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Nonexistence of proper holomorphic maps between certain classical bounded symmetric domains. (English) Zbl 1316.32016

The author proves that, for series of pairs \((\Omega, \Omega ')\) of irreducible bounded symmetric domains of type I, there does not exist any proper holomorphic map \(f:\Omega \to \Omega '\). More precisely the statement is as follows: Let \(D(p,q)\) be the symmetric domain of type I of \(p\) by \(q\) complex matrices \(Z\) such that \(I-\bar Z^t Z\) is positive definite. For integers \(k\geq 0\), \(\ell >0\), there is a positive integer \(N(k,\ell )\) such that, for any integer \(p\geq N(k,\ell )\), there does not exist any proper holomorphic map \(f:D(p+k+\ell ,p-\ell )\to D(p+k,p)\).
The author raises the following questions:
- Let \(\Omega \) and \(\Omega '\) be bounded symmetric domains such that \(\Omega \) is irreducible and of rank \(\geq 2\). Suppose there exists a proper holomorphic map \(f:\Omega \to \Omega '\). Does there always exist a totally holomorphic embedding \(h:\Omega \to \Omega '\) ?
- Suppose there exists a proper holomorphic mapping \(f:\Omega \to \Omega '\) which is not totally geodesic. Is \(\Omega \) necessarily biholomorphically equivalent to a characteristic symmetric subspace of \(\Omega '\) ?

MSC:

32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
32H35 Proper holomorphic mappings, finiteness theorems
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[1] Hwang, J.-M. and Mok, N., Holomorphic maps from rational homogeneous spaces of Picard number 1 onto projective manifolds, Invent. Math., 136, 1999, 209–331. · Zbl 0963.32007 · doi:10.1007/s002220050308
[2] Henkin, G. M. and Novikov, R., Proper mappings of classical domains, Linear and Complex Analysis Problem Book, Lecture Notes in Math., 1043, Springer, Berlin, 1984, 625–627.
[3] Mok, N., Uniqueness theorem of Hermitian metrics of seminegative curvature on quotients of bounded symmetric domains, Ann. Math., 125, 1987, 105–152. · Zbl 0616.53040 · doi:10.2307/1971290
[4] Mok, N., Metric Rigidity Theorems on Hermitian Locally Symmetric Spaces, Series in Pure Math., Vol. 6, World Scientific, Singapore, 1989. · Zbl 0912.32026
[5] Mok, N., Rigidity problems on compact quotients of bounded symmetric domains, AMS/IP Studies in Advanced Mathematics, 39, 2007, 201–247. · Zbl 1129.53025
[6] Mok, N. and Tsai, I.-H., Rigidity of convex realizations of irreducible bounded symmetric domains of rank 2, J. Reine Angew. Math., 431, 1992, 91–122. · Zbl 0765.32017
[7] Siu, Y.-T., The complex analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds, Amer. J. Math., 100, 1978, 197–203. · Zbl 0424.53040 · doi:10.2307/2373880
[8] Tsai, I.-H., Rigidity of proper holomorphic maps between symmetric domains, J. Differ. Geom., 37, 1993, 123–160. · Zbl 0799.32027
[9] Tu, Z.-H., Rigidity of proper holomorphic mappings between equidimensional bounded symmetric domains, Proc. Amer. Math. Soc., 130, 2002, 1035–1042. · Zbl 0999.32007 · doi:10.1090/S0002-9939-01-06383-3
[10] Tu, Z.-H., Rigidity of proper holomorphic mappings between nonequidimensional bounded symmetric domains, Math. Z., 240, 2002, 13–35. · Zbl 1020.32010 · doi:10.1007/s002090100353
[11] Tu, Z.-H., Rigidity of proper holomorphic mappings between bounded symmetric domains, Geometric Function Theory in Several Complex Variables, World Sci. Publishing, River Edge, NJ, 2004, 310–316. · Zbl 1076.32015
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