Searcharxiv⌕ Search

arXiv subjects

Vsevolod Shevchishin

Publications and source records attributed to Vsevolod Shevchishin.

15 recordsLinked to original sources
arXiv2025

Anti-self-dual blowups II

Let $X$ be a closed, oriented four-manifold with $b_2^+ \leq 3$, and suppose $X$ contains a collection of pairwise disjoint embedded $(-2)$-spheres. We prove that there is a Riemannian metric on $X$ such that the Poincare dual of each of these spheres is represented by an anti-self-dual harmonic form. This extends our earlier result for $(-1)$-spheres. The main new ingredient is an application of Eliashberg's $h$-principle for overtwisted contact structures, which we use to construct self-dual harmonic forms on four-orbifolds with prescribed local behaviour near the orbifold singular set.

math.DG↗
arXiv2024

Anti-self-dual blowups

Let $X$ be a closed, oriented four-manifold containing an embedded sphere with self-intersection number $(-1)$. Suppose that $b_2^+(X) \leq 3$. We show that there exists a Riemannian metric on $X$ such that the cohomology class dual to this sphere is represented by an anti-self-dual harmonic form. Furthermore, such a metric can be constructed even when there are multiple disjoint embedded $(-1)$-spheres.

math.DG↗
arXiv2020

Anti-symplectic involutions on rational symplectic 4-manifolds

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.

math.SG↗
arXiv2019

On linear-quadratic Poisson pencils on trivial central extensions of semisimple Lie algebras

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 family of quadratic Poisson structures on $\gl(3)^*$ compatible with the canonical linear Poisson structure and containing the 3-parametric family of quadratic bivectors recently introduced by Vladimir Sokolov. The involutive family of polynomial functions related to the corresponding Poisson pencils contains the hamiltonian of the polynomial form of the elliptic Calogero--Moser system.

math.DG↗
arXiv2019

Symplectic triangle inequality

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

math.SG↗
arXiv2019

Elliptic diffeomorphisms of symplectic 4-manifolds

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.

math.SG↗
arXiv2015

Smoothing 3-dimensional polyhedral spaces

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.

math.DG↗
arXiv2010

Secondary Stiefel-Whitney class and diffeomorphisms of rational and ruled symplectic 4-manifolds

We introduce the secondary Stiefel-Whitney class $\tilde w_2$ of homotopically trivial diffeomorphisms and show that a homotopically trivial symplectomorphism of a ruled 4-manifold is isotopic to identity if and only if the class $\tilde w_2$ vanishes. Using this, we give a detailed description of the combinatorial structure of the diffeotopy group of ruled symplectic 4-manifolds $X$, either minimal or blown-up, and its action on the homology and homotopy groups $H_2(X,\zz)$, $π_1(X)$, and $π_2(X)$.

math.SG↗
arXiv2007

Lagrangian embeddings of the Klein bottle and combinatorial properties of mapping class groups

A proof of non-existence of Lagrangian embeddings of the Klein bottle K in \CP^2 is given. We exploit the existence of a special embedding of K in a symplectic Lefschetz pencil on \CP^2 and study its monodromy. As the main technical tool, we develop the theory of mapping class groups, considered as quotients of special Artin braid groups, and obtain some new results about combinatorial structure of such groups.

math.SG↗
arXiv2000

Pseudoholomorphic curves and the symplectic isotopy problem

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 pseudoholomorphic curves. It is shown that any two symplectically imbedded surfaces Sigma_0, Sigma_1 in CP^2 of the same degree d\le 6 are symplectically isotopic.

math.SG↗
arXiv1999

Complex Curves in Almost-Complex Manifolds and Meromorphic Hulls

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 - to explain our join results on envelopes of meromorphy of real surfaces in complex two-dimensional manifolds. We do this in Chapter IV.

math.CV↗
arXiv1998

Pseudo-holomorphic curves and envelopes of meromorphy of two-spheres in $CP^2$

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 natural holomorphic structure on the pulled back tangent bundle, in which the differential of a ps.-hol. map in an analytic morphism and describe cusps of ps.-hol. curves using Bennequin index.

math.CV↗