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Let |$Y$| be a smooth hypersurface in a projective irreducible holomorphic symplectic manifold X of dimension 2n. The characteristic foliation |$F$| is the kernel of the symplectic form restricted to Y. In this article, we prove that a generic leaf of the characteristic foliation is dense in Y if Y has positive Beauville–Bogomolov–Fujiki square. [ABSTRACT FROM AUTHOR]

Given a holomorphic singular foliation |${\mathcal {F}}$| of |$({\mathbb {C}}^n,0)$|⁠ , we define |$\textrm {Iso}({\mathcal {F}})$| as the group of germs of biholomorphisms on |$({\mathbb {C}}^n,0)$| preserving |${\mathcal {F}}$|⁠ : |$\textrm {Iso}({\mathcal {F}})\!=\!\lbrace \Phi \in \textrm {Diff}({\mathbb {C}}^n,0)\,|\,\Phi ^*({\mathcal {F}})\!=\!{\mathcal {F}}\rbrace $|⁠. The normal subgroup of |$\textrm {Iso}({\mathcal {F}})$|⁠ , of biholomorphisms sending each leaf of |${\mathcal {F}}$| into itself, will be denoted as |$\textrm {Fix}({\mathcal {F}})$|⁠. The corresponding groups of formal biholomorphisms will be denoted as |$\widehat {\textrm {Iso}}({\mathcal {F}})$| and |$\widehat {\textrm {Fix}}({\mathcal {F}})$|⁠ , respectively. The purpose of this paper will be to study the quotients |$\textrm {Iso}({\mathcal {F}})/\textrm {Fix}({\mathcal {F}})$| and |$\widehat {\textrm {Iso}}({\mathcal {F}})/\widehat {\textrm {Fix}}({\mathcal {F}})$|⁠ , mainly in the case of codimension one foliation. [ABSTRACT FROM AUTHOR]

We prove that a logarithmic foliation corresponding to a generic line arrangement of |$d+1 \geq 3$| lines in the complex plane, with pairwise natural and co-prime residues, is a smooth point of the center set of plane foliations (vector fields) of degree |$d$|⁠. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a sub-family of the usual generalized Wronskians, which we call geometric generalized Wronskians. It is well known that one can test linear dependance of holomorphic functions (of several variables) via the identical vanishing of generalized Wronskians. We show that such a statement remains valid if one tests the identical vanishing only on geometric generalized Wronskians. It turns out that geometric generalized Wronskians allow to define intrinsic objects on projective varieties polarized with an ample line bundle: in this setting, the lack of existence of global functions is compensated by global sections of powers of the fixed ample line bundle. Geometric generalized Wronskians are precisely defined so that their local evaluations on such global sections globalize up to a positive twist by the ample line bundle. We then give three applications of the construction of geometric generalized Wronskians: one in intermediate hyperbolicity and two in foliation theory. In intermediate hyperbolicity, we show the algebraic degeneracy of holomorphic maps from |${\mathbb {C}}^{p}$| to a Fermat hypersurface in |$\textbf {P}^{N}$| of degree |$\delta> (N+1)(N-p)$|⁠ : this interpolates between two well-known results, namely for |$p=1$| (first proved via Nevanlinna theory) and |$p=N-1$| (in which case the Fermat hypersurface is of general type). The first application in foliation theory provides a criterion for algebraic integrability of leaves of foliations: our criterion is not optimal in view of current knowledge, but has the advantage of having an elementary proof. Our second application deals with positivity properties of adjoint line bundles of the form |$K_{\mathcal {F}} + L $|⁠ , where |$K_{\mathcal {F}}$| is the canonical bundle of a regular foliation |$\mathcal {F}$| on a smooth projective variety |$X$|⁠ , and where |$L$| is an ample line bundle on |$X$|⁠. [ABSTRACT FROM AUTHOR]

In this article, we give the structure of codimension one foliations with canonical singularities and numerically trivial canonical class on varieties with klt singularities. Building on recent works of Spicer, Cascini—Spicer and Spicer—Svaldi, we then describe the birational geometry of rank two foliations with canonical singularities and canonical class of numerical dimension zero on complex projective three-folds. [ABSTRACT FROM AUTHOR]

We show that any simply connected topological closed |$4$| -manifold punctured along any compact, totally disconnected tame subset |$\Lambda $| admits a continuum of smoothings, which are not diffeomorphic to any leaf of a |$C^{1,0}$| codimension one foliation on a compact manifold. This includes the remarkable case of |$S^4$| punctured along a tame Cantor set. This is the lowest reasonable regularity for this realization problem. These results come from a new criterion for nonleaves in |$C^{1,0}$| regularity. We also include a new criterion for nonleaves in the |$C^2$| -category. Some of our smooth nonleaves are "exotic", that is, homeomorphic but not diffeomorphic to leaves of codimension one foliations on a compact manifold, being the 1st examples in this class. [ABSTRACT FROM AUTHOR]

We describe the basic cohomology ring of the canonical holomorphic foliation on a moment-angle manifold, LVMB-manifold, or any complex manifold with a maximal holomorphic torus action. Namely, we show that the basic cohomology has a description similar to the cohomology ring of a complete simplicial toric variety due to Danilov and Jurkiewicz. This settles a question of Battaglia and Zaffran, who previously computed the basic Betti numbers for the canonical holomorphic foliation in the case of a shellable fan. Our proof uses an Eilenberg–Moore spectral sequence argument; the key ingredient is the formality of the Cartan model for the torus action on a moment-angle manifold. We develop the concept of transverse equivalence as an important tool for studying smooth and holomorphic foliated manifolds. For an arbitrary complex manifold with a maximal torus action, we show that it is transverse equivalent to a moment-angle manifold and therefore has the same basic cohomology. [ABSTRACT FROM AUTHOR]

We study codimension one holomorphic distributions on the projective three-space, analyzing the properties of their singular schemes and tangent sheaves. In particular, we provide a classification of codimension one distributions of degree at most 2 with locally free tangent sheaves and show that codimension one distributions of arbitrary degree with only isolated singularities have stable tangent sheaves. Furthermore, we describe the moduli space of distributions in terms of Grothendieck's Quot-scheme for the tangent bundle. In certain cases, we show that the moduli space of codimension one distributions on the projective space is an irreducible, nonsingular quasi-projective variety. Finally, we prove that every rational foliation and certain logarithmic foliations have stable tangent sheaves. [ABSTRACT FROM AUTHOR]

This work deals with the topological classification of germs of singular foliations on |$(\mathbb{C}^{2},0)$|⁠. Working in a suitable class of foliations we fix the topological invariants given by the separatrix set, the Camacho–Sad indices, and the projective holonomy representations and we compute the moduli space of topological classes in terms of the cohomology of a new algebraic object that we call group-graph. This moduli space may be an infinite-dimensional functional space but under generic conditions we prove that it has finite dimension and we describe its algebraic and topological structure. [ABSTRACT FROM AUTHOR]

Let |$(M,g)$| be a three-dimensional Riemannian manifold. The goal of the paper is to show that if |$P_{0}\in M$| is a nondegenerate critical point of the scalar curvature, then a neighborhood of |$P_{0}$| is foliated by area-constrained Willmore spheres. Such a foliation is unique among foliations by area-constrained Willmore spheres having Willmore energy less than |$32\pi $|⁠ ; moreover, it is regular in the sense that a suitable rescaling smoothly converges to a round sphere in the Euclidean three-dimensional space. We also establish generic multiplicity of foliations and the 1st multiplicity result for area-constrained Willmore spheres with prescribed (small) area in a closed Riemannian manifold. The topic has strict links with the Hawking mass. [ABSTRACT FROM AUTHOR]

We study the deformation theory of pre-symplectic structures, that is, closed 2-forms of fixed rank. The main result is a parametrization of nearby deformations of a given pre-symplectic structure in terms of an |$L_{\infty }$| -algebra, which we call the Koszul |$L_{\infty }$| -algebra. This |$L_{\infty }$| -algebra is a cousin of the Koszul dg Lie algebra associated to a Poisson manifold. In addition, we show that a quotient of the Koszul |$L_{\infty }$| -algebra is isomorphic to the |$L_{\infty }$| -algebra that controls the deformations of the underlying characteristic foliation. Finally, we show that the infinitesimal deformations of pre-symplectic structures and of foliations are both obstructed. [ABSTRACT FROM AUTHOR]

A foliation is said to admit a foliated contact structure if there is a codimension $$1$$ distribution in the tangent space of the foliation such that the restriction to any leaf is contact. We prove a version of the Weinstein conjecture for Reeb vector fields in the presence of an overtwisted leaf. The result is shown to be sharp. [ABSTRACT FROM AUTHOR]

We introduce the notion of Galois holomorphic foliation on the complex projective space as that of foliations whose Gauss map is a Galois covering when restricted to an appropriate Zariski open subset. First, we establish general criteria assuring that a rational map between projective manifolds of the same dimension defines a Galois covering. Then, these criteria are used to give a geometric characterization of Galois foliations in terms of their inflection divisor and their singularities. We also characterize Galois foliations on P2 admitting continuous symmetries, obtaining a complete classification of Galois homogeneous foliations. [ABSTRACT FROM AUTHOR]

Here we state a conjecture concerning a local version of Brunella's alternative: any codimension one foliation in (C³, 0) without germ of invariant surface has a neighborhood of the origin formed by leaves containing a germ of analytic curve at the origin. We prove the conjecture for the class of codimension one foliations whose reduction of singularities is obtained by blowing-up points and curves of equireduction and such that the final singularities are free of saddle-nodes. The concept of "partial separatrix" for a given reduction of singularities has a central role in our argumentations, as well as the quantitative control of the generic Camacho-Sad index in dimension three. The "nodal components" are the only possible obstructions to get such germs of analytic curves. We use the partial separatrices to push the leaves near a nodal component towards compact dicritical divisors, finding in this way the desired analytic curves. [ABSTRACT FROM AUTHOR]

We establish a correspondence between simplicial fans, not necessarily rational, and certain foliated compact complex manifolds called LVMB-manifolds (named after López de Medrano, Verjovsky, Meersseman, Bosio-manifolds). In the rational case, Meersseman and Verjovsky have shown that the leaf space is the usual toric variety. We compute the basic Betti numbers of the foliation for shellable fans. When the fan is in particular polytopal, we prove that the basic cohomology of the foliation is generated in degree 2. We give evidence that the rich interplay between convex and algebraic geometries embodied by toric varieties carries over to our nonrational construction. In fact, our approach unifies rational and nonrational cases. [ABSTRACT FROM AUTHOR]

In this article, we introduce the topological study of codimension-1 foliations which admit contact structures on the leaves. A parametric existence h-principle for foliated contact structures is provided for any cooriented foliation in a closed oriented four-fold. [ABSTRACT FROM AUTHOR]

We introduce a new elliptic operator on null hypersurfaces of 4D Lorentzian manifolds. This operator depends on the first and second fundamental forms of the sections of a foliation of the null hypersurface and its novelty originates from its covariant transformation under change of foliation. It thus provides at any point an elliptic structure intimately connected with the geometry of the null hypersurface, independent of the choice of a specific section through that point. No symmetries or other conditions are imposed on the metric. The spectral properties of this elliptic operator are relevant to the evolution of the wave equation, and in particular, the existence of conservation laws along null hypersurfaces. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the structure of codimension 1 singular holomorphic foliations on ℂ³, 0 without invariant germs of analytic surface. We focus on the socalled CH-foliations, that is, foliations without saddle nodes in 2D sections. Considering a reduction of singularities, we detect the possible existence of "nodal components", which are a higher-dimensional version of the nodal separators in dimension 2. If the foliation is without nodal components, we prove that all the leaves in a neighborhood of the origin contain at least one germ of analytic curve at the origin. We also study the structure of nodal components for the case of "Relatively Isolated CH-foliations" and we show that they cut the dicritical components or they exit the origin through a non-compact invariant curve. This allows us to give a precise statement of a local version of Brunella's alternative: if we do not have an invariant surface, all the leaves contain a germ of analytic curve or it is possible to detect the nodal components in the generic points of the singular curves before doing the reduction of singularities. [ABSTRACT FROM AUTHOR]

Polarized and G-polarized CR manifolds are smooth manifolds endowed with a double structure: a real foliation (given by the action of a Lie group G in the G-polarized case) and a transverse CR distribution (E,J). Polarized means that (E,J) is roughly speaking invariant by . Both structures are therefore linked up. The interplay between them gives to polarized CR-manifolds a very rich geometry. In this paper, we study the properties of polarized and G-polarized manifolds, putting special emphasis on the existence of a local moduli space. [ABSTRACT FROM PUBLISHER]

We consider the stability of Sasaki-extremal metrics under deformations of the transversal complex structure of the foliation induced by the Reeb vector field ξ. Let g be a Sasaki-extremal metric on M, G a compact connected subgroup of the automorphism group of the Sasaki structure, and suppose the reduced scalar curvature satisfies . And consider a G-equivariant deformation of the transversely holomorphic foliation preserving as a smooth foliation. Provided the Futaki invariant relative to G of g is nondegenerate, the existence of Sasaki-extremal metrics is preserved under small variations of and of the Reeb vector in the center of . If G=T⊆Aut(g,ξ) is a maximal torus, the nondegeneracy of the Futaki invariant is automatic. So such deformations provide the easiest examples.When the initial metric g is Sasaki–Einstein and G=T⊆Aut(g,ξ) is a maximal torus a slice of the above family of Sasaki-extremal metrics is Sasaki–Einstein. Thus, for each there is a so that the Sasaki-extremal metric with Reeb vector field ξt is Sasaki–Einstein. We apply this to deformations of toric 3-Sasaki 7-manifolds to obtain new families of Sasaki–Einstein metrics on these manifolds, which are deformations of 3-Sasaki metrics. [ABSTRACT FROM PUBLISHER]

We show that, for any (symmetric) finite generating set of the Torelli group of a closed surface, the probability that a random word is not pseudo-Anosov decays exponentially in the length of the word. [ABSTRACT FROM AUTHOR]

Whether every hyperbolic 3-manifold admits a tight contact structure or not is an open question. Many hyperbolic 3-manifolds contain taut foliations and taut foliations in turn can be perturbed to tight contact structures. The first examples of hyperbolic 3-manifolds without taut foliations were constructed by Roberts, Shareshian, and Stein, and infinitely many of them do not even admit essential laminations as shown by Fenley. In this paper, we construct tight contact structures on a family of 3-manifolds including these examples. These contact structures are described by contact surgery diagrams and their tightness is proved using the contact invariant in Heegaard Floer homology. [ABSTRACT FROM PUBLISHER]

Given a truncated incoming null cone and a truncated outgoing null cone intersecting at a two sphere S with smooth characteristic initial data, a theorem of Rendall shows that the vacuum Einstein equations can be solved in a small neighborhood of S in the future of S. We show that in fact the vacuum Einstein equations can be solved in a neighborhood in the future of the cones, as long as the constraint equations are initially satisfied on the null cones. The proof is based on energy-type estimates and relies heavily on the null structure of the Einstein equations in the double null foliation. [ABSTRACT FROM PUBLISHER]

We show that a classical result of Gromov in symplectic geometry extends to the context of symplectic foliations, which we regard as an h-principle for (regular) Poisson geometry. Namely, we formulate a sufficient cohomological criterion for a regular bivector to be homotopic to a regular Poisson structure, in the spirit of Haefliger’s criterion for homotoping a distribution to a foliation. We give an example to show that this criterion is not too unsharp. [ABSTRACT FROM PUBLISHER]

We give here an explicit example of an algebraic family of foliations of which is topologically trivial but not analytically trivial. This example underlines the necessity of some assumptions in Y. Ilyashenko’s rigidity theorem. [ABSTRACT FROM PUBLISHER]

In this paper we prove the existence of absolutely dicritical foliations for any configuration of exceptional divisor. [ABSTRACT FROM PUBLISHER]

We show that the constant mean curvature hypersurfaces in Hn+1 spanning the boundary of a star-shaped C1,1 domain in S∞n(Hn+1) give a foliation of Hn+1. We also show that if Γ is a closed codimension-1 C2,α submanifold in S∞n(Hn+1) bounding a unique constant mean curvature hypersurface ΣH in Hn+1 with ∂∞ΣH = Γ for any H ∈ (−1,1), then the constant mean curvature hypersurfaces {ΣH} foliate Hn+1. [ABSTRACT FROM PUBLISHER]

The article discusses the nonisotopic legendrain curves different with the invariants from the topologically nontrivial curves. It notes that the proof of the theorem gives an explanation on the contact manifold in many Legendrian curves. It mentions on the theorem by the constructing curves which is peformed on the sequence curves and results to a sequence of non-isotopic contact structures. It also emphasizes on the integrated dimensional singular foliation.

We classify locally finite invariant measures and orbit closure for the action of the mapping class group on the space of measured laminations on a surface. This classification translates to a classification of measures and orbit closures on the space of quadratic differentials invariant under the horospheric foliation. [ABSTRACT FROM PUBLISHER]