Abstract
We study compact complex submanifolds S of quotient manifolds X = Ω/Γ of irreducible bounded symmetric domains by torsion free discrete lattices of automorphisms, and we are interested in the characterization of the totally geodesic submanifolds among compact splitting complex submanifolds S ⊂ X, i.e., under the assumption that the tangent sequence over S splits holomorphically. We prove results of two types. The first type of results concerns S ⊂ X which are characteristic complex submanifolds, i.e., embedding Ω as an open subset of its compact dual manifold M by means of the Borel embedding, the non-zero (1, 0)-vectors tangent to S lift under a local inverse of the universal covering map π: Ω → X to minimal rational tangents of M. We prove that a compact characteristic complex submanifold S ⊂ X is necessarily totally geodesic whenever S is a splitting complex submanifold. Our proof generalizes the case of the characterization of totally geodesic complex submanifolds of quotients of the complex unit ball Bn obtained by Mok (2005). The proof given here is however new and it is based on a monotonic property of curvatures of Hermitian holomorphic vector subbundles of Hermitian holomorphic vector bundles and on exploiting the splitting of the tangent sequence to identify the holomorphic tangent bundle T S as a quotient bundle rather than as a subbundle of the restriction of the holomorphic tangent bundle T X to S. The second type of results concerns characterization of total geodesic submanifolds among compact splitting complex submanifolds S ⊂ X deduced from the results of Aubin (1978) and Yau (1978) which imply the existence of Kähler-Einstein metrics on S ⊂ X. We prove that compact splitting complex submanifolds S ⊂ X of sufficiently large dimension (depending on Ω) are necessarily totally geodesic. The proof relies on the Hermitian-Einstein property of holomorphic vector bundles associated to T S , which implies that endomorphisms of such bundles are parallel, and the construction of endomorphisms of these vector bundles by means of the splitting of the tangent sequence on S. We conclude with conjectures on the sharp lower bound on dim(S) guaranteeing total geodesy of S ⊂ X for the case of the type-I domains of rank 2 and the case of type-IV domains, and examine a case which is critical for both conjectures, viz. on compact complex surfaces of quotients of the 4-dimensional Lie ball, equivalently the 4-dimensional type-I domain dual to the Grassmannian of 2-planes in ℂ4.
Similar content being viewed by others
References
Aubin T. Equations du type de Monge-Ampère sur les variétés kählériennes compactes. Bull Sci Math, 1978, 102: 63–95
Berger M. Sur les groupes d’holonomie homogènes des variét’es connexion affine et des variétés riemanniennes. Bull Sci Math, 1955, 83: 279–330
Bochner S, Yano K. Curvature and Betti Numbers. Princeton: Princeton University Press, 1953
Borel A. On the curvature tensor of the Hermitian symmetric manifolds. Ann Math, 1960, 71: 508–521
Calabi E, Vesentini E. On compact, locally symmetric Kähler manifolds. Ann Math, 1960, 71: 472–507
Deraux M. Forgetful maps between Deligne-Mostow ball quotients. Geom Dedicata, 2011, 150: 377–389
Griffiths P A. Hermitian differential geometry, Chern classes, and positive vector bundles. In: Global Analysis. Tokyo: University Tokyo Press, 1969, 185–251
Helgason S. Differential geometry, Lie groups, and symmetric spaces. New York-London: Academic Press, 1978
Hwang J-M, Mok N. Varieties of minimal rational tangents on uniruled manifolds. In: Several Complex Variables. MSRI publications, vol. 37. Cambridge: Cambridge University Press, 1999, 351–389
Jahnke P. Submanifolds with splitting tangent sequence. Math Z, 2005, 251: 491–507
Kobayashi S. The first Chern class and holomorphic symmetric tensor fields. J Math Soc Japan, 1980, 32: 325–329
Kobayashi S, Ochiai T. Holomorphic structures modeled after compact hermitian symmetric spaces. In: Manifolds and Lie Groups. Progress in Mathematics, vol. 14. Boston: Birkhäuser, 1981, 207–221
Mok N. Uniqueness theorems of Hermitian metrics of seminegative curvature on quotients of bounded symmetric domains. Ann Math, 1987, 125: 105–152
Mok N. Metric Rigidity Theorems on Hermitian Locally Symmetric Manifolds. Singapore-New Jersey-London-Hong Kong: World Scientific, 1989
Mok N. Characterization of certain holomorphic geodesic cycles on quotients of bounded symmetric domains in terms of tangent subspaces. Compos Math, 2002, 132: 289–309
Mok N. On holomorphic immersions into Kähler manifolds of constant holomorphic sectional curvature. Sci China Ser A, 2005, 48: 123–145
Mok N. Geometric structures and substructures on uniruled projective manifolds. In: Foliation Theory in Algebraic Geometry (Simons Symposia). New York: Springer-Verlag, 2016, 103–148
Mustaţă M, Popa M. A new proof of a theorem of A. Van de Ven. Bull Math Soc Sci Math Roumanie (NS), 1997, 40: 49–55
Satake I. Holomorphic imbeddings of symmetric domains into a Siegel space. Amer J Math, 1965, 90: 425–461
Siu Y-T. Complex-analyticity of Harmonic maps, vanishing and Lefschetz theorems. J Differential Geom, 1982, 17: 55–138
Siu Y-T. Lectures on Hermitian-Einstein Metrics for Stable Bundles and Kähler-Einstein Metrics. Basel: Birkhäuser-Verlag, 1987
Van de Ven A. A Property of Algebraic Varieties in Complex Projective Spaces. Bruxelles: Colloque de géom gl, 1958
Weyl H. The Classical Groups. Princeton: Princeton University Press, 1949
Wolf J A. Fine structure of Hermitian symmetric spaces. In: Symmetric Spaces. Pure and Appl Math, vol. 8. New York: Dekker, 1972, 271–357
Yau S-T. On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I. Comm Pure Appl Math, 1978, 31: 339–411
Zhong J-Q. The degree of strong nondegeneracy of the bisectional curvature of exceptional bounded symmetric domains. In: Several Complex Variables. Boston: Birkhäuser, 1984, 127–139
Acknowledgements
This work was supported by the Research Grants Council of Hong Kong of China (Grant No. 17303814), National Natural Science Foundation of China (Grant No. 11501205) and Science and Technology Commission of Shanghai Municipality (Grant No. 13dz2260400).
Author information
Authors and Affiliations
Corresponding author
Additional information
In memory of Professor LU QiKeng (1927–2015)
Rights and permissions
About this article
Cite this article
Mok, N., Ng, S. On compact splitting complex submanifolds of quotients of bounded symmetric domains. Sci. China Math. 60, 1057–1076 (2017). https://doi.org/10.1007/s11425-016-9033-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-016-9033-0
Keywords
- bounded symmetric domains
- tangent sequence
- splitting complex submanifolds
- varieties of minimal rational tangents
- Kähler-Einstein metrics