Rigidity of proper holomorphic mappings between nonequidimensional bounded symmetric domains

Tu, Zhenhan

doi:10.1007/s002090100353

Cited by 37 publications

(28 citation statements)

References 23 publications

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“…In what follows, we are concerned exclusively with the study of proper holomorphic mappings between Cartan domains of type I, although it is possible to generalize the nonexistence results to certain other pairs of irreducible bounded symmetric domains. Concerning D(p, q) we have the following result of Z.-H. Tu [10] which gives the only known nontrivial rigidity and nonexistence results for proper holomorphic maps between bounded symmetric domains in the non-equirank case. We note that the rank defect is always equal to 1 in the result.…”

Section: Lemma 24mentioning

confidence: 95%

“…We will give a proof of the nonexistence result in Theorem 2.3 without using the rigidity result for the case of p ≥ 4 by resorting to the geometry of invariantly geodesic subspaces. This proof will then be generalized in a way that avoids establishing an analogous rigidity result for arbitrarily prescribed rank defects, a problem hitherto unresolved which has remained technically difficult along the line of approach of [10].…”

Section: Lemma 24mentioning

confidence: 97%

“…To start with, we give a proof of the nonexistence result of Z.-H. Tu [10] (cf. Theorem 2.3) for the case of p ≥ 4 which leads to an effective nonexistence result for the case where the rank defect is equal to 2.…”

Section: 1mentioning

confidence: 99%

“…From such a rigidity result Tu deduced also some nonexistence results for proper holomorphic maps f : Ω → Ω where Ω and Ω are again Cartan domains of type I, and where the rank defect is also 1. In this article, we examine further nonexistence results for proper holomorphic maps f : Ω → Ω , where to focus on the discussion we will restrict ourselves to Cartan domains of type I. We will show in this context that there are nonexistence results generalizing those of Z.-H. Tu [10] in which the rank defect is arbitrarily large. More precisely, given any positive integer , we prove that there exist infinite series of pairs of Cartan domains (Ω, Ω ) of type I such that 2 ≤ rank(Ω) < rank(Ω ), and such that the rank defect is equal to , for which there exists no proper holomorphic mapping f : Ω → Ω .…”

Section: Introductionmentioning

confidence: 96%

“…We will call this the non-equirank case, and the difference r − r will be called the rank defect. In the non-equirank case, the only known rigidity result so far is that of Z.-H. Tu [10], where he found examples of Cartan domains Ω, Ω of type I where any proper holomorphic map f : Ω → Ω is necessarily totally geodesic. In this infinite series of examples, the rank defect is always equal to 1.…”

Section: Introductionmentioning

confidence: 97%

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Nonexistence of proper holomorphic maps between certain classical bounded symmetric domains

Mok

2008

Chin. Ann. Math. Ser. B

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The author, motivated by his results on Hermitian metric rigidity, conjectured in [4] that a proper holomorphic mapping f : Ω → Ω from an irreducible bounded symmetric domain Ω of rank ≥ 2 into a bounded symmetric domain Ω is necessarily totally geodesic provided that r := rank(Ω ) ≤ rank(Ω) := r. The Conjecture was resolved in the affirmative by I.-H. Tsai [8]. When the hypothesis r ≤ r is removed, the structure of proper holomorphic maps f : Ω → Ω is far from being understood, and the complexity in studying such maps depends very much on the difference r − r, which is called the rank defect. The only known nontrivial non-equidimensional structure theorems on proper holomorphic maps are due to Z.-H. Tu [10], in which a rigidity theorem was proven for certain pairs of classical domains of type I, which implies nonexistence theorems for other pairs of such domains. For both results the rank defect is equal to 1, and a generalization of the rigidity result to cases of higher rank defects along the line of arguments of [10] has so far been inaccessible. In this article, the author produces nonexistence results for infinite series of pairs of (Ω, Ω ) of irreducible bounded symmetric domains of type I in which the rank defect is an arbitrarily prescribed positive integer. Such nonexistence results are obtained by exploiting the geometry of characteristic symmetric subspaces as introduced by N. Mok and I.-H Tsai [6] and more generally invariantly geodesic subspaces as formalized in [8]. Our nonexistence results motivate the formulation of questions on proper holomorphic maps in the non-equirank case.

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Section: Lemma 24mentioning

confidence: 95%

Section: Lemma 24mentioning

confidence: 97%

Section: 1mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 97%

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