Deformation rigidity of the rational homogeneous space associated to a long simple root. (English) Zbl 1008.32012

Hwang, Jun-Muk; Mok, Ngaiming

Ann. Sci. Éc. Norm. Supér. (4) 35, No. 2, 173-184 (2002).

Summary: As a continuation of our previous works [Jun-Muk Hwang, J. Reine Angew. Math. 486, 153-163 (1997; Zbl 0876.53030), the authors, Invent. Math. 131, No. 2, 393-418 (1998; Zbl 0902.32014)] we study the conjecture on the rigidity under Kähler deformation of the complex structure of rational homogeneous spaces \(G/P\) of Picard number 1, confirming its validity whenever \(G/P\) is associated to a long simple root. For these rational homogeneous spaces the minimal \(G\)-invariant holomorphic distribution \(D\) is spanned by varieties of minimal rational tangents, and, excepting the symmetric and the contact cases, the complex structure of \(G/P\) is completely determined by the nilpotent symbol algebra of the weak derived differential system of \(D\). The problem is reduced, in a sense, to the invariance of this nilpotent symbol algebra under Kähler deformation. In our earlier works in relation to the question of the integrability of distributions spanned by varieties of minimal rational tangents we have established identities on Lie brackets using integral surfaces arising from pencils of rational curves. In the case on hand, at a point \(o\in G/P\) we prove that the nilpotent symbol algebra at \(o\) is nothing other than the universal Lie algebra generated by \(D_o\) subject to these identities on Lie brackets, by verifying that they correspond to finiteness condition in the Serre presentation of the simple Lie algebra \(G\).

MSC:

32M15	Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
32J27	Compact Kähler manifolds: generalizations, classification
58H15	Deformations of general structures on manifolds

Keywords:

rigidity under Kähler deformation; rational homogeneous spaces; Picard number 1

Citations:

Zbl 0876.53030; Zbl 0902.32014

References:

[1]	Bourbaki N. , Groupes et Algèbres de Lie, Ch. 7-8 , Hermann , Paris , 1975 . MR 453824
[2]	Grothendieck A. , Sur la classification des fibrés holomorphes sur la sphère de Riemann , Amer. J. Math. 79 ( 1957 ) 121 - 138 . MR 87176 | Zbl 0079.17001 · Zbl 0079.17001 · doi:10.2307/2372388
[3]	Hwang J.-M. , Rigidity of homogeneous contact manifolds under Fano deformation , J. Reine Angew. Math. 486 ( 1997 ) 153 - 163 . MR 1450754 | Zbl 0876.53030 · Zbl 0876.53030 · doi:10.1515/crll.1997.486.153
[4]	Hwang J.-M. , Stability of tangent bundles of low dimensional Fano manifolds with Picard number 1 , Math. Ann. 312 ( 1998 ) 599 - 606 . MR 1660263 | Zbl 0932.14025 · Zbl 0932.14025 · doi:10.1007/s002080050237
[5]	Hwang J.-M. , Mok N. , Rigidity of irreducible Hermitian symmetric spaces of the compact type under Kähler deformation , Invent. Math. 131 ( 1998 ) 393 - 418 . MR 1608587 | Zbl 0902.32014 · Zbl 0902.32014 · doi:10.1007/s002220050209
[6]	Hwang J.-M. , Mok N. , Holomorphic maps from rational homogeneous spaces of Picard number 1 onto projective manifolds , Invent. Math. 136 ( 1999 ) 209 - 231 . MR 1681093 | Zbl 0963.32007 · Zbl 0963.32007 · doi:10.1007/s002220050308
[7]	Hwang J.-M. , Mok N. , Varieties of minimal rational tangents on uniruled projective manifolds , in: Schneider M. , Siu Y.-T. (Eds.), Several Complex Variables , MSRI Publications , 37 , Cambridge University Press , 2000 , pp. 351 - 389 . MR 1748609 | Zbl 0978.53118 · Zbl 0978.53118
[8]	Humphreys J. , Introduction to Lie Algebras and Representation Theory , Grad. Texts Math. , 9 , Springer , 1972 . MR 323842 | Zbl 0254.17004 · Zbl 0254.17004
[9]	Kac V. , Infinite Dimensional Lie Algebras , Cambridge University Press , 1990 . MR 1104219 | Zbl 0716.17022 · Zbl 0716.17022
[10]	Kodaira K. , On stability of compact submanifolds of complex manifolds , Amer. J. Math. 85 ( 1963 ) 79 - 94 . MR 153033 | Zbl 0173.33101 · Zbl 0173.33101 · doi:10.2307/2373187
[11]	Kollár J. , Rational Curves on Algebraic Varieties , Ergebnisse Math. 3 Folge , 32 , Springer-Verlag , 1996 . MR 1440180 | Zbl 0877.14012 · Zbl 0877.14012
[12]	Mok N. , Metric Rigidity Theorems on Hermitian Locally Symmetric Manifolds , Series in Pure Math. , 6 , World Scientific , 1989 . MR 1081948 | Zbl 0912.32026 · Zbl 0912.32026
[13]	Morimoto T. , Geometric structures on filtered manifolds , Hokkaido Math. J. 22 ( 1993 ) 263 - 347 . MR 1245130 | Zbl 0801.53019 · Zbl 0801.53019
[14]	Serre J.-P. , Complex Semisimple Lie Algebras , Springer-Verlag , 1987 . MR 914496 | Zbl 0628.17003 · Zbl 0628.17003
[15]	Tanaka N. , On the equivalence problems associated with simple graded Lie algebras , Hokkaido Math. J. 8 ( 1979 ) 23 - 84 . MR 533089 | Zbl 0409.17013 · Zbl 0409.17013
[16]	Yamaguchi K. , Differential systems associated with simple graded Lie algebras , in: Progress in Differential Geometry , Adv. Study Pure Math. , 22 , 1993 , pp. 413 - 494 . MR 1274961 | Zbl 0812.17018 · Zbl 0812.17018