Study transverse Dolbeault cohomology for almost complex structures.
problem Understanding cohomology of transverse structures on manifolds.
method Define transverse Dolbeault cohomology, extend transverse complex structure, introduce involutive limit distribution.
result Cohomology spaces of (p,0) for almost complex structures coincide with transverse Dolbeault cohomology.
Paper develops theory of transverse generalized complex structures and proves a key lemma.
problem Proving equivalent conditions to the basic ddJ-lemma. method Describing transverse symplectic structure and relating the lemma to the Lefschetz map.
result Justified approach and proved equivalent conditions to the basic ddJ-lemma. Regular Jacobi structures on line bundles lead to generalized contact bundles.
problem Understanding the relationship between Jacobi structures and generalized contact bundles.
method Investigating weakly regular Jacobi structures and their transverse complex structures.
result Conditions for a pair of a regular Jacobi structure and a transverse complex structure to form a generalized contact structure.
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.
We answer the natural question: when are a regular Poisson structure along with a complex structure transverse to its symplectic leaves induced by generalized complex structure? The leafwise symplectic form and transverse complex structure determine an obstruction class in a certain cohomology, which vanishes if and on…
The study provides obstructions and examples for p-symplectic structures on complex manifolds.
problem Obstructing the existence of p-symplectic structures on compact complex manifolds. method Analyzing properties of (p,p)-transverse forms and 2p-forms. result Found obstructions and examples of p-symplectic structures on compact complex manifolds. Classifies complex Dirac structures with invariants and local structure.
problem Classifying complex Dirac structures.
method Introducing invariants (order, type), proving existence and splitting theorems.
result Pointwise classification and local structure of complex Dirac structures.
The transversal twistor space of a foliation F of an even codimension is the bundle ZF of the complex structures of the fibers of the transversal bundle of F. On ZF, there exists a foliation F' by covering spaces of the leaves of F, and any Bott connection of F produces an ordered pair (I,J) of transversal almost compl…
Proves immediate transversality for conic singularities.
problem Transversality issues in Morse complexes with conic singularities.
method Proves immediate transversality for conic singularities in Morse complexes.
result Immediate transversality holds for conic singularities.
The article details Fukaya's A∞-structure of Morse complexes.
problem Understanding the A∞-structure of Morse complexes.
method Detailed exposition of Fukaya's A∞-structure, emphasizing transversality arguments.
result The A∞-structure is homotopically independent of choices.
We consider manifolds equipped with a foliation F of codimension 4q, and an almost quaternionic structure Q on the transversal bundle of F. After discussing conditions of projectability and integrability of Q, we study the transversal twistor space ZF which, by definition, consists of the…
We prove that a compact smooth 4-manifold admits generalized complex structures of odd type if and only if it has a transversely holomorphic 2-foliation. Consequently, there exist generalized complex structures of odd type on a circle bundle over a closed Seifert fibered 3-manifold.
We study a number of local and global classification problems in generalized complex geometry. In the first topic, we characterize the local structure of generalized complex manifolds by proving that a generalized complex structure near a complex point arises from a holomorphic Poisson structure. In the proof we use a …
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.
The paper explores families of almost complex structures and transverse (p,p)-forms.
problem Understanding and producing families of almost p-Kähler structures.
method Producing families of almost p-Kähler structures (Jt,Ωt) on $\C^3$, $\C^4$, and T6. result The almost complex structures Jt cannot be locally compatible with any symplectic form for teq0. New theorems on Hodge numbers and Kähler structures derived from complex differential forms.
problem Deformation invariance and local stability of Hodge numbers and Kähler structures.
method Using the exponential operator and power series method, the approach focuses on d-closed extensions and foliated cases. result Local stabilities of transversely p-Kähler structures and new theorems on Hodge numbers. Simplified method for L∞ algebra of Dirac structures.
problem Deformation of Dirac structures.
method Simplified method for L∞ algebra. result Canonical L∞-isomorphism of L∞ algebras. Study the geometry of embedding spaces for almost complex manifolds.
problem Investigate the embeddability of compact almost complex manifolds into foliated algebraic varieties.
method Introduce a more general category of universal embedding spaces and elucidate their geometric structure.
result Characterize the integrability locus of integrable almost complex structures.
Study of BGG sequences on foliated manifolds with transverse parabolic geometry.
problem Analysis of BGG sequences on foliated manifolds with transverse parabolic structures.
method Filtered calculus and transversal index theory for filtered manifolds.
result Derived curved BGG sequences for foliated manifolds with transverse parabolic geometry.
New method constructs stable generalized complex structures.
problem Creating stable generalized complex structures.
method Developed Gompf-Thurston symplectic techniques adapted to Lie algebroids, used to construct stable structures from log-symplectic structures.
result Introduced boundary Lefschetz fibrations to obtain stable structures from genus one Lefschetz fibrations.
No projective structure found on foliations of elliptic curves.
problem Existence of transversely projective structures on foliations.
method Analyzing foliations on the product of two elliptic curves.
result Generic nonsingular turbulent foliations do not have transversely projective structures.
We consider a consider the case of a compact manifold M, together with the following data: the action of a compact Lie group H and a smooth H-invariant distribution E, such that the H-orbits are transverse to E. These data determine a natural equivariant differential form with generalized coefficients J(E,X) whose prop…
We study the transverse Poisson structure to adjoint orbits in a complex semi-simple Lie algebra. The problem is first reduced to the case of nilpotent orbits. We prove then that in suitably chosen quasi-homogeneous coordinates the quasi-degree of the transverse Poisson structure is -2. In the particular case of {\emph…
The paper studies Hodge structures on contact manifolds and their cohomology.
problem Analyzing Hodge structures transversal to Reeb foliations.
method Applying general results about Hodge structures to contact forms and Reeb vector fields.
result Differential complexes of basic forms are canonically isomorphic under certain conditions.
We study Riemannian foliations whose transverse Levi-Civita connection ∇ has special holonomy. In particular, we focus on the case where Hol(∇) is contained either in SU(n) or in Sp(n). We prove a Weitzenbock formula involving complex basic forms on Kähler foliations and we apply this formula for pointing…
By use of a variety of techniques (most based on constructions of quasipositive knots and links, some old and others new), many smooth 3-manifolds are realized as transverse intersections of complex surfaces in complex 3-space with strictly pseudoconvex 5-spheres. These manifolds not only inherit interesting intrinsic …
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.
Geometric compactification for complex structures on Lie groups.
problem Compactifying moduli stack of complex structures on Lie groups.
method Describes a geometric compactification using CR structures transverse to a real foliation.
result Extra points represent CR structures transverse to a real foliation.
We study blow-ups in generalized complex geometry. To that end we introduce the concept of holomorphic ideal, which allows one to define a blow-up in the category of smooth manifolds. We then investigate which generalized complex submanifolds are suitable for blowing up. Two classes naturally appear; generalized Poisso…
Overview of algebraic geometry for almost complex manifolds.
problem Developing algebraic geometry for almost complex manifolds without genericity.
method Reviewing results based on pseudoholomorphic maps and intersection theory.
result Introduction of birational morphism between almost complex manifolds.
New theorem allows transverse links to be braided with rational book structure.
problem Transverse isotopy of links in contact 3-manifolds.
method Generalization of Pavelescu's argument for rational open book decomposition.
result Every transverse link can be isotoped to a braid with rational open book.
Study on transverse knots and their neighborhoods, proving unique standard neighborhoods and destabilization results.
problem Understanding the structure and properties of transverse knots and their neighborhoods.
method Proves unique standard neighborhoods and structure theorems for non-loose Legendrian knots through destabilization results.
result Finds a manifold with infinite tight contact structures, up to contactomorphism, without Giroux torsion.
The Hitchin-Kobayashi correspondence is extended to foliated Riemannian manifolds.
problem Proving a Hitchin-Kobayashi correspondence for foliated manifolds.
method Adapting Uhlenbeck and Yau's proof to the foliated setting, defining stability for foliated bundles, and relating to higher-dimensional instantons.
result Foliated Hermitian-Einstein connections correspond to polystability, with implications for higher-dimensional instantons.
Study on symplectic Dirac operators on foliations, estimating eigenvalues.
problem Estimating eigenvalues of transversely symplectic Dirac operators.
method Analysis of transversely symplectic structures and use of Weitzenbock formula.
result Estimation of lower bounds for eigenvalues of transversely symplectic Dirac operators.
Study contact instantons on Sasakian 5-manifolds with Calabi-Yau structures.
problem Anti-self-dual contact instantons on Sasakian 5-manifolds with transverse Calabi-Yau structures.
method Singularity data of leaf spaces and computation of moduli spaces.
result Explicit computation of complex dimensions of moduli spaces for 95 orbifold K3 surfaces.
Smooth maps in o-minimal structures are mostly transverse.
problem Transversality of smooth definable maps in o-minimal structures.
method Definable smooth version of Thom transversality theorem, proving nowhere density of non-transverse maps, and a definable version of Trotman's theorem.
result Non-transverse maps are nowhere dense in the definable smooth topology.
Constructs the moduli space of super J-holomorphic curves.
problem Defines and constructs the moduli space of super J-holomorphic curves.
method Uses component fields of a super differential equation and a transversality argument.
result Constructs the moduli space of super J-holomorphic curves as a smooth subsupermanifold.
We show that a transverse link in a contact structure supported by an open book decomposition can be transversely braided. We also generalize Markov's theorem on when the closures of two braids represent (transversely) isotopic links.
New transverse links solve contact manifold problems.
problem Contact manifold structures.
method Transverse links in 3-sphere contact structures.
result All contact 3-manifolds are contact branched covers over a transverse link.
Given a contact manifold $M_#$ together with a transversal infinitesimal automorphism ξ, we show that any local leaf space M for the foliation determined by ξ naturally carries a conformally symplectic (cs-) structure. Then we show that the Rumin complex on $M_#$ descends to a complex of differential operators on…
A generalized complex manifold which satisfies the ∂∂-lemma admits a Hodge decomposition in twisted cohomology. Using a Courant algebroid theoretic approach we study the behavior of the Hodge decomposition in smooth and holomorphic families of generalized complex manifolds. In particular we …
This is a survey article on a known generalization of Dirac-type operators to transverse operators called basic Dirac operators on Riemannian foliations, which are smooth foliations that have a transverse geometric structure. Construction of these operators requires the additional structure of what is called a bundle-l…
Study jets of flat partial connections in foliations.
problem Characterize and understand flat partial connections in foliations.
method Define and apply jets to flat partial connections in smooth foliations and locally free sheaves, focusing on codimension one and arbitrary codimension foliations.
result Define and apply jets to characterize transversely affine and projective structures in foliations.
Simple perturbation of Vafa-Witten equations leads to transversality.
problem Transversality of Vafa-Witten moduli space.
method Simple perturbation of Vafa-Witten equations, proving transversality for SU(2) or SO(3) structure groups. result For generic perturbation parameter, the full rank part of the moduli space satisfies transversality.
New method computes transverse knot invariant using braids.
problem Compute transverse invariant of knots and links.
method Define new combinatorial complex to compute knot Floer homology.
result Identify BRAID invariant of transverse knots and links.
The paper proves transversality for special Lagrangian submanifolds in a 6D manifold.
problem Counting special Lagrangian submanifolds in higher dimensions.
method Proving transversality for the moduli space of perturbed special Lagrangian submanifolds using a Lagrange multipliers problem.
result The moduli space is generically a set of isolated points.
We study transversality for Lipschitz-Fredholm maps in the context of bounded Fréchet manifolds. We show that the set of all Lipschitz-Fredholm maps of a fixed index between Fréchet spaces has the transverse stability property. We give a straightforward extension of the Smale transversality theorem by using the general…
Study on stability of Sasaki structures under deformations.
problem Stability of Sasaki structures under transverse holomorphic deformations.
method Analysis of transverse Kähler holonomy groups and stability properties.
result Stability of ${\oldmathcal S}$ under certain conditions on Sasaki manifolds.