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**On the validity or failure of gap rigidity for certain pairs of bounded symmetric domains.**
*(English)*
Zbl 1085.32010

The authors characterize pairs \((\Omega,D)\) of bounded symmetric domains, \(D\) a totally-geodesic submanifold of \(\Omega\), by specific rigidity properties. Let \(\Gamma\) be a discrete properly discontinuous subgroup of Aut \(\Omega\) and \(S\subset\Omega/\Gamma\) a compact complex submanifold such that \(S\) is \(\varepsilon\)-geodesic, i.e. the norm of the second fundamental form of \(S\subset X\) is uniformly bounded by \(\varepsilon >0\). In a former paper [Ann. Math. Stud. 137, 79–117 (1995; Zbl 0848.32027)], the authors proved that \(S\) can be locally approximated by a uniquely determined isomorphism class of totally-geodesic complex submanifolds \(D\subset\Omega\) if the pinching constant \(\epsilon\) is sufficiently small. Briefly, \(S\) can be modelled on the pair \((\Omega,D)\).

The pair \((\Omega,D)\) is called gap rigid, if, for sufficiently small \(\varepsilon >0\), every \(\varepsilon\)-geodesic compact complex submanifold \(S\subset\Omega/\Gamma\) modelled on \((\Omega,D)\) is totally geodesic, independent of \(\Gamma\). The authors also introduce a stronger version of gap rigidity in algebro-geometric terms. Let \(K\) be the isotropy group of \(o\in\Omega\). By definition, gap rigidity in the Zariski topology holds for \((\Omega,D)\) if there exists a \(K\)-invariant Zariski-open subset \({\mathcal O}_o\) in Grass(dim\((D),T_o(\Omega))\) such that \([T_o(D)]\in{\mathcal O}_o\) and such that, independent of \(\Gamma\), every compact complex manifold \(S\subset\Omega/\Gamma\) with \(\dim S=\dim D\) is totally geodesic whenever each tangent space of \(S\) lifts to an element of \({\mathcal O}_o\).

On the one hand it is shown that gap rigidity fails already in the complex topology for the pair \((\triangle^2,\triangle\times\{0\})\) and therefore fails in general for any bounded symmetric domain of rank \(\geq 2\). On the other hand the authors prove that gap rigidity in the Zariski topology holds for the pair \((\Omega,D)\), \(\Omega\) irreducible, if \({\mathcal O}_o\) can be chosen such that Grass(dim\((D),T_o(\Omega))\backslash{\mathcal O}_o\) is a hypersurface.

The authors discuss thoroughly several examples where this criterion applies and give explicit descriptions of the hypersurfaces. This part of the paper is closely related to former results of the author, see Ph. Eyssidieux [J. Reine Angew. Math. 490, 155–212 (1997; Zbl 0886.32013), Lond. Math. Soc. Lect. Note Ser. 264, 71–92 (1999; Zbl 0952.32014)], N. Mok [Compos. Math. 132, No. 3, 289–309 (2002; Zbl 1013.32013)].

The pair \((\Omega,D)\) is called gap rigid, if, for sufficiently small \(\varepsilon >0\), every \(\varepsilon\)-geodesic compact complex submanifold \(S\subset\Omega/\Gamma\) modelled on \((\Omega,D)\) is totally geodesic, independent of \(\Gamma\). The authors also introduce a stronger version of gap rigidity in algebro-geometric terms. Let \(K\) be the isotropy group of \(o\in\Omega\). By definition, gap rigidity in the Zariski topology holds for \((\Omega,D)\) if there exists a \(K\)-invariant Zariski-open subset \({\mathcal O}_o\) in Grass(dim\((D),T_o(\Omega))\) such that \([T_o(D)]\in{\mathcal O}_o\) and such that, independent of \(\Gamma\), every compact complex manifold \(S\subset\Omega/\Gamma\) with \(\dim S=\dim D\) is totally geodesic whenever each tangent space of \(S\) lifts to an element of \({\mathcal O}_o\).

On the one hand it is shown that gap rigidity fails already in the complex topology for the pair \((\triangle^2,\triangle\times\{0\})\) and therefore fails in general for any bounded symmetric domain of rank \(\geq 2\). On the other hand the authors prove that gap rigidity in the Zariski topology holds for the pair \((\Omega,D)\), \(\Omega\) irreducible, if \({\mathcal O}_o\) can be chosen such that Grass(dim\((D),T_o(\Omega))\backslash{\mathcal O}_o\) is a hypersurface.

The authors discuss thoroughly several examples where this criterion applies and give explicit descriptions of the hypersurfaces. This part of the paper is closely related to former results of the author, see Ph. Eyssidieux [J. Reine Angew. Math. 490, 155–212 (1997; Zbl 0886.32013), Lond. Math. Soc. Lect. Note Ser. 264, 71–92 (1999; Zbl 0952.32014)], N. Mok [Compos. Math. 132, No. 3, 289–309 (2002; Zbl 1013.32013)].

Reviewer: Eberhard Oeljeklaus (Bremen)