On the topology of compact smooth three-dimensional Levi-flat hypersurfaces

We study topological conditions that must be satisfied by a compactC ^∞ Levi-flat hypersurface in a two-dimensional complex manifold, as well as related questions about the holonomy of Levi-flat hypersurfaces. As a consequence of our work, we show that no two-dimensional complex manifold admits a subdomain Ω with compact nonemptyC ^∞ boundary such that Ω ? ?².     

Existence of dicritical singularities of Levi-flat hypersurfaces and holomorphic foliations

We study holomorphic foliations tangent to singular real-analytic Levi-flat hypersurfaces in compact complex manifolds of complex dimension two. We give some hypotheses to guarantee the existence of dicritical singularities of these objects. As consequence, we give some applications to holomorphic foliations tangent to real-analytic Levi-flat hypersurfaces with singularities in \(\mathbb {P}^2\).     

Algebraic Levi-Flat Hypervarieties in Complex Projective Space

We study singular real-analytic Levi-flat hypersurfaces in complex projective space. We define the rank of an algebraic Levi-flat hypersurface and study the connections between rank, degree, and the type and size of the singularity. In particular, we study degenerate singularities of algebraic Levi-flat hypersurfaces. We then give necessary and sufficient conditions for a Levi-flat hypersurface to be a pullback of a real-analytic curve in ℂ via a meromorphic function. Among other examples, we construct a nonalgebraic semianalytic Levi-flat hypersurface with compact leaves that is a perturbation of an algebraic Levi-flat variety.     

Boundary value problems and J-complex curves

We give a solution to the problem of filling by a Levi-flat hypersurface for a class of totally real tori in ?² equipped with a certain almost complex structure.     

On Levi-flat hypersurfaces with given boundary in ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Let S ⊂ ℂ ⁿ be a compact connected 2-codimensional submanifold. If n ⩾ 3, essentially local conditions and the assumption: every complex point of S is elliptic imply the existence of a projection in ℂ ⁿ of a Levi-flat (2n−1)-subvariety whose boundary is S (Dolbeault, Tomassini, Zaitsev, 2005). We extend the result when S is homeomorphic to a sphere and has one hyperbolic point. For n = 2 many results are known since the 1980’s and a new result with a very technical hypothesis is announced. Dedicated to Professor LU QiKeng on the occasion of his 80th birthday     

A contractible Levi-flat hypersurface which is a determining set for pluriharmonic functions

We find a real analytic Levi-flat hypersurface in C² containing a bounded contractible domain which is a determining set for pluriharmonic functions.     

Q-Complete domains with corners in {\mathbb{P}^n} and extension of line bundles

We show that if a compact set X in ${\mathbb P^n}$ is laminated by holomorphic submanifolds of dimension q, then ${\mathbb P^n{\setminus}X}$ is (q + 1)-complete with corners. Consider a manifold U, q-complete with corners. Let ${\mathcal N}$ be a holomorphic line bundle in the complement of a compact in U. We study when ${\mathcal N}$ extends as a holomorphic line bundle in U. We give applications to the non existence of some Levi-flat foliations in open sets in ${\mathbb P^n}$ . The results apply in particular when U is a Stein manifold of dimension n ≥ 3, then every holomorphic line bundle in the complement of a compact extends holomorphically to U.     

Levi-Flat Hypersurfaces Tangent to Projective Foliations

Let M?? ⁿ be a singular real-analytic Levi-flat hypersurface tangent to a codimension-one holomorphic foliation \(\mathcal{F}\) on ? ⁿ . For n≥3, we give sufficient conditions to guarantee the existence of degenerate singularities in M, (in the sense of Segre varieties) and as a consequence we prove that \(\mathcal{F}\) can be defined by a global closed meromorphic 1-form.     

Compact Kähler manifolds with nonpositive bisectional curvature

Let (M ⁿ , g) be a compact Kähler manifold with nonpositive bisectional curvature. We show that a finite cover is biholomorphic and isometric to a flat torus bundle over a compact Kähler manifold N ^k with c ₁ <  0. This confirms a conjecture of Yau. As a corollary, for any compact Kähler manifold with nonpositive bisectional curvature, the Kodaira dimension is equal to the maximal rank of the Ricci tensor. We also prove a global splitting result under the assumption of certain immersed complex submanifolds.     

Real quadrics in C n , complex manifolds and convex polytopes

In this paper, we investigate the topology of a class of non-Kähler compact complex manifolds generalizing that of Hopf and Calabi-Eckmann manifolds. These manifolds are diffeomorphic to special systems of real quadrics C ⁿ which are invariant with respect to the natural action of the real torus (S ¹) ⁿ onto C ⁿ . The quotient space is a simple convex polytope. The problem reduces thus to the study of the topology of certain real algebraic sets and can be handled using combinatorial results on convex polytopes. We prove that the homology groups of these compact complex manifolds can have arbitrary amount of torsion so that their topology is extremely rich. We also resolve an associated wall-crossing problem by introducing holomorphic equivariant elementary surgeries related to some transformations of the simple convex polytope. Finally, as a nice consequence, we obtain that affine non-Kähler compact complex manifolds can have arbitrary amount of torsion in their homology groups, contrasting with the Kähler situation.     

Real hypersurfaces of a complex space form

In this paper we are interested in obtaining a condition under which a compact real hypersurface of a complex projective space CP ⁿ is a geodesic sphere. We also study the question as to whether the characteristic vector field of a real hypersurface of the complex projective space CP ⁿ is harmonic, and show that the answer is in negative.     

L 2-\overline{\partial}-Cohomology groups of some singular complex spaces

Let X be a pure n-dimensional (where n≥2) complex analytic subset in ? ^N with an isolated singularity at 0. In this paper we express the L ²-(0,q)- $\overline{\partial}$ -cohomology groups for all q with 1≤q≤n of a sufficiently small deleted neighborhood of the singular point in terms of resolution data. We also obtain identifications of the L ²-(0,q)- $\overline{\partial}$ -cohomology groups of the smooth points of X, in terms of resolution data, when X is either compact or an open relatively compact complex analytic subset of a reduced complex space with finitely many isolated singularities.     

Almost complex structures on quaternion-Kähler manifolds and inner symmetric spaces

We prove that compact quaternionic-Kähler manifolds of positive scalar curvature admit no almost complex structure, even in the weak sense, except for the complex Grassmannians Gr₂(?ⁿ⁺²). We also prove that irreducible inner symmetric spaces M ⁴ⁿ of compact type are not weakly complex, except for spheres and Hermitian symmetric spaces.     

SVEP and compact perturbations

Let H be a complex separable infinite dimensional Hilbert space. In this paper, we prove that an operator T acting on H is a norm limit of those operators with single-valued extension property (SVEP for short) if and only if T^?, the adjoint of T, is quasitriangular. Moreover, if T^? is quasitriangular, then, given an ε>0, there exists a compact operator K on H with ‖K‖<ε such that T+K has SVEP. Also, we investigate the stability of SVEP under (small) compact perturbations. We characterize those operators for which SVEP is stable under (small) compact perturbations.     

Chebyshev Polynomials on Compact Sets

In connection with a problem of H. Widom it is shown that if a compact set K on the complex plane contains a smooth Jordan arc on its outer boundary, then the minimal norm of monic polynomials of degree n?=?1,2,... is at least (1?+?β)cap(K) ⁿ with some β?>?0, where cap(K) ⁿ would be the theoretical lower bound. It is also shown that the rate (1?+?o(1))cap(K) ⁿ is possible only for compact for which the unbounded component of the complement is simply connected. A related result for sets lying on the real line is also proven.     

A Runge Approximation Theorem for Pseudo-Holomorphic Maps

The Runge approximation theorem for holomorphic maps is a fundamental result in complex analysis, and, consequently, many works have been devoted to extend it to other spaces (e.g. maps between certain algebraic varieties or complex manifolds). This article presents such a result for pseudo-holomorphic maps from a compact Riemann surface to a compact almost-complex manifold M, given that the manifold M admits many pseudo-holomorphic maps from ${\mathbb {C}{\rm P}^1}$ which can be thought of as local approximations of the Laurent expansion az +br ²/z. This result specializes to some compact algebraic varieties (e.g. rationally connected projective varieties). An application to Lefschetz fibrations is presented.     

Real minimal hypersurfaces in complex two-plane Grassmannians

We give a pinching condition for compact minimal hypersurfaces in complex two-plane Grassmannians G ₂(? ^m+2) in terms of sectional curvature and the squared norm of the shape operator.     

Compact open sets in dual spaces and projections in group algebras of [FC]? groups

The structure of a compact open set in the dual of an [FC]^? group G, a locally compact group with relatively compact conjugacy classes, is given in terms of certain subsets which arise somewhat naturally. The support in the dual of a projection in L ¹(G) is a compact open set. Therefore, knowledge of the structure of such sets helps in identifying and constructing projections. We describe explicitly the compact open sets and construct projections for some illustrative examples.     

Properties of P-sets and trapped compact convex sets

New properties of P-sets, which constitute a large class of convex compact sets in ? ⁿ that contains all convex polyhedra and strictly convex compact sets, are obtained. It is shown that the intersection of a P-set with an affine subspace is continuous in the Hausdorff metric. In this theorem, no assumption of interior nonemptiness is made, unlike in other known intersection continuity theorems for set-valued maps. It is also shown that if the graph of a set-valued map is a P-set, then this map is continuous on its entire effective set rather than only on the interior of this set. Properties of the so-called trapped sets are also studied; well-known Jung’s theorem on the existence of a minimal ball containing a given compact set in ? ⁿ is generalized. As is known, any compact set contains n + 1 (or fewer) points such that any translation by a nonzero vector takes at least one of them outside the minimal ball. This means that any compact set is trapped in the minimal ball. Compact sets trapped in any convex compact sets, rather than only in norm bodies, are considered. It is shown that, for any compact set A trapped in a P-set M ? ? ⁿ , there exists a set A ⁰ ? A trapped in M and containing at most 2n elements. An example of a convex compact set M ? ? ⁿ for which such a finite set A ⁰ ? A does not exist is given.     

Covering properties on X2⧹Δ, W-sets,and compact subsets of Σ-products

Let Δ ? X¹ be the diagonal. In the first part of this paper, we show that a compact space X is Corson compact (resp., Eberlein compact; compact metric) if and only if X²?Δ is metalindelöf (resp., σ-metacompact; paracompact). In the second part of the paper, we investigate the notion of a W-set in a space X, which is defined in terms of an infinite game. We show that a compact space X is Corson compact if and only if X has a W-set diagonal, and that a compact scattered space X is strong Eberlein compact if and only if each point of X is a W-set in X.