125,050 research outputs found
N. Mok - Linearly saturated subvarieties on uniruled projective manifolds: Complex analytic and differential geometry 2017
No full textLet Z and X be uniruled projective manifolds of Picard number 1 such that the respective variety of minimal rational tangents (VMRT) at a general point satisfies a nondegeneracy condition on the second fundamental form. In 2001 Hwang and Mok established the equidimensional Cartan-Fubini extension principle, according to which a germ of VMRT-preserving holomorphic map f : (Z, z0) → (X, x0) must necessarily extend to a biholomorphism F : Z→X. In 2010, Hong and Mok extended this to the nonequidimensional case for germs of holomorphic immersions between uniruled projective manifolds, allowing dim(Z) < dim(X), by proving that f must necessarily extend to a rational map provided that a certain relative version of the nondegeneracy condition on the second fundamental form is satisfied. Very recently, Mok and Zhang developed the theory of geometric substructures by considering germs of complex submanifolds of (S, x0) ↪ (X, x0) and introducing geometric substructures on S by taking intersections of the VMRTs of X with projectivized tangent spaces of S. We introduced a new relative nondegeneracy condition related to thesecond fundamental form and proved the extendibility of the germ (S, x0) to a projective subvariety Y ⊂ X under the assumption that X is uniruled by lines, i.e., by rational curves whose homology classes are the positive generator of H2(X,Z)∼=Z. We achieved this by recoveringYas the image under a tautological map of a certain universal family of chains of minimal rational curves. The existence of the latter family is obtained by means of analytic continuation of the Thullen type for germs of holomorphic substructures
On the admissible pairs of rational homogeneous manifolds of Picard number 1 and geometric structures defined by their varieties of minimal rational tangents
No full textIn a series of works, Jun-Muk Hwang and Ngaiming Mok have developed a geometric theory of uniruled projective manifolds, especially those of Picard Number 1, relying on the study of Varieties of Minimal Rational Tangents (VMRT) from both the algebro-geometric and the G-structure perspectives. Based on this theory, Ngaiming Mok and Jaehyun Hong studied the standard embedding between two Rational Homogeneous Spaces (RHS) associated to long simple roots which are of different dimensions. In this thesis, I consider admissible pairs of RHS (X0, X) of Picard number 1 and locally closed complex submanifolds S ⊂ X inheriting VMRT sub-structures modeled on X0 = G0/P0 ⊂ X = G/P de_ned by taking intersections of VMRT of X with tangent space of S. Moreover, if any such S modeled on (X0, X) is necessarily the image of a standard embedding i : X0 → X, (X0, X) is said to be rigid. In this thesis, it is proved that an admissible pair (X0, X) is rigid whenever X is associated to a long simple root and X0 is non-linear and de_ned by a marked Dynkin sub-diagram. In the case of the pair (S0, S) of compact Hermitian Symmetric Spaces (cHSS), all the admissible pairs (S0, S) are completely classified. Based on this classification, a sufficient condition for the pair (S0, S) to be non-rigid is established through explicitly constructing a submanifold S ⊂ S such that S can never be obtained from the image of any standard embedding i : S0 → S. Besides, the term special pair is coined for those (S0; S) sorted out through classification, and the algebraicity of submanifolds modeled on special pairs is confirmed by checking a modified form of the non-degeneracy condition defined by Hong and Mok is satisfied. However, the question as to whether these special pairs are rigid, as pointed out in this thesis, remains to be investigated. Finally, pairs of hyperquadrics (Q^n, Q^m) are studied separately. Since non-rigidity is trivial, in these cases it is interesting to establish a characterization of the standard embedding i : Q^n→Q^m under some stronger condition. In this thesis, the latter problem is solved in terms of the partial vanishing of second fundamental forms.published_or_final_versionMathematicsDoctoralDoctor of Philosoph
On singularities of generically immersive holomorphic maps between complex hyperbolic space forms
No full textIn 1965, Feder proved using a cohomological identity that any holomorphic immersion t: Pn→Pm between complex projective spaces is necessarily a linear embedding whenever m < 2n. In 1991, Cao-Mok adapted Feder’s identity to study the dual situation of holomorphic immersions between compact complex hyperbolic space forms, proving that any holomorphic immersion f : X→Y from an n-dimensional compact complex hyperbolic space form X into any m-dimensional complex hyperbolic space form Y must necessarily be totally geodesic provided that m < 2n. We study in this article singularity loci of generically injective holomorphic immersions between complex hyperbolic space forms. Under dimension restrictions, we show that the open subset U over which the map is a holomorphic immersion cannot possibly contain compact complex-analytic sub-varieties of large dimensions which are in some sense sufficiently deformable. While in the finite-volume case it is enough to apply the arguments of Cao-Mok, the main input of the current article is to introduce a geometric argument that is completely local. Such a method applies to f: X→Y in which the complex hyperbolic space form X is possibly of infinite volume. To start with we make use of the Ahlfors-Schwarz Lemma, as motivated by recent work of Koziarz-Mok, and reduce the problem to the local study of contracting leafwise holomorphic maps between open subsets of complex unit balls. Rigidity results are then derived from a commutation formula on the complex Hessian of the holomorphic map.postprin
Holomorphic Isometries between irreducible bounded symmetric domains with respect to the Bergman metric
No full textThe study of holomorphic isometries between K¨ahler manifolds with real-analytic potential functions dated back to Bochner and Calabi. Especially, in his seminal work on holomorphic isometries in 1953 in which the diastasis was introduced, Calabi established results on existence, uniqueness and analytic
continuation of holomorphic isometries into PN , 1 ≤ N ≤ ∞, from which one derives analytic continuation of germs of holomorphic isometries f between bounded domains with respect to the Bergman metric, and the question remained as to whether analytic continuation persists across the boundary. In 2012, the author solved the problem of boundary extension in a very general context, proving in particular that Graph(f) extends to an affine algebraic variety provided that Bergman kernels are rational functions, which applies in particular to the case of germs of holomorphic isometries from the complex unit ball Bn
into bounded symmetric domains Ω in their standard embeddings. In 2016 the author published examples
of holomorphic isometric embeddings of higher dimensional complex unit balls into irreducible bounded symmetric domains Ω. Images of such isometries are intersections of Ω with cones of minimal rational curves passing through a vertex lying on Reg(∂Ω). In the case where Ω is a Lie sphere (i.e., a type-IV domain), Chan-Mok classified all holomorphic isometric embeddings of complex unit balls into Ω (the codimension 1 cases being also classified by Uppmeier-Wang-Zhang and Xiao-Yuan). When Ω is an irreducible bounded symmetric domain of rank 2 other than a Lie sphere, Mok-Yang proved the uniqueness of holomorphic isometric embeddings of complex unit balls of maximal admissible dimenion into Ω modulo reparametrization. The proof relies on the use of a “duality principle” leading to the determination of isomorphism types of tangent spaces of images of holomorphic isometric embeddings, the method of reconstruction of uniruled projective varieties by means of varieties of minimal rational tangents (VMRTs) and the construction of essentially smooth neighborhoods of certain minimal rational
curves by means of the “Thickening Lemma” in the recent work of Mok-Zhang on geometric substructures modelled on pairs of VMRTs.
N-Terminal Acetylation-Targeted N-End Rule Proteolytic System: The Ac/N-End Rule Pathway
No full textAlthough N-terminal acetylation (Nt-acetylation) is a pervasive protein modification in eukaryotes, its general functions in a majority of proteins are poorly understood. In 2010, it was discovered that Nt-acetylation creates a specific protein degradation signal that is targeted by a new class of the N-end rule proteolytic system, called the Ac/N-end rule pathway. Here, we review recent advances in our understanding of the mechanism and biological functions of the Ac/N-end rule pathway, and its crosstalk with the Arg/N-end rule pathway (the classical N-end rule pathway).112011Ysciescopuskc
From transcendence to algebraicity: techniques of analytic continuation on bounded symmetric domains and their dual compact Hermitian symmetric spaces
No full textAnalytic continuation is a central issue in Several Complex Vari-ables, starting with the Hartogs Phenomenon. We examine the applicationsof techniques of analytic continuation in Complex Geometry for irreduciblebounded symmetric domains Ω and their dual Hermitian symmetric spaces ofthe compact typeS, and their ramifications to the geometric theory of unir-uled projective manifolds. As a starting point, in the case where rank(S)≥2 we recall a proof of Ochiai’s theorem (1970) for analytic continuation of flatS-structure using Hartogs extension, and its generalization to the Cartan-Fubini extension principle of Hwang-Mok (2001) in the geometric theory ofuniruled projective manifolds basing on varieties of minimal rational tangents(VMRTs). Applying methods of algebraic extension in CR-geometry of Web-ster and Huang, and Ochiai’s theorem, we give the proof of Mok-Ng (2012)that under a nondegeneracy assumption, a germ of measure-preserving holo-morphic mapf: (Ω,λdμΩ; 0)→(Ω,dμΩ; 0)×···×(Ω,dμΩ; 0), wheredμΩdenotes the Bergman volume form andλ >0 is a real constant, is necessarilya totally geodesic diagonal embedding, answering in the affirmative a problemof Clozel-Ullmo stemming from a problem in Arithmetic Dynamics regard-ing Hecke correspondences. The proof involves Alexander’s Theorem for thecomplex unit ballBn,n≥2, in the rank-1 case and a new Alexander-type ex-tension theorem for the case of irreducible bounded symmetric domains Ω ofrank≥2 for germs of holomorphic maps preserving the regular part Reg(∂Ω)of the boundary. In another direction we explain the non-equidimensionalCartan-Fubini extension principle of Hong-Mok (2010) and its application tothe characterization of smooth Schubert varieties in rational homogeneousmanifolds of Picard number 1 (Hong-Mok 2013). Finally, we consider theproblem of analytic continuation of subvarieties of uniruled projective man-ifolds (X,K) equipped with a VMRT-structure (e.g. irreducible Hermitiansymmetric spacesSof the compact type) under the assumption that thesubvariety inherits a sub-VMRT structure by taking intersections of VMRTswith tangent spaces, and establish a principle of analytic continuation (Mok-Zhang 2015) by a parametrized Thullen extension of sub-VMRT structuresalong chains of rational curves
Geometric structures and substructures on uniruled projective manifolds
No full textWith J.-M. Hwang the speaker has developed a geometric theory of uniruled projective manifolds (X) modeled on varieties of minimal rational tangents (mathcal{C}_x(X) subset Bbb PT_x(X)), alias VMRTs. Generalizing works of Hwang-Mok, Hong-Mok considered pairs ((X_0;X)) of uniruled projective manifolds, and established a non-equidimensional Cartan-Fubini Extension Principle (2010) in terms of a certain non-degeneracy condition on the second fundamental form for a pair ((mathcal B subsetmathcal A)) consisting of a VMRT (mathcal A ,) and a linear section (mathcal B ,) of (mathcal A). The latter has led to the characterization of standard embeddings (i: G_0/P_0 hookrightarrow G/P) between rational homogeneous manifolds of Picard number 1 by Hong-Mok (2010) in the long-root and non-linear cases and by Hong-Park (2011) in the short-root cases and in the cases of linear subspaces with identifiable exceptions. The argument therein involving parallel transport of VMRTs has also been applied by Hong-Mok (2013) to establish homological rigidity for certain smooth Schubert cycles.
Recently in a joint work with Y. Zhang we have established a stronger rigidity phenomenon for sub-VMRT structures, where in place of a germ of mapping (f: (X_0;0) o (X;0)) we consider a germ of submanifold ((S;0) subset (X;0)) for a uniruled projective manifold (X) equipped with a minimal rational component (mathcal K). Defining a sub-VMRT structure by taking intersections (mathcal C_x(X) cap Bbb PT_x(S)) we have obtained sufficient conditions for (S) to extend to a rationally saturated projective subvariety (Z subset X). In the rational homogeneous case the method yields a strengthening of the results of Hong-Mok and Hong-Park. For instance, if a germ of submanifold ((S;0) subset (X;0)) inherits by intersecting VMRTs with projectivized tangent subspaces a Grassmann structure of rank (ge 2), then (S) in fact extends to a sub-Grassmannian in its standard embedding
R. Mok A.L. Loveridge, Theories of Labour Market Segmentation, 1979
No full textCasassus-Montero Cecilia. R. Mok A.L. Loveridge, Theories of Labour Market Segmentation, 1979. In: Sociologie du travail, 22ᵉ année n°3, Juillet-septembre 1980. pp. 358-361
Rigidity of certain admissible pairs of rational homogeneous spaces of Picard number 1 which are not of the subdiagram type
No full textRecently, Mok and Zhang (2019) introduced the notion of admissible pairs (X0, X) of rational homogeneous spaces of Picard number 1 and proved rigidity of admissible pairs (X0, X) of the subdiagram type whenever X0 is nonlinear. It remains unsolved whether rigidity holds when (X0, X) is an admissible pair NOT of the subdiagram type of nonlinear irreducible Hermitian symmetric spaces such that (X0, X) is nondegenerate for substructures. In this article we provide sufficient conditions for confirming rigidity of such an admissible pair. In a nutshell our solution consists of an enhancement of the method of propagation of sub-VMRT (varieties of minimal rational tangents) structures along chains of minimal rational curves as is already implemented in the proof of the Thickening Lemma of Mok and Zhang (2019). There it was proven that, for a sub-VMRT structure ω¯¯¯:C(S)→S on a uniruled projective manifold (X,K) equipped with a minimal rational component and satisfying certain conditions so that in particular S is “uniruled” by open subsets of certain minimal rational curves on X, for a “good” minimal rational curve ℓ emanating from a general point x ∈ S, there exists an immersed neighborhood Nℓ of ℓ which is in some sense “uniruled” by minimal rational curves. By means of the Algebraicity Theorem of Mok and Zhang (2019), S can be completed to a projective subvariety Z ⊂ X. By the author’s solution of the Recognition Problem for irreducible Hermitian symmetric spaces of rank ⩾ 2 (2008) and under Condition (F), which symbolizes the fitting of sub-VMRTs into VMRTs, we further prove that Z is the image under a holomorphic immersion of X0 into X which induces an isomorphism on second homology groups. By studying ℂ*-actions we prove that Z can be deformed via a one-parameter family of automorphisms to converge to X0 ⊂ X. Under the additional hypothesis that all holomorphic sections in Γ(X0, Tx∣x0) lift to global holomorphic vector fields on X, we prove that the admissible pair (X0, X) is rigid. As examples we check that (X0, X) is rigid when X is the Grassmannian G(n, n) of n-dimensional complex vector subspaces of W ≅ ℂ2n, n ⩾ 3, and when X0 ⊂ X is the La grangian Grassmannian consisting of Lagrangian vector subspaces of (W, σ) where σ is an arbitrary symplectic form on W.link_to_subscribed_fulltex
Nonexistence of proper holomorphic maps between certain classical bounded symmetric domains
No full textThe 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
- …
