Nonexistence of Proper Holomorphic Maps Between Certain Classical Bounded Symmetric Domains |
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Authors: | Ngaiming MOK |
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Institution: | Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong, China |
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Abstract: | The author, motivated by his results on Hermitian metric rigidity, conjectured in 4] that a proper holomorphic mapping ƒ: 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 ƒ: 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.
Project supported by a CERG of the Research Grants Council of Hong Kong, China. |
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Keywords: | Proper holomorphic maps Bounded symmetric domains Char-acteristic symmetric subspaces Invariantly geodesic subspaces Rank defects |
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