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A characterization of varieties whose universal cover is a bounded symmetric domain without ball factors. (English) Zbl 1295.32039

This is a kind of converse result of [S. Kobayashi and T. Ochiai, Prog. Math. 14, 207–222 (1981; Zbl 0498.53038)]. See especially Theorem (1.3) therein.
In [loc. cit.], Kobayashi and Ochiai found that the curvature tensor for the compact Hermitian symmetric space \(S\) is a holomorphic endomorphism for \(T_S \otimes T_S^V\), see in [loc. cit., page 219] the paragraph after the proof of Lemma 5.1. There the curvature is defined by \(R(v, \bar{w})=[v, \bar{w}]\in k\) with \(v\in T_ =p_{+}\) and \(\bar{w} \in \bar{T}_o =p_{-}\), notice that the Lie algebra for the holomorphic automorphism group is \(g=p_{-}+k+p_{+}\) with \(k\) the reductive part of the isotropic parabolic Lie algebra \(k+p_{-}\). Since every bounded symmetric domain is an open set of its compact dual, this also gives a holomorphic endomorphism on the bounded symmetric domains and their quotients. For the construction, see also in [loc. cit.], Lemmas 2.9, 2.10 and the paragraph after the proof of Lemma 2.10 on page 212.
The authors then prove Theorem 1.2 using the first Mok characteristic cone of the endomorphism in the tangent space in the description.
The authors’ abstract: “We give two characterizations of varieties whose universal cover is a bounded symmetric domain without ball factors in terms of the existence of a holomorphic endomorphism \(\sigma\) of the tensor product \(T\otimes T^{V}\) of the tangent bundle \(T\) with the cotangent bundle \(T^{V}\). To such a curvature type tensor \(\sigma\) one associates the first Mok characteristic cone \(CS\), obtained by projecting on \(T\) the intersection of \(\ker(\sigma)\) with the space of rank 1 tensors. The simpler characterization requires that the projective scheme associated to \(CS\) be a finite union of projective varieties of given dimensions and codimensions in their linear spans which must be skew and generate.”
The authors could give a little bit more details about the Kobayashi-Ochiai construction in their own language and explain the \(X^{\tau}\) in Corollary 1.4.

MSC:

32Q30 Uniformization of complex manifolds
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
32Q20 Kähler-Einstein manifolds
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
32J27 Compact Kähler manifolds: generalizations, classification

Citations:

Zbl 0498.53038
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References:

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