On holomorphic immersions into kähler manifolds of constant holomorphic sectional curvature

We study holomorphic immersions f:X →M from a complex manifoldX into a Kähler manifold of constant holomorphic sectional curvatureM, i.e. a complex hyperbolic space form, a complex Euclidean space form, or the complex projective space equipped with the Fubini-Study metric. ForX compact we show that the tangent sequence splits holomorphically if and only iff is a totally geodesic immersion. ForX not necessarily compact we relate an intrinsic cohomological invariantp(X) onX, viz. the invariant defined by Gunning measuring the obstruction to the existence of holomorphic projective connections, to an extrinsic cohomological invariant(f) measuring the obstruction to the holomorphic splitting of the tangent sequence. The two invariantsp(X) and?(f) are related by a linear map on cohomology groups induced by the second fundamental form. In some cases, especially whenX is a complex surface andM is of complex dimension 4, under the assumption thatX admits a holomorphic projective connection we obtain a sufficient condition for the holomorphic splitting of the tangent sequence in terms of the second fundamental form.     

On the singularity of Quillen metrics

Let π: X → S be a holomorphic map from a compact Kähler manifold (X,g _X ) to a compact Riemann surface S. Let Σ_π be the critical locus of π and let Δ  =  π(Σ_π) be the discriminant locus. Let (ξ, h _ξ) be a holomorphic Hermitian vector bundle on X. We determine the singularity of the Quillen metric on det Rπ_*ξ near Δ with respect to g _X | _TX/S and h _ξ.     

Isotropic Lagrangian submanifolds in the homogeneous nearly Kähler S<Superscript>3</Superscript> × S<Superscript>3</Superscript>

We show that isotropic Lagrangian submanifolds in a 6-dimensional strict nearly Kähler manifold are totally geodesic. Moreover, under some weaker conditions, a complete classification of the J-isotropic Lagrangian submanifolds in the homogeneous nearly Kähler S³ × S³ is also obtained. Here, a Lagrangian submanifold is called J-isotropic, if there exists a function λ, such that g((?h)(v, v, v), Jv) = λ holds for all unit tangent vector v.     

Thom-Sebastiani properties of Kohn-Rossi cohomology of compact connected strongly pseudoconvex CR manifolds

Let X_1 and X_2 be two compact connected strongly pseudoconvex embeddable Cauchy-Riemann(CR) manifolds of dimensions 2m-1 and 2n-1 in C~(m+1)and C~(n+1), respectively. We introduce the ThomSebastiani sum X = X_1 ⊕X_2which is a new compact connected strongly pseudoconvex embeddable CR manifold of dimension 2m+2n+1 in C~(m+n+2). Thus the set of all codimension 3 strongly pseudoconvex compact connected CR manifolds in Cn+1for all n 2 forms a semigroup. X is said to be an irreducible element in this semigroup if X cannot be written in the form X_1 ⊕ X_2. It is a natural question to determine when X is an irreducible CR manifold. We use Kohn-Rossi cohomology groups to give a necessary condition of the above question. Explicitly,we show that if X = X_1 ⊕ X_2, then the Kohn-Rossi cohomology of the X is the product of those Kohn-Rossi cohomology coming from X_1 and X_2 provided that X_2 admits a transversal holomorphic S~1-action.     

Certain condition on the second fundamental form of CR submanifolds of maximal CR dimension of complex hyperbolic space

Studying the condition \({h(FX,Y)-h(X,FY)=g(FX,Y)\eta, 0\ne\eta\in T^\perp(M)}\) on the almost contact structure F and on the second fundamental form h of n-dimensional real submanifolds M of complex hyperbolic space \({\mathbb {CH}^{\frac{n+p}{2}}}\) when their maximal holomorphic tangent subspace is (n ? 1)-dimensional, we obtain the complete classification of such submanifolds M and we characterize certain model spaces in complex hyperbolic space.     

Screen Pseudo-Slant Lightlike Submanifolds of Indefinite Sasakian Manifolds

In this paper, we introduce the notion of screen pseudo-slant lightlike submanifolds of indefinite Sasakian manifolds giving characterization theorem with some non-trivial examples of such submanifolds. Integrability conditions of distributions D ₁, D ₂ and RadTM on screen pseudo-slant lightlike submanifolds of indefinite Sasakian manifolds have been obtained. Further, we obtain necessary and sufficient conditions for foliations determined by above distributions to be totally geodesic. We also study mixed geodesic screen pseudo-slant lightlike submanifolds of indefinite Sasakian manifolds.     

Compact Manifolds Covered by a Torus

Let X be a compact complex manifold which is the image of a complex torus by a holomorphic surjective map A→X. We prove that X is Kähler and that up to a finite étale cover, X is a product of projective spaces by a torus.     

Λ-modules and holomorphic Lie algebroid connections

Let X be a complex smooth projective variety, and G a locally free sheaf on X. We show that there is a one-to-one correspondence between pairs (Λ, Ξ), where Λ is a sheaf of almost polynomial filtered algebras over X satisfying Simpson’s axioms and \( \equiv :Gr\Lambda \to Sym \bullet _{\mathcal{O}_X } \mathcal{G}\) is an isomorphism, and pairs (L, Σ), where L is a holomorphic Lie algebroid structure on \(\mathcal{G}\) and Σ is a class in F ¹ H ²(L, ?), the first Hodge filtration piece of the second cohomology of L.As an application, we construct moduli spaces of semistable flat L-connections for any holomorphic Lie algebroid L. Particular examples of these are given by generalized holomorphic bundles for any generalized complex structure associated to a holomorphic Poisson manifold.     

On the sectionwise connectedness of a contingent

Let X be a real normed space and let f: ? → X be a continuous mapping. Let T _f (t ₀) be the contingent of the graph G(f) at a point (t ₀, f(t ₀)) and let S ⁺ ? (0,∞) × X be the “right” unit hemisphere centered at (0, 0 _X ). We show that

1. 1.
   If dimX < ∞ and the dilation D(f, t ₀) of f at t ₀ is finite then T _f (t ₀) ∩ S ⁺ is compact and connected. The result holds for \(T_f (t_0 ) \cap \overline {S^ + } \) even with infinite dilation in the case f: [0,∞) → X.
    
2. 2.
   If dimX = ∞, then, given any compact set F ? S ⁺, there exists a Lipschitz mapping f: ? → X such that T _f (t ₀) ∩ S ⁺ = F.
    
3. 3.
   But if a closed set F ? S ⁺ has cardinality greater than that of the continuum then the relation T _f (t ₀) ∩ S ⁺ = F does not hold for any Lipschitz f: ? → X.
    

     

Classification of Lagrangian Surfaces of Curvature e in Non-flat Lorentzian Complex Space Form M12（4ε）

It is well known that a totally geodesic Lagrangian surface in a Lorentzian complex space form M12（4ε） of constant holomorphic sectional curvature 4s is of constant curvature 6. A natural question is ＂Besides totally geodesic ones how many Lagrangian surfaces of constant curvature εin M12（46） are there？＂ In an earlier paper an answer to this question was obtained for the case e = 0 by Chen and Fastenakels. In this paper we provide the answer to this question for the case ε≠0. Our main result states that there exist thirty-five families of Lagrangian surfaces of curvature ε in M12（4ε） with ε ≠ 0. Conversely, every Lagrangian surface of curvature ε≠0 in M12（4ε） is locally congruent to one of the Lagrangian surfaces given by the thirty-five families.     

On the finite principal bundles

Let G be a connected linear algebraic group defined over \({\mathbb C}\). Fix a finite dimensional faithful G-module V ₀. A holomorphic principal G-bundle E _G over a compact connected Kähler manifold X is called finite if for each subquotient W of the G-module V ₀, the holomorphic vector bundle E _G (W) over X associated to E _G for W is finite. Given a holomorphic principal G-bundle E _G over X, we prove that the following four statements are equivalent: (1) The principal G-bundle E _G admits a flat holomorphic connection whose monodromy group is finite. (2) There is a finite étale Galois covering \({f: Y \longrightarrow X}\) such that the pullback f*E _G is a holomorphically trivializable principal G-bundle over Y. (3) For any finite dimensional complex G-module W, the holomorphic vector bundle E _G (W) = E ×  ^G W over X, associated to the principal G-bundle E _G for the G-module W, is finite. (4) The principal G-bundle E _G is finite.     

A Generalization of Strassen’s Functional LIL

Let X ₁,X ₂,… be a sequence of i.i.d. mean zero random variables and let S _n denote the sum of the first n random variables. We show that whenever we have with probability one, lim?sup? _n→∞|S _n |/c _n =α ₀<∞ for a regular normalizing sequence {c _n }, the corresponding normalized partial sum process sequence is relatively compact in C[0,1] with canonical cluster set. Combining this result with some LIL type results in the infinite variance case, we obtain Strassen type results in this setting.     

A heat equation on some adic completions of ℚ and ultrametric analysis

For each finite set S of prime numbers there exists a unique completion ? _S of ?, which is a second countable, locally compact and totally disconnected topological ring. This topological ring has a natural ultrametric that allows to define a pseudodifferential operator D ^α and to study an abstract heat equation on the Hilbert space L ²(? _S ). The fundamental solution of this equation is a normal transition function of a Markov process on ? _S . The techniques developed provides a general framework for these kind of problems on different ultrametric groups.     

Degeneracy of holomorphic curves in surfaces

LetX be a complex projective algebraic manifold of dimension 2 and let D₁, ..., D_u be distinct irreducible divisors onX such that no three of them share a common point. Let\(f:{\mathbb{C}} \to X\backslash ( \cup _{1 \leqslant i \leqslant u} D_i )\) be a holomorphic map. Assume thatu ? 4 and that there exist positive integers n₁, ... ,n_u,c such that n_in_J D _i.D_j) =c for all pairsi,j. Thenf is algebraically degenerate, i.e. its image is contained in an algebraic curve onX.     

On the hyperbolicity of flows

Let X be a C~1 vector field on a compact boundaryless Riemannian manifold M(dim M≥2),and A a compact invariant set of X.Suppose that A has a hyperbolic splitting,i.e.,T∧M = E~sX E~u with E~s uniformly contracting and E~u uniformly expanding.We prove that if,in addition,A is chain transitive,then the hyperbolic splitting is continuous,i.e.,A is a hyperbolic set.In general,when A is not necessarily chain transitive,the chain recurrent part is a hyperbolic set.Furthermore,we show that if the whole manifold M admits a hyperbolic splitting,then X has no singularity,and the flow is Anosov.     

Axioms of spheres in lightlike geometry of submanifolds

We prove that if an indefinite Kaehler manifold \(\bar {M}\) with lightlike submanifolds satisfies the axioms of holomorphic 2r-spheres, axioms of holomorphic 2r-planes, axioms of transversal r-spheres and axioms of transversal r-planes, then it is an indefinite complex space form.     

Grauert’s line bundle convexity,reduction and Riemann domains

We consider a convexity notion for complex spaces X with respect to a holomorphic line bundle L over X. This definition has been introduced by Grauert and, when L is analytically trivial, we recover the standard holomorphic convexity. In this circle of ideas, we prove the counterpart of the classical Remmert’s reduction result for holomorphically convex spaces. In the same vein, we show that if H⁰(X,L) separates each point of X, then X can be realized as a Riemann domain over the complex projective space Pⁿ, where n is the complex dimension of X and L is the pull-back of O(1).     

On canonical embeddings of complex projective spaces in real projective spaces

In this paper, we construct a natural embedding \(\sigma :\mathbb{C}P_\mathbb{R}^{n} \to \mathbb{R}P^{n^2 + 2n} \) of the complex projective space ?P ⁿ considered as a 2n-dimensional, real-analytic manifold in the real projective space \(\mathbb{R}P^{n^2 + 2n} \). The image of the embedding σ is called the ?P ⁿ-surface. To construct the embedding, we consider two equivalent approaches. The first approach is based on properties of holomorphic bivectors in the realification of a complex vector space. This approach allows one to prove that a ?P-surface is a flat section of a Grassman manifold. In the second approach, we use the adjoint representation of the Lie group U(n + 1) and the canonical decomposition of the Lie algebra u(n). This approach allows one to state a gemetric characterization of the canonical decomposition of the Lie algebra u(n). Moreover, we study properties of the embedding constructed. We prove that this embedding determines the canonical Kähler structure on ?P _? ⁿ . In particular, the Fubini-Study metric is exactly the first fundamental form of the embedding and the complex structure on ?P _? ⁿ is completely defined by its second fundamental form; therefore, this embedding is said to be canonical. Moreover, we describe invariant and anti-invariant completely geodesic submanifolds of the complex projective space.     

Derived category of toric varieties with small Picard number

This paper aims to construct a full strongly exceptional collection of line bundles in the derived category D ^b (X), where X is the blow up of ? ^n?r ×? ^r along a multilinear subspace ? ^n?r?1×? ^r?1 of codimension 2 of ? ^n?r ×? ^r . As a main tool we use the splitting of the Frobenius direct image of line bundles on toric varieties.