Study of affine and projective structures on foliated complex manifolds.
problem Formalizing and analyzing affine and projective structures on foliations.
method Formalizing concepts, providing local normal forms, proving index formulae, classifying structures.
result Compact algebraic manifolds of even dimension do not admit foliated projective structures.
Study on complex tori foliations and flat geometries.
problem Understanding turbulent foliations on compact complex tori.
method Defined and analyzed smooth turbulent foliations on compact complex tori.
result All transversely holomorphic Cartan geometries are flat.
Classifies actions on complex space forms with Lagrangian orbits.
problem Classifying actions on complex space forms with Lagrangian orbits.
method Classifies holomorphic isometric actions on complex space forms.
result Only examples are Lagrangian affine subspace foliations of complex Euclidean spaces and Lagrangian horocycle foliations of complex hyperbolic spaces.
Study foliations from complex ball to another via harmonic maps.
problem Rigidity of complex ball quotients.
method Lattice-equivariant harmonic map of small rank.
result Rigidity of complex ball quotients proven.
Survey on holomorphic structures on complex manifolds.
problem Holomorphic foliations with transverse holomorphic Cartan geometries.
method Analyzes G-structures and Cartan geometries on compact complex manifolds.
result Foliated holomorphic Cartan geometries on compact complex manifolds.
Classifies foliations of complex and quaternionic projective spaces.
problem Classifying isoparametric foliations of complex and quaternionic projective spaces.
method Investigating projections of inhomogeneous isoparametric foliations of the 31-sphere under Hopf fibrations.
result Solved the last remaining open cases in the classification.
Complex foliation cohomology is shown to be simple.
problem Analyzing cohomology of a specific complex foliation.
method Explicitly computed the Dolbeault cohomology in degree 1.
result Foliated Dolbeault cohomology in degree 1 is isomorphic to C.
In this paper, we study stability for harmonic foliations on locally conformal Kähler manifolds with complex leaves. We also discuss instability for harmonic foliations on compact submanifolds immersed in Euclidean spaces and compact homogeneous spaces.
The study introduces new foliations and structures on complex manifolds.
problem Understanding transverse Kähler structures on complex manifolds.
method Introducing holomorphic foliations and developing differential graded and bigraded algebras.
result Obtains quasi-isomorphic complexes to de Rham and Dolbeault complexes, similar to compact Kähler manifolds.
Motivated by Gray's work on tube formulae for complex submanifolds of complex projective space equipped with the Fubini-Study metric, Riemannian foliations of projective space are studied. We prove that there are no complex Riemannian foliations of any open subset of Pn of codimension one. As a consequence …
Irreducible isoparametric foliations of arbitrary codimension q on complex projective spaces CP^n are classified, except if n=15 and q=1. Remarkably, there are noncongruent examples that pull back under the Hopf map to congruent foliations on the sphere. Moreover, there exist many inhomogeneous isoparametric foliations…
Study complex foliations using D-modules and Lie subalgebras.
problem Measure irregularity of complex foliations.
method Use D-modules and Lie subalgebras to associate integers.
result Integers measure the irregularity of foliations.
Survey on the topology of singular foliations in complex 2-space.
problem Understanding the topology of singular foliations in complex 2-space.
method Overview and survey of existing research.
result Overview of current knowledge on foliation singularities.
Explains complex analytic invariants of vector fields and foliations.
problem Integrating theories of singular varieties and foliations.
method Expository discussion of invariants.
result Introduces connections between complex analytic singular varieties and foliations.
5 homogeneous convex foliations of degree 4 found on complex projective plane.
problem Classifying reduced convex foliations on complex projective plane.
method Analyzing homogeneous foliations of degree 4.
result 5 homogeneous convex foliations of degree four identified.
14 homogeneous convex foliations of degree 5 found on complex projective plane.
problem Classifying homogeneous convex foliations of degree 5 on the complex projective plane.
method Analyzing properties of specific foliations and using automorphisms of PC2 to establish results. result Every reduced convex foliation of degree 5 is linearly conjugated to one of two specific foliations.
Study of Riemannian foliations with bounded geometry, proving leafwise Hodge decomposition.
problem Understanding Riemannian foliations with bounded geometry.
method Leafwise Hodge decomposition and associated smoothing operators.
result Extension of Novikov differential complex to leafwise version.
We prove that, up to isometric congruence, there are exactly 2n+1 homogeneous polar foliations of the complex hyperbolic space. We also give an explicit description of each of these foliations.
Holomorphic foliations on complex manifolds without invariant complete intersections.
problem Holomorphic foliations on complex manifolds without invariant complete intersections.
method Study of holomorphic normal bundle properties and their implications on invariant sets.
result If the holomorphic normal bundle is Griffiths positive, the foliation does not admit a compact invariant set that is a complete intersection of k smooth real hypersurfaces. This paper deals with the question of analytic continuation of holonomy germs of holomorphic foliations. We prove that for a quasi-minimal Riccati foliation of the complex projective plane, any holonomy germ of the foliation between complex projective lines can be analytically continued along a generic Brownian path.
After gluing foliated complex manifolds, we derive a preparation-like theorem for singularities of codimension one foliations and planar vector fields (in the real or complex setting). Without computation, we retrieve and improve results of Levinson-Moser for functions, Dufour-Zhitomirskii for non degenerate codimensio…
The aim of this paper is to describe complex foliations on Kahler surfaces.
The paper proves that complex hypersurfaces in the ball are foliations.
problem Characterizing complete complex hypersurfaces in the ball.
method Analyzing noncritical holomorphic functions and submersions.
result Every complex hypersurface in the ball is part of a nonsingular holomorphic foliation.
The paper proves isomorphisms between two complexes related to singular foliations.
problem Understanding the isomorphisms between two complexes associated with singular foliations.
method Analyzing the quotient map and proving isomorphisms in specific cases.
result Isomorphisms between the complexes of differential forms on the leaf space and basic differential forms on the manifold.
Study of complex diffeomorphisms and their groups to find first integrals for holomorphic foliations.
problem Finding first integrals for holomorphic foliations under irreducibility conditions.
method Investigate groups of complex diffeomorphisms with irreducibility property and apply to foliations.
result Existence of first integrals for foliations under certain irreducibility conditions.
Solves generalized Kazdan-Warner equations on foliated manifolds.
problem Existence and uniqueness of solutions to generalized Kazdan-Warner equations on foliated manifolds.
method Extends theorem to compact foliated manifolds, provides examples of PDEs.
result Solves the transverse Hitchin equation and its generalizations.
The current article studies certain problems related to complex cycles of holomorphic foliations with singularities in the complex plane. We focus on the case when polynomial differential one-form gives rise to a foliation by Riemann surfaces. In this setting, a complex cycle is defined as a nontrivial element of the f…
Study automorphisms of complex foliations in 3D.
problem Computing automorphisms of complex foliations in specific cases.
method Analyzing automorphisms of 3D Reeb components obtained by Hopf construction.
result Almost complete description of leafwise holomorphic automorphisms.
This manuscript is an introduction to the theory of holomorphic foliations on the complex projective plane. Historically the subject has emerged from the theory of ODEs in the complex domain and various attempts to solve Hilbert's 16th Problem, but with the introduction of complex algebraic geometry, foliation theory a…
Equivalences between conformal foliations on Euclidean 3-space, Hermitian structures on Euclidean 4-space, shear-free ray congruences on Minkowski 4-space, and holomorphic foliations on complex 4-space are explained geometrically and twistorially; these are used to show that 1) any real-analytic complex-valued …
Formula for foliations' singularities in complex projective spaces.
problem Counting singularities of foliations on complex projective spaces.
method Global residue formula for logarithmic indices of foliations with isolated singularities.
result Formula for the number of singularities in the complement of the invariant divisor on complex projective spaces.
Holomorphic discs cover a ball in complex space.
problem Covering a ball in complex space with holomorphic discs.
method Showed a nonsingular holomorphic foliation by complete discs.
result The open unit ball in complex space admits a foliation by complete discs.
Classifies foliations on open Kähler manifolds with explicit curvature control.
problem Classifying foliations on open Kähler manifolds.
method Study of infinitesimal model associated to canonical connection for explicit curvature control.
result Complete classification of foliations on irreducible Hermitian symmetric spaces of compact type.
Extends Masur's divergence theorem to complex tori and Kummer surfaces.
problem Establishing uniquely ergodic horizontal foliations for geodesic flows on moduli spaces.
method Defined and calculated horizontal foliations and geodesic flows on moduli spaces of Kähler metrics.
result Proved that horizontal foliations are uniquely ergodic if geodesic flows are recurrent.
Holomorphic foliations found in ball space with unique properties.
problem Finding holomorphic foliations in the ball space.
method Proving existence of nonsingular holomorphic foliations by closed complex hypersurfaces.
result First example of a holomorphic foliation with complete and incomplete leaves.
We obtain a local classification of complex homothetic foliations on Kaehler manifolds by complex curves. This is used to construct almost Kaehler, Ricci-flat metrics subject to additional curvature properties.
We study Riemannian foliations with complex leaves on Kaehler manifolds. The tensor T, the obstruction to the foliation be totally geodesic, is interpreted as a holomorphic section of a certain vector bundle. This enables us to give classification results when the manifold is compact.
Study conformal foliations on Lie groups, finding new families and harmonic morphisms.
problem Classifying conformal foliations on Lie groups with minimal leaves.
method Analyzing left-invariant foliations generated by specific subgroups.
result New multi-dimensional families of Lie groups with conformal foliations.
Classifies foliated complex manifolds using marked fans.
problem Understanding foliated complex manifolds similar to toric varieties.
method Classifies by marked fans and describes cohomology algebras.
result Basic cohomology and Dolbeault cohomology algebras described in terms of marked fans.
Study of flat Legendre transforms on complex projective plane.
problem Characterizing homogeneous foliations with flat dual web.
method Effective criteria for flatness of dual d-web, explicit examples, classification results. result Up to automorphism of P2, there are 11 homogeneous foliations of degree 3 with flat dual web. We study the moduli space of CR-projective complex foliated tori. We describe it in terms of isotropic subspaces of Grassmannian and we show that it is a normal complex analytic space.
It is known that all but finitely many leaves of a measured foliated 2-complex of thin type are quasi-isometric to an infinite tree with at most two topological ends. We show that if the foliation is cooriented, and the associated R-tree is self-similar, then a typical leaf has exactly one topological end. We also cons…
Proves Dolbeault cohomology conjecture for complex nilmanifolds foliated by tori.
problem Computing Dolbeault cohomology of complex nilmanifolds.
method Generalizes previous methods by assuming foliation in toroidal groups.
result Conjecture holds in real dimension up to six.
Study leafwise holomorphic automorphisms of Reeb components.
problem Characterize automorphisms of Reeb components of leafwise complex foliations.
method Analyze Hopf construction and boundary holonomy properties.
result Determine structure of leafwise holomorphic automorphisms.
We study the space of deformations of a smooth foliation of the 5-sphere by complex manifolds
The paper proves residue formulas for logarithmic foliations on non-compact manifolds.
problem Analyzing logarithmic foliations on non-compact complex manifolds.
method Proves Baum-Bott type formula for residue.
result Provides a Poincaré-Hopf type theorem and optimal description for foliations.
We explain an error in our paper "A smooth foliation of the 5-sphere by complex surfaces", Ann. Math 156 (2002), p.915-930.
Quadratic differentials induce spiralling foliations on Riemann surfaces.
problem Understanding the structure of foliations induced by quadratic differentials.
method Introduced a space of measured foliations and used harmonic maps to real trees.
result Any measured foliation is realized by a quadratic differential with second order poles at marked points.