arXiv research

We show that the Hodge numbers of Sasakian manifolds are invariant under arbitrary deformations of the Sasakian structure. We also present an upper semi continuity Theorem for the dimensions of kernels of a smooth family of transversely elliptic operators on manifolds with transversely Riemannian foliations. We use thi…

Study on stability of Sasaki structures under deformations.

Simplicial sets deformation retract onto transverse simplices.

Constructs deformations of Vaisman manifolds preserving foliations.

Deformations of the Reeb flow of a Sasakian manifold as transversely Kähler flows may not admit compatible Sasakian metrics anymore. We show that the triviality of the (0,2)-component of the basic Euler class characterizes the existence of compatible Sasakian metrics for given small deformations of the Reeb flow as tra…

The deformation theory of a Dirac structure is controlled by a differential graded Lie algebra which depends on the choice of an auxiliary transversal Dirac structure; if the transversal is not involutive, one obtains an $L_\infty$ algebra instead. We develop a simplified method for describing this $L_\infty$ algebra a…

Study deformations of Calabi-Yau foliations using Kuranishi spaces.

There is a natural way to deform a Killing foliation with non-closed leaves, due to Ghys and Haefliger--Salem, into a closed foliation, i.e., a foliation whose leaves are all closed. Certain transverse geometric and topological properties are preserved under these deformations, as previously shown by the authors. For i…

Given a manifold M with a submanifold N, the deformation space D(M,N) is a manifold with a submersion to R whose zero fiber is the normal bundle, and all other fibers are equal to M. This article uses deformation spaces to study the local behavior of various geometric structures associated with singular foliations, wit…

New theorems on Hodge numbers and Kähler structures derived from complex differential forms.

In this paper we investigate the possibility to obtain locally new Sasaki-Einstein metrics on the space $T^{1,1}$ considering a deformation of the standard metric tensor field. We show that from the geometric point of view this deformation leaves transverse and the leafwise metric intact, but changes the orthogonal com…

We classify simply connected compact Sasaki manifolds of dimension $2n+1$ with positive transverse bisectional curvature. In particular, the Kähler cone corresponding to such manifolds must be bi-holomorphic to $\C^{n+1}\backslash \{0\}$. As an application we recover the Mori-Siu-Yau theorem on the Frankel conjecture a…

Cataclysm deformations study Anosov representations and their convergence.

We consider Lagrangian-like submanifolds in certain even-dimensional 'symplectic-like' Poisson manifolds. We show, under suitable transversality hypotheses, that the pair consisting of the ambient Poisson manifold and the submanifold has unobstructed deformations and that the deformations automatically preserve the Lag…

We give a necessary and sufficient condition for lagrangians in a symplectic vector bundle to be deformed stably into transversal lagrangians. In the case of three lagrangians, we show that the associated Grothendieck group can be identified with a Hermitian K-theory group.

Sphere theorems extended to Riemannian foliations with new results on curvature and leaf spaces.

In this article we introduce a family of transverse invariants arising from the deformations of Khovanov homology. This family includes the invariants introduced by Plamenevskaya and by Lipshitz, Ng, and Sarkar. Then, we investigate the invariants arising from Bar-Natan's deformation. These invariants, called $β$-invar…

We show that a compact manifold admitting a Killing foliation with positive transverse curvature fibers over finite quotients of spheres or weighted complex projective spaces, provided that the singular foliation defined by the closures of the leaves has maximal dimension. This result is obtained by deforming the folia…

We produce the first examples of closed, tight contact 3-manifolds which become overtwisted after performing admissible transverse surgeries. Along the way, we clarify the relationship between admissible transverse surgery and Legendrian surgery. We use this clarification to study a new invariant of transverse knots - …

We express Witten's deformation of Morse functions using deformation to the normal cone and $C^*$-modules. This allows us to obtain asymptotics of the `large eigenvalues'. Our methods extend to Morse functions along a foliation. We construct the Witten deformation using any generic function on an arbitrary foliation on…

We deform a minimal disk in $\mathbb{R}^4$ with a branch point into symplectic minimally immersed disks with only transverse double points.

Extending the work of G. Székelyhidi and T. Brönnle to Sasakian manifolds we prove that a small deformation of the complex structure of the cone of a constant scalar curvature Sasakian manifold admits a constant scalar curvature structure if it is K-polystable. This also implies that a small deformation of the complex …

The paper studies quantization on symplectic manifolds with real polarizations, comparing different quantization methods.

O. Plamenevskaya associated to each transverse knot K an element of the Khovanov homology of K. In this paper, we give two refinements of Plamenevskaya's invariant, one valued in Bar-Natan's deformation of the Khovanov complex and another as a cohomotopy element of the Khovanov spectrum. We show that the first of these…

We develop some foundations for the study of Kahler-Einstein metrics with cone singularities transverse to a divisor. The main goal is a treatment of the deformation of the cone angle.

Invariant obstructs separating coassociative 4-folds.

Motivated by questions of deformations/moduli in foliation theory, we investigate the structure of some groups of diffeomorphisms preserving a foliation. We give an example of a $C^\infty$ foliation whose diffeomorphism group is not a Lie group in any reasonable sense. On the positive side, we prove that the automorphi…

New cohomology $η$ for deformed Sasaki-Einstein manifolds derived from Dolbeault cohomology.

The abstract discusses transforming Vaisman metrics into Kähler-Einstein structures.

This paper shows that an arbitrary generic submanifold in a complex manifold can be deformed into a 1-parameter family of generic submanifolds satisfying strong nondegeneracy conditions. The proofs use a careful analysis of the jet spaces of embeddings satisfying certain nondegeneracy properties, and also make use of t…

Using the method of Witten deformation, we express the basic index of a transversal Dirac operator over a Riemannian foliation as the sum of integers associated to the critical leaf closures of a given foliated bundle map.

Study infinitesimal deformations of Lie algebroid pairs.

This work is a continuation of our previous paper arXiv:1812.06473 where we have constructed ${\cal N}=2$ supersymmetric Yang-Mills theory on 4D manifolds with a Killing vector field with isolated fixed points. In this work we expand on the mathematical aspects of the theory, with a particular focus on its nature as a …

Extends quantization theory to mixed polarizations using transverse differential operators.

New gauge preserves Einstein metrics' interactions, proving rigidity on negatively curved manifolds.

In this paper we will make use of the Mackaay-Vaz approach to the universal $\mathfrak{sl}_3$-homology to define a family of cycles (called $β_3$-invariants) which are transverse braid invariants. This family includes Wu's $ψ_{3}$-invariant. Furthermore, we analyse the vanishing of the homology classes of the $β_3$-inv…

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…

Let $M$ be a compact Riemannian manifold endowed with an isometric action of a compact Lie group. The method of the Witten deformation is used to compute the virtual representation-valued equivariant index of a transversally elliptic, first order differential operator on $M$. The multiplicities of irreducible represent…

In this paper we study compact Sasaki manifolds in view of transverse Kähler geometry and extend some results in Kähler geometry to Sasaki manifolds. In particular we define integral invariants which obstruct the existence of transverse Kähler metric with harmonic Chern forms. The integral invariant $f_1$ for the first…

Cataclysm deformations study Anosov representations, leading to new formulas and non-open sets.

Study transverse metric expansion on null hypersurfaces, proving uniqueness for Killing horizons.

Quantizes symplectic manifolds with toric singularities using Toeplitz operators.

Given an Anosov representation $ρ\colon π_1(S) \to \PSL_{n}(\mathbb{R})$ and a maximal geodesic lamination $λ$ in a surface $S$, we construct shear deformations along the leaves of the geodesic lamination $λ$ endowed with a certain flag decoration, that is provided by the associated flag curve $\mathcal{F}_ρ\colon \Sin…

We present natural and general ways of building Lie groupoids, by using the classical procedures of blowups and of deformations to the normal cone. Our constructions are seen to recover many known ones involved in index theory. The deformation and blowup groupoids obtained give rise to several extensions of $C^*$-algeb…

The paper explores families of almost complex structures and transverse (p,p)-forms.

We study the action of the group of contact diffeomorphisms on CR deformations of compact three-dimensional CR manifolds. Using anisotropic function spaces and an anisotropic structure on the space of contact diffeomorphisms, we establish the existence of local transverse slices to the action of the contact diffeomorph…

We consider smooth 1-parameter families of plane curves tangent to a semicubic parabola, when the curvature radius of their curves at the tangency point vanishes at the cusp point. We find the $\A$-normal form of these families, their envelopes and local patterns near the cusp. We obtain a new codimension 2 singularity…

Let M be a cusped 3-manifold, and let T be an ideal triangulation of M. The deformation variety D(T), a subset of which parameterises (incomplete) hyperbolic structures obtained on M using T, is defined and compactified by adding certain projective classes of transversely measured singular codimension-one foliations of…