Related papers: Rigidity properties of holomorphic Legendrian sing…

Suppose $K$ is a knot in a 3-manifold $Y$, and that $Y$ admits a pair of distinct contact structures. Assume that $K$ has Legendrian representatives in each of these contact structures, such that the corresponding Thurston-Bennequin…

We explain a connection between the algebraic and geometric properties of groups of contact transformations, open book decompositions, and flexible Legendrian embeddings. The main result is that, if a closed contact manifold $(V, \xi)$ has…

We prove rigidity and vanishing theorems for several holomorphic Euler characteristics on complex contact manifolds admitting holomorphic circle actions preserving the contact structure. Such vanishings are reminiscent of those of LeBrun…

In this note we show that $+1$-contact surgery on distinct Legendrian knots frequently produces contactomorphic manifolds. We also give examples where this happens for $-1$-contact surgery. As an amusing corollary we find overtwisted…

Let $M$ be a surface with a Riemannian metric and $UM$ the unit tangent bundle over $M$ with the canonical contact sub-Riemannian structure $D$ on $UM$. In this paper, the complete local classification of singularities, under the Legendre…

This paper describes a characterization of tightness of closed contact 3-manifolds in terms of supporting open book decompositions. The main result is that tightness of a closed contact 3-manifold is preserved under Legendrian surgery.

In this paper we study the variability and rigidity of secondary characteristic classes which arise from flat connections on a manifold. Considering the connection as a Lie-algebra valued one-form, we study the characteristic map from Lie…

Standard singularity theorems are proven in Lorentzian manifolds of arbitrary dimension n if they contain closed trapped submanifolds of arbitrary co-dimension. By using the mean curvature vector to characterize trapped submanifolds, a…

We show that for a big class of contact manifolds the groups of order $\leq n$ invariants (with values in an arbitrary Abelian group) of Legendrian, of transverse and of framed knots are canonically isomorphic. On the other hand for an…

In this note, we show that transverse knots have unique standard neighborhoods and prove a structure theorem about non-loose Legendrian knots. We also prove a finiteness result for transverse knots in a tight contact manifold. The common…

A theorem of Mumford states that, on complex surfaces, any normal isolated singularity whose link is diffeomorphic to a sphere is actually a smooth point. While this property fails in higher dimensions, McLean asks whether the contact…

We construct a family of examples of Legendrian subvarieties in some projective spaces. Although most of them are singular, a new example of smooth Legendrian variety in dimension 8 is in this family. The 8-fold has interesting properties:…

This paper is devoted to the study of various aspects of deformations of log pairs, especially in connection to questions related to the invariance of singularities and log plurigenera. In particular, using recent results from the minimal…

It is given the diffeomorphism classification on generic singularities of tangent varieties to curves with arbitrary codimension in a projective space. The generic classifications are performed in terms of certain geometric structures and…

We give a brief survey of the concept of birational rigidity, from its origins in the two-dimensional birational geometry, to its current state. The main ingredients of the method of maximal singularities are discussed. The principal…

We establish a full $h-$principle ($C^0-$close, relative, parametric) for the simplification of singularities of Lagrangian and Legendrian fronts. More precisely, we prove that if there is no homotopy theoretic obstruction to simplifying…

We use the Ozsv\'ath-Szab\'o contact invariants to distinguish between tight contact structures obtained by Legendrian surgeries on stabilized Legendrian links in tight contact 3-manifolds. We also discuss the implication of our result on…

We overview some recent results on Fano varieties giving evidence of their rigid nature under small deformations.

We present classification results for exceptional Legendrian realisations of torus knots. These are the first results of that kind for non-trivial topological knot types. Enumeration results of Ding-Li-Zhang concerning tight contact…

We show that a null-homologous transverse knot K in the complement of an overtwisted disk in a contact 3-manifold is the boundary of a Legendrian ribbon if and only if it possesses a Seifert surface S such that the self-linking number of K…