Heat equation and ergodic theorems for Riemann surface laminations

We study the dynamics of possibly singular foliations by Riemann surfaces. The main examples are holomorphic foliations by Riemann surfaces in projective varieties. We introduce the heat equation relative to a positive ${\partial\overline\partial}$ -closed current and apply it to the directed currents associated with Riemann surface laminations possibly with singularities. This permits to construct the heat diffusion with respect to various Laplacians that could be defined almost everywhere with respect to the ${\partial\overline\partial}$ -closed current. We prove two kinds of ergodic theorems for such currents: one associated to the heat diffusion and one of geometric nature close to Birkhoff??s averaging on orbits of a dynamical system. Here the averaging is on hyperbolic leaves and the time is the hyperbolic time. The heat diffusion theorem with respect to a harmonic measure is also developed for real laminations.     

Transfinite diameter notions in ℂ N and integrals of Vandermonde determinants

We provide a general framework and indicate relations between the notions of transfinite diameter, homogeneous transfinite diameter, and weighted transfinite diameter for sets in ? ^N . An ingredient is a formula of Rumely (A Robin formula for the Fekete–Leja transfinite diameter, Math. Ann. 337 (2007), 729–738) which relates the Robin function and the transfinite diameter of a compact set. We also prove limiting formulas for integrals of generalized Vandermonde determinants with varying weights for a general class of compact sets and measures in ? ^N . Our results extend to certain weights and measures defined on cones in ? ^N .     

Frobenius theorem for foliations on singular varieties

We generalize Frobenius singular theorem due to Malgrange, for a large class of codimension one holomorphic foliations on singular analytic subsets of ℂ ^N . This research was partially supported by Pronex.     

Holomorphic vector fields with totally degenerate zeroes

A zero set of a holomorphic vector field is totally degenerate, if the endomorphism of the conormal sheaf induced by the vector field is identically zero. By studying a class of foliations generalizing foliations of C^*-actions, we show that if a projective manifold admits a holomorphic vector field with a smooth totally degenerate zero component,then the manifold is stably birational to that component of the zero set.When the vector field has an isolated totally degenerate zero, we prove that the manifold is rational. This is a special case of Carrell's conjecture.     

Approximately holomorphic geometry and estimated transversality on 2-calibrated manifolds

The notion of 2-calibrated structure, generalizing contact structures, smooth taut foliations, etc., is defined. Approximately holomorphic geometry as introduced by S. Donaldson for symplectic manifolds is extended to 2-calibrated manifolds. An estimated transversality result that enables to study the geometry of such manifolds is presented. To cite this article: A. Ibort, D. Martínez Torres, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

A Strong Oka Principle for Embeddings of Some Planar Domains into ℂ×ℂ∗

Gromov, in his seminal 1989 paper on the Oka principle, introduced the notion of an elliptic manifold and proved that every continuous map from a Stein manifold to an elliptic manifold is homotopic to a holomorphic map. We show that a much stronger Oka principle holds in the special case of maps from certain open Riemann surfaces called circular domains into ?×?^?, namely that every continuous map is homotopic to a proper holomorphic embedding. An important ingredient is a generalization to ?×?^? of recent results of Wold and Forstneri? on the long-standing problem of properly embedding open Riemann surfaces into ?², with an additional result on the homotopy class of the embeddings. We also give a complete solution to a question that arises naturally in Lárusson’s holomorphic homotopy theory, of the existence of acyclic embeddings of Riemann surfaces with abelian fundamental group into 2-dimensional elliptic Stein manifolds.     

Christoffel functions and universality on the boundary of the ball

We establish asymptotics for Christoffel functions, and universality limits, associated with multivariate orthogonal polynomials, on the boundary of the unit ball in ? ^d .     

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 \(\mathbb {P}^2\).     

Foliations with a Kupka component on algebraic manifolds

We consider codimension one holomorphic foliations in complex projective manifolds of dimension at least 3, having a compact Kupka component and represented by integrable holomorphic sections of the bundleTM ^*L, whereL denotes a very ample holomorphic line bundle. We will show that, if the transversal type is not the radial vector field andH ¹ (M,) = 0, then the foliation has a meromorphic first integral.Supported by Conacyt: 3398-E 9307     

Vector Bundles Near Negative Curves: Moduli and Local Euler Characteristic

We study moduli of vector bundles on a two-dimensional neighbourhood Z _k of an irreducible curve ? ? ?¹ with ?² = ?k and give an explicit construction of their moduli stacks. For the case of instanton bundles, we stratify the stacks and construct moduli spaces. We give sharp bounds for the local holomorphic Euler characteristic of bundles on Z _k and prove existence of families of bundles with prescribed numerical invariants. Our numerical calculations are performed using a Macaulay 2 algorithm, which is available for download at http://www.maths.ed.ac.uk/~s0571100/Instanton/.     

The Kernel Bundle of a Holomorphic Fredholm Family

Let 𝒴 be a smooth connected manifold, Σ ? ? an open set and (σ, y) → 𝒫 _y (σ) a family of unbounded Fredholm operators D ? H ₁ → H ₂ of index 0 depending smoothly on (y, σ) ∈ 𝒴 × Σ and holomorphically on σ. We show how to associate to 𝒫, under mild hypotheses, a smooth vector bundle 𝒦 → 𝒴 whose fiber over a given y ∈ 𝒴 consists of classes, modulo holomorphic elements, of meromorphic elements φ with 𝒫 _y φ holomorphic. As applications we give two examples relevant in the general theory of boundary value problems for elliptic wedge operators.     

Partial regularity for homogeneous complex Monge–Ampere equations

In this Note, we establish a new partial regularity theory on certain homogeneous complex Monge–Ampere equations. This partial regularity theory is obtained by studying foliations by holomorphic disks and their relation to these equations. To cite this article: X.X. Chen, G. Tian, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Deformation of holomorphic maps onto the blow-up of the projective plane

Let S be the blow-up of the projective plane at d distinct points and be any surjective holomorphic map from a compact complex manifold S^′. We will show that all deformations of ψ come from automorphisms of S if d?3. The result is optimal in the sense that it is not true if d?2. The strategy of the proof is to use the infinitesimal automorphisms of the web geometry on S arising from the natural foliations of S induced by the pencils of the lines through the blow-up centers.     

Surfaces de la classe VII0 et automorphismes

In this Note we construct surfaces S of class VII containing a global spherical shell starting with a Hénon automorphism H(x,y)=(x²+c−ay,x) of C². For a special choice of the parameter a we show the existence of a non-trivial holomorphic vector field X on S. This contributes to the (open) problem of classifying all compact complex surfaces with a positive dimensional automorphism group. For general results concerning vector fields and foliations on surfaces containing a global spherical shell we refer to our note [2].     

Boundedness for iterated commutators on the mixed norm spaces

This paper studies the iterated commutators on mixed norm spaces L²(?) characterizing the conjugate holomorphic symbols for which the corresponding iterated commutators are bounded by using the Bergman geometry, properties of holomorphic functions and related analysis.     

Existence and stability of foliations by <Emphasis Type="Italic">J</Emphasis>-holomorphic spheres

We study the existence and stability of holomorphic foliations in dimension greater than 4 under perturbations of the underlying almost-complex structure. An example is given to show that, unlike in dimension 4, J-holomorphic foliations are not stable under large perturbations of almost-complex structure.     

The Noether–Fano inequalities for codimension one singular holomorphic foliations

The idea of the proof of the classical Noether–Fano inequalities can be adapted to the domain of codimension one singular holomorphic foliations of the projective space. We obtained criteria for proving that the degree of a foliation on the plane is minimal in the birational class of the foliation and for the non-existence of birational symmetries of generic foliations (except automorphisms). Moreover, we give several examples of birational symmetries of special foliations illustrating our results.      

Difféomorphismes holomorphes Anosov

We classify the holomorphic diffeomorphisms of complex projective varieties with an Anosov dynamics and holomorphic stable and unstable foliations: The variety is finitely covered by a compact complex torus and the diffeomorphism corresponds to a linear transformation of this torus.

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Difféomorphismes holomorphes Anosov