Characterization of certain holomorphic geodesic cycles on quotients of bounded symmetric domains in terms of tangent subspaces

Let Ω be an irreducible bounded symmetric domain and Γ ⊂ Aut(Ω) be a torsion-free discrete group of automorphisms, X:= Ω/Γ. We study the problem of algebro-geometric and differential-geometric characterizations of certain compact holomorphic geodesic cycles S ⊂ X. We treat special cases of the problem, pertaining to a situation in which S is a compact holomorphic curve, and to the case where Ω is a classical domain dual to the hyperquadric. In both cases we consider algebro-geometric characterizations in terms of tangent subspaces. As a consequence we derive effective pinching theorems where certain complex submanifolds S ⊂ X are proven to be totally geodesic whenever their scalar curvatures are pinched between certain computed universal constants, independent of the volume of the submanifold S, giving new examples of the gap phenomenon for the characterization of compact holomorphic geodesic cycles.published_or_final_versio

Extension of germs of holomorphic isometries up to normalizing constants with respect to the Bergman metric

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Nonexistence of proper holomorphic maps between certain classical bounded symmetric domains

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. © 2008 Editorial Office of CAM (Fudan University) and Springer-Verlag Berlin Heidelberg.postprin

On the asymptotic behavior of holomorphic isometries of the Poincaré disk into bounded symmetric domains

In this article we study holomorphic isometries of the Poincare disk into bounded symmetric domains. Earlier we solved the problem of analytic continuation of germs of holomorphic maps between bounded domains which are isometries up to normalizing constants with respect to the Bergman metric, showing in particular that the graph V0 of any germ of holomorphic isometry of the Poincaré disk Δ into an irreducible bounded symmetric domain Ω {double subset} ℂN in its Harish-Chandra realization must extend to an affine-algebraic subvariety V C ⊂ = ℂ×ℂN = ℂN+1, and that the irreducible component of V∩(Δ×Ω) containing V0 is the graph of a proper holomorphic isometric embedding F: Δ → Ω. In this article we study holomorphic isometric embeddings which are asymptotically geodesic at a general boundary point b ∈ δΔ. Starting with the structural equation for holomorphic isometries arising from the Gauss equation, we obtain by covariant differentiation an identity relating certain holomorphic bisectional curvatures to the boundary behavior of the second fundamental form a of the holomorphic isometric embedding. Using the nonpositivity of holomorphic bisectional curvatures on a bounded symmetric domain, we prove that ∥ρ∥ must vanish at a general boundary point either to the order 1 or to the order 1/2 called a holomorphic isometry of the first resp. second kind. We deal with special cases of non-standard holomorphic isometric embeddings of such maps, showing that they must be asymptotically totally geodesic at a general boundary point and in fact of the first kind whenever the target domain is a Cartesian product of complex unit balls. We also study the boundary behavior of an example of holomorphic isometric embedding from the Poincare disk into a Siegel upper half-plane by an explicit determination of the boundary behavior of holomorphic sectional curvatures in the directions tangent to the embedded Poincare disk, showing that the map is indeed asymptotically totally geodesic at a general boundary point and of the first kind. For the metric computation we make use of formulas for symplectic geometry on Siegel upper half-planes. © 2009 Wuhan Institute of Physics and Mathematics.postprin

On the Zariski closure of a germ of totally geodesic complex submanifold on an arithmetic variety

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On Fano manifolds with NEF tangent bundles admitting 1-dimensional varieties of minimal rational tangents

Let X be a Fano manifold of Picard number 1 with numerically effective tangent bundle. According to the principal case of a conjecture of Campana-Peternell's, X should be biholomorphic to a rational homogeneous manifold G/p, where G is a simple Lie group, and P ⊂ G is a maximal parabolic subgroup. In our opinion there is no overriding evidence for the Campana-Peternell Conjecture for the case of Picard number i to be valid in its full generality. As part of a general programme that the author has undertaken with Jun-Muk Hwang to study uniruled projective manifolds via their varieties of minimal rational tangents, a new geometric approach is adopted in the current article in a special case, consisting of (a) recovering the generic variety of minimal rational tangents Cx, and (b) recovering the structure of a rational homogeneous manifold from Cx. The author proves that, when b4(X) = 1 and the generic variety of minimal rational tangents is 1-dimensional, X is biholomorphic to the projective plane ℙ2, the 3-dimensional hyperquadric Q3, or the 5-dimensional Fano homogeneous contact manifold of type G2, to be denoted by K(G2). The principal difficulty is part (a) of the scheme. We prove that Cx ⊂ ℙTx(X) is a rational curve of degrees ≤ 3, and show that d = 1 resp. 2 resp. 3 corresponds precisely to the cases of X = ℙ2 resp. Q3 resp. K(G2). Let K be the normalization of a choice of a Chow component of minimal rational curves on X. Nefness of the tangent bundle implies that K is smooth. Furthermore, it implies that at any point x ∈ X, the normalization Kx of the corresponding Chow space of minimal rational curves marked at x is smooth. After proving that Kx is a rational curve, our principal object of study is the universal family U of K, giving a double fibration ρ : U → K, μ : U → X, which gives ℙ1-bundles. There is a rank-2 holomorphic vector bundle V on K whose projectivization is isomorphic to ρ : U → K. We prove that V is stable, and deduce the inequality d ≤ 4 from the inequality c1 2(V) ≤ 4c2(V) resulting from stability and the existence theorem on Hermitian-Einstein metrics. The case of d = 4 is ruled out by studying the structure of the curvature tensor of the Hermitian-Einstein metric on V in the special case where c1 2(V) = 4c2(V).published_or_final_versio

Recognizing Certain Rational Homogeneous Manifolds of Picard Number 1 from their Varieties of Minimal Rational Tangents

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Holomorphic isometries of Bm into bounded symmetric domains arising from linear sections of minimal embeddings of their compact duals

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Rigidity of irreducible Hermitian symmetric spaces of the compact type under Kähler deformation

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Projective manifolds dominated by abelian varieties

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