Curves on the Oeljeklaus-Toma manifolds

The Oeljeklaus-Toma manifolds are complex non-Kähler manifolds constructed by Oeljeklaus and Toma from certain number fields and generalizing the Inoue surfaces S _m . We prove that the Oeljeklaus-Toma manifolds contain no compact complex curves.     

Oka maps

We prove that for a holomorphic submersion of reduced complex spaces, the basic Oka property implies the parametric Oka property. It follows that a stratified subelliptic submersion, or a stratified fiber bundle whose fibers are Oka manifolds, enjoys the parametric Oka property.     

Stability of Strongly Gauduchon Manifolds Under Modifications

In our previous works on deformation limits of projective and Moishezon manifolds, we introduced and made crucial use of the notion of strongly Gauduchon metrics as a reinforcement of the earlier notion of Gauduchon metrics. Using direct and inverse images of closed positive currents of type (1,1) and regularization, we now show that compact complex manifolds carrying strongly Gauduchon metrics are stable under modifications. This stability property, known to fail for compact Kähler manifolds, mirrors the modification stability of balanced manifolds proved by Alessandrini and Bassanelli.     

Hodge structures on cohomology algebras and geometry

We study restrictions on cohomology algebras of compact Kähler manifolds, imposed by the presence of a polarized Hodge structure on cohomology groups, compatible with the cup-product, but not depending on the h ^p,q numbers or the symplectic structure. To illustrate the effectiveness of these restrictions, we give a number of examples of compact symplectic manifolds satisfying the formality condition, the Lefschetz property and having commutative or trivial π ₁, but not having the cohomology algebra of a compact Kaehler manifold. We also prove a stability theorem for these restrictions : if a compact Kähler manifold is homeomorphic to a product X × Y, with one summand satisfying b ₁ = 0, then the cohomology algebra of each summand carries a polarized Hodge structure.     

Deformation limits of projective manifolds: Hodge numbers and strongly Gauduchon metrics

This paper is intended to start a series of works aimed at proving that if in a (smooth) complex analytic family of compact complex manifolds all the fibres, except one, are supposed to be projective, then the remaining (limit) fibre must be Moishezon. A new method of attack, whose starting point originates in Demailly’s work, is introduced. While we hope to be able to address the general case in the near future, two important special cases are established here: the one where the Hodge numbers h ^0,1 of the fibres are supposed to be locally constant and the one where the limit fibre is assumed to be a strongly Gauduchon manifold. The latter is a rather weak metric assumption giving rise to a new, rather general, class of compact complex manifolds that we hereby introduce and whose relevance to this type of problems we underscore.     

Properties of manifolds with skew-symmetric torsion and special holonomy

In this paper we provide examples of hypercomplex manifolds which do not carry HKT structures, thus answering a question in Grantcharov and Poon (Comm. Math. Phys. 213 (2000) 19). We also prove that the existence of an HKT structure is not stable under small deformations. Similarly we provide examples of compact complex manifolds with vanishing first Chern class which do not admit a Hermitian structure whose Bismut connection has restricted holonomy in SU(n), thus providing a counter-example to the conjecture in Gutowski et al. (Deformations of generalized calibrations and compact non-Kähler manifolds with vanishing first Chern class, math.DG/0205012, Asian J. Math., to appear). Again we prove that such a property is not stable under small deformations.     

Exact Lagrangians in plumbings

Consider a Stein manifold M obtained by plumbing cotangent bundles of manifolds of dimension greater than or equal to 3 at points. We prove that the Fukaya category of closed exact Spin Lagrangians with vanishing Maslov class in M is generated by the compact cores of the plumbing. As applications, we classify exact Lagrangian spheres in A ₂-Milnor fibres of arbitrary dimension, derive constraints on exact Lagrangian fillings of Legendrian unknots in disk cotangent bundles, and prove that the categorical equivalence given by the spherical twist in a homology sphere is typically not realised by any compactly supported symplectomorphism.     

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.     

Compact complex manifolds bimeromorphic to locally conformally Kähler manifolds

We study compact complex manifolds bimeromorphic to locally conformally Kähler (LCK) manifolds. This is an analogy of studying a compact complex manifold bimeromorphic to a Kähler manifold. We give a negative answer for a question of Ornea, Verbitsky, Vuletescu by showing that there exists no LCK current on blow ups along a submanifold (dim \(\ge 1\)) of Vaisman manifolds. We show that a compact complex manifold with LCK currents satisfying a certain condition can be modified to an LCK manifold. Based on this fact, we define a compact complex manifold with a modification from an LCK manifold as a locally conformally class C (LC class C) manifold. We give examples of LC class C manifolds that are not LCK manifolds. Finally, we show that all LC class C manifolds are locally conformally balanced manifolds.     

Geometry of compact complex manifolds with maximal torus action

We study the geometry of compact complex manifolds M equipped with a maximal action of a torus T = (S ¹) ^k . We present two equivalent constructions that allow one to build any such manifold on the basis of special combinatorial data given by a simplicial fan Σ and a complex subgroup H ? T _? = (?*) ^k . On every manifold M we define a canonical holomorphic foliation F and, under additional restrictions on the combinatorial data, construct a transverse Kähler form ω _F . As an application of these constructions, we extend some results on the geometry of moment-angle manifolds to the case of manifolds M.     

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.     

Higher dimensional knot spaces for manifolds with vector cross products

Vector cross product structures on manifolds include symplectic, volume, G₂- and Spin(7)-structures. We show that the knot spaces of such manifolds have natural symplectic structures, and relate instantons and branes in these manifolds to holomorphic disks and Lagrangian submanifolds in their knot spaces.For the complex case, the holomorphic volume form on a Calabi-Yau manifold defines a complex vector cross product structure. We show that its isotropic knot space admits a natural holomorphic symplectic structure. We also relate the Calabi-Yau geometry of the manifold to the holomorphic symplectic geometry of its isotropic knot space.     

Suites d’Applications Méromorphes Multivaluées et Courants Laminaires

Let F_n: X₁ → X₂ be a sequence of (multivalued) meromorphic maps between compact Kähler manifolds. We study the asymptotic distribution of preimages of points by F_n and, for multivalued self-maps of a compact Riemann surface, the asymptotic distribution of repelling fixed points. Let (Z_n) be a sequence of holomorphic images of ?^s in a projective manifold. We prove that the currents, defined by integration on Z_n, properly normalized, converge to currents which satisfy some laminarity property. We also show this laminarity property for the Green currents, of suitable bidimensions, associated to a regular polynomial automorphism of ?^k or an automorphism of a projective manifold.     

REDUCTIVE COMPACT HOMOGENEOUS CR MANIFOLDS

We define and investigate a class of compact homogeneous CR manifolds, that we call $ \mathfrak{n} $ -reductive. They are orbits of minimal dimension of a compact Lie group K ₀ in algebraic affine homogeneous spaces of its complexification K. For these manifolds we obtain canonical equivariant fibrations onto complex flag manifolds, generalizing the Hopf fibration $ {S^3}\to \mathbb{C}{{\mathbb{P}}^1} $ . These fibrations are not, in general, CR submersions, but satisfy the weaker condition of being CR-deployments; to obtain CR submersions we need to strengthen their CR structure by lifting the complex stucture of the base.     

W v paths on the torus

We say that a cell complex has theW _v property provided any two vertices can be joined by a path that never returns to a facet once it leaves it. The boundaries of convex polytopes have been shown to have theW _v property for polytopes of dimensions at most 3. We extend the three-dimensional result to polyhedral maps on the torus by showing that all such maps have theW _v property. We also show that theW _v property for all polyhedral maps on manifolds of a given genus is equivalent to the property that for all maps on manifolds of that genus each two faces lie in a subcomplex that is a cell.     

Proper Modifications of Generalized <Emphasis Type="Italic">p</Emphasis>-Kähler Manifolds

In this paper, we consider a proper modification \(f : \tilde{M} \rightarrow M\) between complex manifolds, and study when a generalized p-Kähler property goes back from M to \(\tilde{M}\). When f is the blow-up at a point, every generalized p-Kähler property is conserved, while when f is the blow-up along a submanifold, the same is true for \(p=1\). For \(p=n-1\), we prove that the class of compact generalized balanced manifolds is closed with respect to modifications, and we show that the fundamental forms can be chosen in the expected cohomology class. We also get some partial results in the non-compact case; finally, we end the paper with some examples of generalized p-Kähler manifolds.     

Transgression on hyperkähler manifolds and generalized higher torsion forms

We propose a generalization of the Hodge dd_c-lemma to the case of hyperk?hler manifolds. As an application we derive a global construction of the fourth order transgression of the Chern character forms of hyperholomorphic bundles over compact hyperk?hler manifolds. In Section 3 we consider the fourth order transgression for the infinite-dimensional bundle arising from local families of hyperk?hler manifolds. We propose a local construction of the fourth order transgression of the Chern character form. We derive an explicit expression for the arising hypertorsion differential form. Its zero-degree part may be expressed in terms of the Laplace operators defined on the fibres of the local family.     

Stable bundles on hypercomplex surfaces

A hypercomplex manifold is a manifold equipped with three complex structures I, J, K satisfying the quaternionic relations. Let M be a 4-dimensional compact smooth manifold equipped with a hypercomplex structure, and E be a vector bundle on M. We show that the moduli space of anti-self-dual connections on E is also hypercomplex, and admits a strong HKT metric. We also study manifolds with (4,4)-supersymmetry, that is, Riemannian manifolds equipped with a pair of strong HKT-structures that have opposite torsion. In the language of Hitchin’s and Gualtieri’s generalized complex geometry, (4,4)-manifolds are called “generalized hyperkähler manifolds”. We show that the moduli space of anti-self-dual connections on M is a (4,4)-manifold if M is equipped with a (4,4)-structure.     

Une caractérisation des variétés complexes compactes parallélisables admettant des structures affines

We classify complex compact parallelizable manifolds which admit flat torsion free holomorphic affine connections. We exhibit complex compact manifolds admitting holomorphic affine connections, but no flat torsion free holomorphic affine connections. To cite this article: S. Dumitrescu, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

A note on compact Kähler manifolds with nef tangent bundles

In this paper, we study the compact Kähler manifolds whose tangent bundles are numerically effective and whose anti-Kodaira dimensions are equal to one. LetX be a compact Kähler manifold with nef tangent bundle and semiample anti-canonical bundle. We prove that κ(?K _X )=1 if and only if there exists a finite étale coverY→X such thatY??¹×A, whereA is a complex torus. As a consequence, we are able to improve upon a result of T. Fujiwara [3, 4].