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We prove that if $X---> X^+$ is a threefold terminal flip, then $c_1(X).c_2(X)\leq c_1(X^+).c_2(X^+)$ where $c_1(X)$ and $c_2(X)$ denote the Chern classes. This gives the affirmative answer to a Question by Xie \cite{Xie2}. We obtain the…

In any positive CR-dimension and CR-codimension we provide a construction of real-analytic holomorphically nondegenerate CR-submanifolds, which are $C^\infty$ CR-equivalent, but are inequivalent holomorphically. As a corollary, we provide…

On a real analytic 5-dimensional CR-generic submanifold M^5 in C^4 of codimension 3, hence of CR dimension 1, which enjoys the generically satisfied nondegeneracy condition that Lie brackets up to length 3 of T^{1,0}M generate CTM, a…

This short paper gives a constraint on Chern classes of closed strictly pseudoconvex CR manifolds (or equivalently, closed holomorphically fillable contact manifolds) of dimension at least five. We also see that our result is ''optimal''…

Let $(X,T^{1,0}X)$ be a $(2n+1+d)$-dimensional compact CR manifold with codimension $d+1$, $d\geq1$, and let $G$ be a $d$-dimensional compact Lie group with CR action on $X$ and $T$ be a globally defined vector field on $X$ such that…

We derive an explicit formula for the well-known Chern-Moser-Weyl tensor for nondegenerate real hypersurfaces in complex space in terms of their defining functions. The formula is considerably simplified when applying to "pluriharmonic…

A closed subscheme of codimension two $T \subset P^2$ is a quasi complete intersection (q.c.i.) of type $(a,b,c)$ if there exists a surjective morphism $\mathcal{O} (-a) \oplus \mathcal{O} (-b) \oplus \mathcal{O} (-c) \to \mathcal{I} _T$.…

We investigate CR-manifolds which are tubes M:= F x iV over general bases F in a real vector space V and characterize the k-nondegeneracy of M in terms of the real affine geometry of F. We give a method for an explicit computation of the…

It is proved that a germ of a real analytic CR map from a smooth real-analytic minimal CR manifold M to an essentially finite real-algebraic generic submanifold M' of P^N of the same CR-dimension extends as a holomorphic correspondence…

We study a germ of real analytic $n$-dimensional submanifold of ${\mathbf C}^n$ that has a complex tangent space of maximal dimension at a CR singularity. Under the condition that its complexification admits the maximum number of deck…

We develop a holonomy reduction procedure for general Cartan geometries. We show that, given a reduction of holonomy, the underlying manifold naturally decomposes into a disjoint union of initial submanifolds. Each such submanifold…

Many quantum invariants of knots and 3-manifolds (e.g. Jones polynomials) are special cases of the Witten-Reshetikhin-Turaev 3D TQFT. The latter is in turn a part of a larger theory - the Crane-Yetter 4D TQFT. In this work, we compute the…

We consider the 4-dimensional $\mathcal{N}=1$ Lie superconformal algebra and search for completely "symmetric" (in the graded sense) 3-index invariant tensors. The solution we find is unique and we show that the corresponding invariant…

In this article the notion of ultradifferentiable CR manifold is introduced and an ultradifferentiable regularity result for finitely nondegenerate CR mappings is proven. Here ultradifferentiable means with respect to Denjoy-Carleman…

In this paper, we show that an irreducible proper complex equifocal submanifold of codimension greater than one in a symmetric space of non-compact type. The proof is performed by showing the homogeneity of the lift of the complexification…

A general theory of rigid completely integrable analytic partial differential equations is endeavoured. The tube over the light cone in C^3 is shown to be the unique model (up to biholomorphisms) having CR automorphism group of maximal…

Let M,M' be smooth real hypersurfaces in N-dimensional space and assume that M is k-nondegenerate at a point p in M. We prove that holomorphic mappings that extend smoothly to M, sending a neighborhood of p in M diffeomorphically into M'…

We define a class of generic CR submanifolds of $C^n$ of real codimension $d$, with $d$ in $1, ..., n-1$, called the Bloom-Graham model graphs, whose graphing functions are partially decoupled in their dependence on the variables in the…

It is shown that any smooth closed orientable manifold of dimension $2k + 1$, $k \geq 2$, admits a smooth polynomially convex embedding into $\mathbb C^{3k}$. This improves by $1$ the previously known lower bound of $3k+1$ on the possible…

We extend finite dimensional Chern-Simons theory to certain infinite dimensional principal bundles with connections, in particular to the frame bundle $FLM\to LM$ over the loop space of a Riemannian manifold $M$. Chern-Simons forms are…