arXiv subjects

Mykola Matviichuk

Publications and source records attributed to Mykola Matviichuk.

8 recordsLinked to original sources

arXiv2026

Deformations of T-log-symplectic log-canonical Poisson structures and symmetric Poisson CGL extensions

For a complex algebraic torus $\mathbb{T}$, we study $\mathbb{T}$-invariant Poisson deformations of a $\mathbb{T}$-log-symplectic log-canonical Poisson structure $π_0$ on $\mathbb{C}^n$. We show that every $\mathbb{T}$-invariant first-order deformation of $π_0$ with linearly independent $(\mathbb{C}^\times)^n$-weights is unobstructed. For a special class of $π_0$ defined by the so-called symmetric $\mathbb{T}$-action data, we show that $π_0$ can be canonically deformed to symmetric $\mathbb{T}$-Poisson CGL extensions (of $\mathbb{C}$) as defined by K. Goodearl and M. Yakimov. As a consequence, we classify all symmetric $\mathbb{T}$-Poisson CGL extensions in terms of their log-canonical terms $π_0$ and the second $\mathbb{T}$-invariant Poisson cohomology of $π_0$. We further characterize, among all symmetric Poisson CGL extensions, those of Cartan type, i.e., those associated to sequences of simple roots in the root systems of symmetrizable generalized Cartan matrices. In particular, we prove that the standard Poisson structures on Bott-Samelson cells and generalized Schubert cells for semi-simple complex Lie groups are the (uniquely determined) maximal normalized admissible deformations of their log-canonical terms. Finally, for any symmetric $\mathbb{T}$-Poisson CGL extension $π$ with log-canonical term $π_0$, we present an explicit formula expressing the initial mutation matrix in the Goodearl-Yakimov theory on cluster algebras associated to $π$ in terms of the $(\mathbb{C}^\times)^n$-weights of the second $\mathbb{T}$-invariant Poisson cohomology of $π_0$.

math.SG↗

arXiv2026

Weighted blowups and 3d Poisson desingularizations

We establish existence of functorial orbifold reductions of singularities for Poisson subvarieties in smooth Poisson threefolds. Namely, we show that with enough weighted blowups, one can reduce the singularities of such Poisson subvarieties to certain simple, explicit, local normal forms: Du Val surface singularities where the Poisson structure is locally Jacobian, and plane curves lying in the vanishing locus of a particular linear Poisson structure. The proof combines Abramovich--Temkin--Włodarczyk and McQuillan's recent approach to resolution of singularities for varieties via weighted blowups with some new normal forms for three-dimensional Poisson brackets derived via Poisson cohomology. Along the way, we describe necessary and sufficient conditions for a polyvector field to lift to the weighted blowup of an orbifold along a suborbifold, generalizing criteria of Polishchuk for unweighted blowups of Poisson structures on smooth varieties.

math.AG↗

arXiv2024

Creating quantum projective spaces by deforming q-symmetric algebras

We construct a large collection of "quantum projective spaces", in the form of Koszul, Calabi-Yau algebras with the Hilbert series of a polynomial ring. We do so by starting with the toric ones (the q-symmetric algebras), and then deforming their relations using a diagrammatic calculus, proving unobstructedness of such deformations under suitable nondegeneracy conditions. We then prove that these algebras are identified with the canonical quantizations of corresponding families of quadratic Poisson structures, in the sense of Kontsevich. In this way, we obtain the first broad class of quadratic Poisson structures for which his quantization can be computed explicitly, and shown to converge, as he conjectured in 2001.

math.QA↗

arXiv2023

Elliptic log symplectic brackets on projective bundles

Let $\mathsf{X}$ be the product of a complex projective space and a polydisc. We study Poisson brackets on $\mathsf{X}$ that are log symplectic, that is, generically symplectic and such that the inverse two-form has only first order poles. We propose a method of constructing such Poisson brackets that additionally are elliptic, in a precise sense. Our method relies on the local Torelli theorem for log symplectic manifolds of Pym, Schedler and the author, and uses combinatorics of smoothing diagrams. We demonstrate effectiveness of the method on a series of examples, recovering, in particular, all log symplectic cases of elliptic Feigin-Odesskii Poisson brackets $q_{n,k}$ on $\mathbb{P}^{n-1}$.

math.AG↗

arXiv2023

Elliptic zastava

We study the elliptic zastava spaces, their versions (twisted, Coulomb, Mirkovic local spaces, reduced) and relations with monowalls moduli spaces and Feigin-Odesskii moduli spaces of $G$-bundles with parabolic structure on an elliptic curve.

math.AG↗

arXiv2022

Holonomic Poisson geometry of Hilbert schemes

We undertake a detailed study of the geometry of Bottacin's Poisson structures on Hilbert schemes of points in Poisson surfaces, i.e. smooth complex surfaces equipped with an effective anticanonical divisor. We focus on three themes that, while logically independent, are linked by the interplay between (characteristic) symplectic leaves and deformation theory. Firstly, we construct the symplectic groupoids of the Hilbert schemes and develop the classification of their symplectic leaves, using the methods of derived symplectic geometry. Secondly, we establish local normal forms for the Poisson brackets, and combine them with a toric degeneration argument to verify that Hilbert schemes satisfy our recent conjecture characterizing holonomic Poisson manifolds in terms of the geometry of the modular vector field. Finally, using constructible sheaf methods, we compute the space of first-order Poisson deformations when the anti-canonical divisor is reduced and has only quasi-homogeneous singularities. (The latter is automatic if the surface is projective.) Along the way, we find a tight connection between the Poisson geometry of the Hilbert schemes and the finite-dimensional Lie algebras of affine transformations, which is mediated by syzygies. In particular, we find that the Hilbert scheme has a natural subvariety that serves as a global counterpart of the nilpotent cone, and we prove that the Lie algebras of affine transformations have holonomic dual spaces -- the first such series of Lie algebras to be discovered.

math.AG↗

arXiv2021

A local Torelli theorem for log symplectic manifolds

We establish a local model for the moduli space of holomorphic symplectic structures with logarithmic poles, near the locus of structures whose polar divisor is normal crossings. In contrast to the case without poles, the moduli space is singular: when the cohomology class of a symplectic structure satisfies certain linear equations with integer coefficients, its polar divisor can be partially smoothed, yielding adjacent irreducible components of the moduli space that correspond to possibly non-normal crossings structures. These components are indexed by combinatorial data we call smoothing diagrams, and amenable to algorithmic classification. Applying the theory to four-dimensional projective space, we obtain a total of 40 irreducible components of the moduli space, most of which are new. Our main technique is a detailed analysis of the relevant deformation complex (the Poisson cohomology) as an object of the constructible derived category.

math.AG↗

arXiv2011

On the dynamics of subcontinua of a tree

Given a tree map $f:T\to T$, we study the dynamics of subcontinua of $T$ under action of $f$. In particular, we prove that a subcontinuum of $T$ is either asymptotically periodic or asymptotically degenerate. As an application of this result, we show that zero topological entropy of the system $(T,f)$ implies zero topological entropy of its functional envelope (endowed with the Hausdorff metric).

math.DS↗