arXiv subjects
Publications and source records attributed to Ilya Kossovskiy.
In this paper, we provide two (significantly different) proofs of the well known Strong Ramadanov Conjecture for the class of real ellipsoids in the complex $2$-space.
In our earlier work \cite{KZ}, we introduced an analytic regularizability theory for smooth strictly pseudoconvex hypersurfaces in complex space. That is, we found a necessary and sufficient condition for a hypersurface to be CR-equivalent to an analytic target. The condition amount to the holomorphic extension property for a smooth function on a totally real submanifold, both the function and the submanifold being uniquely associated with the given hypersurface. In the present paper, we develop our method further. First, we extend the result in \cite{KZ} to hypersurfaces of finite (possibly low) smoothness. Second, we introduce a new tool for studying CR hypersurfaces in low regularity called {\em regularizing $(0,1)$ sections}. Using the latter key tool, we solve the open problem of checking the {\em sphericity} of a strictly pseudoconvex hypersurface in $\mathbb C^{2}$ in {low regularity}, precisely in the regularity $C^k,\,2\leq k< 7$ which is {\em not} covered by the classical Cartan-Tanaka-Chern-Moser theory. As an application of our theory, we deduce the sphericity of a strictly pseudoconvex hypersurface in $\mathbb C^{2}$ of regularity $C^6$ with vanishing Cartan-Chern CR-curvature.
For a smooth, non-degenerate locally integrable structure of hypersurface type on a manifold $M$, we provide necessary and sufficient conditions for it to be equivalent, near a point, to a real-analytic locally integrable structure (the analytic regularizability), generalizing a recent result of Zaitsev and the first author. First, we discover, in our setting, a (previously unknown) invariant CR submanifold $Σ$ in $M$ of hypersurface type, which we call the central submanifold. We prove that the analytic regularizability of $M$ is equivalent to that of the associated CR manifold $Σ$. Furthermore, as a byproduct of our construction, we show that the central manifold construction reduces the whole (smooth or analytic) equivalence problem for nondegenerate structures with the Levi positivity condition to that of the associated central manifolds, i.e. to CR geometry. Second, we make use of a classical construction due to Marson and show that sufficient for the analytic regularizability of $M$ is the analytic regularizability of the CR manifold $\tilde M$ associated with $M$ in the sense of Marson. We show applications of both regularizability conditions to classes of locally integrable structures.
$2$-nondegenerate real hypersurfaces in complex manifolds play an important role in CR-geometry and the theory of Hermitian Symmetric Domains. In this paper, we construct a complete convergent normal form for everywhere $2$-nondegenerate real-analytic hypersurfaces in complex $3$-space. We do so by developing the homological approach of Chern-Moser in the $2$-nondegenerate setting. This seems to be the first such construction for hypersurfaces of infinite Catlin multitype. Our approach is based on using a rational (nonpolynomial) model for everywhere $2$-nondegenerate hypersurfaces, which is the local realization due to Fels-Kaup of the well known tube over the light cone. As an application, we obtain, in the spirit of Chern-Moser theory, a criterion for the local sphericity (i.e. local equivalence to the model) for a $2$-nondegenerate hypersurface in terms of its normal form. As another application, we obtain an explicit description of the moduli space of everywhere $2$-nondegenerate hypersurfaces.
We classify singular holomorphic vector fields in two-dimensional complex space admitting a (Levi-nonflat) real-analytic invariant 3-fold through the singularity. In this way, we complete the classification of infinitesimal symmetries of real-analytic Levi-nonflat hypersurfaces in complex two-space. The classification of holomorphic vector fields obtained in the paper has very interesting overlaps with the recent Lombardi-Stolovitch classification theory for holomorphic vector fields at a singularity. In particular, we show that most of the resonances arising in Lombardi-Stolovitch theory do not occur under the presence of (Levi-nonflat) integral manifolds.
We introduce a class of uniformly $2$-nondegenerate CR hypersurfaces in $\mathbb{C}^N$, for $N>3$, having a rank $1$ Levi kernel. The class is first of all remarkable by the fact that for every $N>3$ it forms an {\em explicit} infinite-dimensional family of everywhere $2$-nondegenerate hypersurfaces. To the best of our knowledge, this is the first such construction. Besides, the class an infinite-dimensional family of non-equivalent structures having a given constant nilpotent CR symbol for every such symbol. Using methods that are able to handle all cases with $N>5$ simultaneously, we solve the equivalence problem for the considered structures whose symbol is represented by a single Jordan block, classify their algebras of infinitesimal symmetries, and classify the locally homogeneous structures among them. We show that the remaining considered structures, which have symbols represented by a direct sum of Jordan blocks, can be constructed from the single block structures through simple linking and extension processes.
We investigate the problem of holomorphic algebraizibility for real hypersurfaces in complex space. We introduce a new invariant of a (real-analytic) Levi-nondegenerate hypersurface called {\em the jet transcendence degree}. Using this invariant, we solve in the negative the Conjecture of Huang, Ji and Yau on the algabraizability of real hypersurfaces with algebraic syzygies.
We provide regularity results for CR-maps between real hypersurfaces in complex spaces of different dimension with a Levi-degenerate target. We address both the real-analytic and the smooth case. Our results allow immediate applications to the study of proper holomorphic maps between Bounded Symmetric Domains.
We prove that if two real-analytic hypersurfaces in $\mathbb C^2$ are equivalent formally, then they are also $C^\infty$ CR-equivalent at the respective point. As a corollary, we prove that all formal equivalences between real-algebraic Levi-nonflat hypersurfaces in $\mathbb C^2$ are algebraic (in particular are convergent). The result is obtained by using the recent {\em CR - DS technique}, connecting degenerate CR-manifolds and Dynamical Systems, and employing subsequently the {\em multisummability theory} of divergent power series used in the Dynamical Systems theory.
We develop a classification theory for real-analytic hypersurfaces in $\mathbb C^2$ in the case when the hypersurface is of {\em infinite type} at the reference point. This is the remaining, not yet understood case in $\mathbb C^2$ in the {\it Problème local}, formulated by H.\,Poincaré in 1907 and asking for a complete biholomorphic classification of real hypersurfaces in complex space. One novel aspect of our results, appearing in this revised version, is a notion of {\em smooth normal forms} for real-analytic hypersurfaces. We rely fundamentally on the recently developed CR -- DS technique in CR-geometry.
Locally homogeneous strictly pseudoconvex hypersurfaces in $\mathbb C^2$ were classified by E.\,Cartan in 1932. In this work, we complete the classification of locally homogeneous strictly pseudoconvex hypersurfaces in $\mathbb C^3$.
For a smooth strictly pseudoconvex hypersurface in a complex manifold, we give a necessary and sufficient condition for being CR-diffeomorphic to a real-analytic CR manifold. Our condition amounts to a holomorphic extension property for the canonically associated function expressing $2$-jets of the formal Segre varieties in terms of their $1$-jets. We also express this condition in equivalent terms for a Fefferman type determinant
We solve the local equivalence problem for second order (smooth or analytic) ordinary differential equations. We do so by presenting a {\em complete convergent normal form} for this class of ODEs. The normal form is optimal in the sense that it is defined up to the automorphism group of the model (flat) ODE $y"=0$. For a generic ODE, we also provide a unique normal form. By doing so, we give a solution to a problem which remained unsolved since the work of Arnold. The method can be immediately applied to important classes of second order ODEs, in particular, the Painlevé equations. As another application of the convergent normal form, we discover distinguished curves associated with a differential equation that we call {\em chains}.
In this paper, we provide {\em effective} results on the non-embeddability of real-analytic hypersurfaces into a hyperquadric. We show that, for any $N >n \geq 1$, the defining functions $φ(z,\bar z,u)$ of all real-analytic hypersurfaces $M=\{v=φ(z,\bar z,u)\}\subset\mathbb C^{n+1}$ containing Levi-nondegenerate points and locally transversally holomorphically embeddable into some hyperquadric $\mathcal Q\subset\mathbb C^{N+1}$ satisfy an {\em universal} algebraic partial differential equation $D(φ)=0$, where the algebraic-differential operator $D=D(n,N)$ depends on $n, N$ only. To the best of our knowledge, this is the first effective result characterizing real-analytic hypersurfaces embeddable into a hyperquadric of higher dimension. As an application, we show that for every $n,N$ as above there exists $μ=μ(n,N)$ such that a Zariski generic real-analytic hypersurface $M\subset\mathbb C^{n+1}$ of degree $\geq μ$ is not transversally holomorphically embeddable into any hyperquadric $\mathcal Q\subset\mathbb C^{N+1}$. We also provide an explicit upper bound for $μ$ in terms of $n,N$. To the best of our knowledge, this gives the first effective lower bound for the CR-complexity of a Zariski generic real-algebraic hypersurface in complex space of a fixed degree.
In this work, we obtain an unexpected geometric characterization of sphericity of a real-analytic Levi-nondegenerate hypersurface $M\subset\mathbb C^{2}$. We prove that $M$ is spherical if and only if its Segre\,(-Webster) varieties satisfy an elementary combinatorial property, identical to a property of straight lines on the plane and known in Projective Geometry as the {\em Desargues Theorem}.
One of effective ways to solve the equivalence problem and describe moduli spaces for real submanifolds in complex space is the normal form approach. In this survey, we outline some normal form constructions in CR-geometry and formulate a number of open problems.
In our recent work [25] we showed that $C^\infty$ CR-diffeomorphisms of real-analytic Levi-nonflat hypersurfaces in $\mathbb C^{2}$ are not analytic in general. This result raised again the question on the nature of CR-maps of real-analytic hypersurfaces. In this paper, we give a complete picture of what CR-maps actually are. First, we discover an analytic continuation phenomenon for CR-diffeomorphisms which we call the {\em sectorial analyticity property}. It appears to be the optimal regularity property for CR-diffeomorphisms in general. We emphasize that such type of extension never appeared previously in the literature. Second, we introduce the class of {\em Fuchsian type hypersurfaces} and prove that (infinitesimal generators of) CR-automorphisms of a Fuchsian type hypersurface are still analytic. In particular, this solves a problem formulated in [28]. Finally, we prove a regularity result for {\em formal} CR-automorphisms of Fuchsian type hypersurfaces.
We study the holomorphic equivalence problem for finite type hypersurfaces in $\mathbb C^2$. We discover a geometric condition, which is sufficient for the existence of a natural convergent normal form for a finite type hypersurface. We also provide an explicit construction of such a normal form. As an application, we construct a canonical connection for a large class of finite type hypersurfaces. To the best of our knowledge, this gives the first construction of an invariant connection for Levi-degenerate hypersurfaces in $\mathbb C^2$.