold \(S \subset \text{D} ^{IV} _n / \Gamma\) to be totally geodesically embedded.
The article concludes with a result on the borderline cases of both conjectures. It establishes that any compact splitting complex surface \(S \subset \text{D} ^{IV} _3 / \Gamma\) is biholomorphic to a compact torsion-free discrete quotient \({\mathbb B} ^1 \times {\mathbb B}^1 / \Gamma _o\) of the bi-disc \({\mathbb B}^1 \times {\mathbb B}^1\). If \(S \subset \text{D} ^{IV} _4 / \Gamma\) is a compact splitting complex surface then either \((S, g | _S) \hookrightarrow (X, g)\) is totally geodesic or for any point \(x \in S\) and any holomorphic normal vector \(v\) to \(S\) at \(x\) there is a minimal rational curve on \(Z^{IV} _4 :=\text{SO} (6) / \text{SO} (2) \times \text{SO} (4)\), which is tangent to \(v\) at \(x\). Any totally geodesic surface \(S \subset \text{D} ^{IV} _4 / \Gamma\) is biholomorphic to a compact torsion-free discrete quotient \({\mathbb B}^1 \times {\mathbb B}^1 / \Gamma _o\) of the bi-disc \({\mathbb B}^1 \times {\mathbb B}^1\) or to a compact torsion-free discrete quotient \({\mathbb B}^2 / \Sigma\) of the complex \(2\)-ball \({\mathbb B}^2 = \text{SU} (2,1) / \text{S} ( \text{U} _2 \times \text{U} _1)\).

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
32Q20 Kähler-Einstein manifolds
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
53C35 Differential geometry of symmetric spaces
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