Vsevolod Shevchishin

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• Professor
• Professor at University of Warmia and Mazury in Olsztyn

Consider a closed oriented four-manifold $X$ that contains an embedded sphere of self-intersection number $(-1)$. Suppose that $b_2^+(X) \leq 3$. A Riemannian metric on $X$ is found such that the cohomology class dual to this sphere is represented by an anti-self-dual harmonic form. Moreover, such a metric exists even for multiple disjoint embedded...

This is an expanded version of the talk given by the first author at the conference “Topology, Geometry, and Dynamics: Rokhlin – 100”. The purpose of this talk was to explain our current results on the classification of rational symplectic 4-manifolds equipped with an anti-symplectic involution. A detailed exposition will appear elsewhere.

This is an expanded version of the talk given be the first author at the conference "Topology, Geometry, and Dynamics: Rokhlin - 100". The purpose of this talk was to explain our current results on classification of rational symplectic 4-manifolds equipped with an anti-symplectic involution. Detailed exposition will appear elsewhere.

We prove a non-squeezing result for Lagrangian embeddings of the real projective plane into blow-ups of the symplectic ball.

The paper is devoted to quadratic Poisson structures compatible with the canonical linear Poisson structures on trivial 1-dimensional central extensions of semisimple Lie algebras. In particular, we develop the general theory of such structures and study related families of functions in involution. We also show that there exists a 10-parametric fam...

We prove a non-squeezing result for Lagrangian embeddings of the real projective plane into blow-ups of the symplectic ball.

We show that symplectically embedded $(-1)$-tori give rise to certain elements in the symplectic mapping class group of $4$-manifolds. An example is given where such elements are proved to be of infinite order.

We show that symplectically embedded $(-1)$-tori give rise to certain elements in the symplectic mapping class group of $4$-manifolds. An example is given where such elements are proved to be of infinite order.

We show that 3-dimensional polyhedral manifolds with nonnegative curvature in the sense of Alexandrov can be approximated by nonnegatively curved 3-dimensional Riemannian manifolds.

We obtain, in local coordinates, the explicit form of the two-dimensional, super-integrable systems of Matveev and Shevchishin involving cubic integrals. This enables us to determine for which values of the parameters these systems are indeed globally defined on the two-sphere.

We prove a version of the reflection principle for pseudoholomorphic disks with boundary on totally real submanifolds in almost-complex manifolds. Furthermore, we give a proof of the Gromov compactness theorem for pseudoholomorphic curves with boundary on immersed totally real submanifolds. As a corollary we show that a complex disk can be attached...

We describe all local Riemannian metrics on surfaces whose geodesic flows are superintegrable with one integral linear in momenta and one integral cubic in momenta.We also show that some of these metrics can be extended to S2. This gives us new examples of Hamiltonian systems on the sphere with integrals of degree three in momenta, and the first ex...

We introduce the secondary Stiefel-Whitney clas w 2 of homotopically tri-vial diffeomorphisms and show that a homotopically trivial symplectomorphism of a ruled 4-manifold is isotopic to identity if and only if the clas w 2 vanishes. Using this, we describe the combinatorial structure of the diffeotopy group of ruled symplectic 4-manifolds X, eithe...

We describe all local Riemannian metrics on surfaces whose geodesic flows are superintegrable with one integral linear in momenta and one integral cubic in momenta. We also show that some of these metrics can be extended to the 2-sphere. This gives us new examples of Hamiltonian systems on the sphere with integrals of degree three in momenta, and t...

We construct differential invariants that vanish if and only if the geodesic flow of a 2-dimensional metric admits an integral of 3rd degree in momenta with a given Birkhoff-Kolokoltsov 3-codifferential.

We prove the vanishing of the secondary Stiefel-Whitney class $w~_2$ of homotopically trivial symplectomorphisms of an irrational ruled symplectic 4-manifold X, either minimal or blown-up. We also show that any value of $w~_2$ can be realised by an appropriate diffeomorphism. This proves the existence of at least $2^{2g}$ different deformation clas...

Abstract: (russian) Доказано отсутствие лагранжевых вложений бутылки Клейна K в CP^2. Рассматриваются специальные вложения K в симплектический пучок Лефшеца pr : X → S^2 и изучается его монодромия. В качестве основного технического средства используется развитая в работе комбинаторная теория групп классов отображений. Показано, что если гомологичес...

We prove that any two irreducible cuspidal Hurwitz curves $C_0$ and $C_1$ (or more generally, curves with A-type singularities) in the Hirzebruch surface $F_N$ with coinciding homology classes and sets of singularities are regular homotopic; and symplectically regular homotopic if $C_0$ and $C_1$ are symplectic with respect to a compatible symplect...

In this paper we prove the non-existence of Lagrangian embeddings of the Klein bottle K in CP². We exploit the existence of a special embedding of K in a symplectic Lefschetz pencil pr: X → S² and study its monodromy. As the main technical tool, we develop the combinatorial theory of mapping class groups. The obtained results allow us to show that...

We prove the existence of primitive curves and positivity of intersections of $J$-complex curves for Lipschitz-continuous almost complex structures. These results are deduced from the Comparison Theorem for $J$-holomorphic maps in Lipschitz structures, previously known for $J$ of class $C^{1, Lip}$. We also give the optimal regularity of curves in...

We prove that the locus of irreducible nodal curves on a given Hirzebruch surface F_k of given linear equivalency class and genus g is irreducible.

For a given singularity of a plane curve we consider the locus of nodal deformations of the singularity with the given number of nodes and describe possible components of the locus. As applications, we solve the local symplectic isotopy for nodal curves in a neighborhood of a given pseudoholomorphic curve without multiple components and prove the u...

A ∂-symplectic structure on a complex manifold M of complex dimension2n is given by a smooth ∂-closed (2, 0)-form ω such thatωn is nonvanishing. We prove that a version of the Darboux theorem isvalid for such a structure: locally ω can be represented as∑ i=1n ∂ f i ∧ ∂ f n +i for appropriate smooth complex valuedfunctions f 1, ..., f 2n . We also p...

We consider the local behavior of Sobolev connections in a neighborhood of a singularity of codimension 2 and determine sufficient conditions for existence and local constancy of the limit holonomy of such connection. We prove that every Sobolev connection on an mdimensional manifold with locally Lm/2-integrable curvature and trivial limit holonomy...

For a given singularity of a plane curve we consider the locus of nodal deformations of the singularity with the given number of nodes and describe possible components of the locus. As applications, we solve the local symplectic isotopy for nodal curves in a neighborhood of a given pseudoholomorphic curve without multiple components and prove the u...

It is shown that two braids represent transversally isotopic links if and only if one can pass from one braid to another by conjugations in braid groups, positive Markov moves, and their inverses.

The deformation problem for pseudoholomorphic curves and related geometrical properties of the total moduli space of pseudoholomorphic curves are studied. A sufficient condition for the saddle point property of the total moduli space is established. The local symplectic isotopy problem is formulated and solved for the case of imbedded pseudoholomor...

This are the notes of a course, given by the first author for the Graduiertenkollegs (=graduate students) at the Ruhr-University Bochum, in December 1997. These lectures pursued two main tasks: FIRST - to give a systematic and self-contained introduction to the Gromov theory of pseudoholomorphic curves. This is done in Chapters I,II,III. SECOND - t...

The purpose of this paper is to give the proof of two related results. The first is the Gromov compactness theorem for J -complex curves with boundary (but without boundary conditions on maps), and the second is an improvement of the removing of a point singularity theorem. An almost complex structure J in both theorems is supposed to be of class C...

We give a proof of the Gromov compactness theorem using the language of stable curves (i.e. cusp-curve of Gromov, or stable maps of Kontsevich and Manin) in general setting: An almost complex structure on a target manifold is only continuous and can vary; the curves are only assumed to have fixed ``topological type'', in particular they can be non-...

We study the envelopes of meromorphy of neighborhoods of symplectically immersed two-spheres in complex K\"ahler surfaces using the Gromov's theory of pseudoholomorphic curves. The construction of a complete family of holomorphic deformations of a non-compact complex curve in a complex manifold, parametrized by a finite codimension analytic subset...

The paper discusses the properties of the envelopes of meromorphy of neighbourhoods of symplectically immersed two-spheres in complex Kahler surfaces. The method used to study the envelopes of meromorphy is based on Gromov's theory of pseudoholomorphic curves. The exposition includes a construction of a complete family of holomorphic deformations o...

We show that under mild boundary conditions the moduli space of non-compact curves on a complex surface is (locally) an analytic subset of a ball in a Banach manifold, defined by {\it finitely} many holomorphic function.

We prove that the envelope of meromorphy of any imbedded symplectic sphere in $CP^2$ coincides with the whole $CP^2$. As a tool for the proof we use the Gromov theory of pseudo-holomorphic curves. Several results in this subject, such as adjunction formula, smoothness of moduli space in the neighborhood of a cusp-curve are improved. We introduce a...

For holomorphic line bundles with L2-bounded curvature we prove an theorem on extension across an analytic subset and construct a counterexample to extension across a smooth noncomplex submanifold of codimension 2.

We construct a natural complex structure on the moduli space of Riemann surfaces with boundary consisting of a finite number of punctures and circles and with marked points on boundary circles. We also give a description of the tangent space to the moduli space in terms of holomorphic objects associated to the corresponding Riemann surface.