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Jakub Koncki

Publications and source records attributed to Jakub Koncki.

9 recordsLinked to original sources
arXiv2026

Degenerations of multisingularities and Artin algebras

We study the degeneration hierarchy of commutative, associative, finite-dimensional complex Artin algebras. Instead of studying degenerations in the Hilbert scheme, we introduce a singularity-theoretic notion of degeneration based on the correspondence between singularities of stable map germs and local algebras. This leads to a natural partially ordered set, the stable hierarchy, in which one algebra degenerates to another if nearby singularities realize the latter. Our first main result is that, in a wide range of dimensions, this hierarchy can be determined purely from symmetry data, namely from the automorphism groups of the algebras or singularities. The key tool is the theory of certain equivariant characteristic classes called Thom polynomials of multisingularities, established by Kazarian. Suitable substitutions into these polynomials completely characterize the hierarchy. As a consequence, the computation of degeneration posets becomes algorithmic in nature. In our second main result, we prove that our singularity-theoretic hierarchy extends the algebraic hierarchy obtained from deformation theory. While deformation theory requires the dimension (rank) to be fixed, our hierarchy generalizes this framework by comparing algebras of varying dimensions.

math.AG↗
arXiv2026

The multiplicative structure of the K-theoretical McKay correspondence for the Hilbert scheme of points in the complex plane

We consider the K-theory of the Hilbert scheme of points in the complex plane, which under McKay correspondence is isomorphic to the space of symmetric functions $Λ^n$. We prove a formula conjectured by Boissière for the endomorphism of $Λ^n$ induced by multiplication by the classes of the Adams powers of the tautological bundle. We describe the structure constants for the multiplication on $Λ^n$ induced by the tensor product in K-theory.

math.AG↗
arXiv2025

Higher characteristic classes of multisingularity loci

A map between manifolds induces stratifications of both the source and the target according to the occurring multisingularities. In this paper, we study universal expressions-called higher Thom polynomials-that describe the Segre-Schwartz-MacPherson class of such multisingularity loci. We prove a Structure Theorem reducing these Thom polynomials to the data of a linear series associated with each multisingularity. The series corresponding to the empty multisingularity, referred to as the Master Series, plays a distinguished role. Motivated by connections with geometric representation theory, we further prove an Interpolation Theorem that allows Thom polynomials to be computed algorithmically within Mather's range of nice dimensions. As an application, we derive an explicit formula for the image Milnor number of quasihomogeneous germs, providing one side of the celebrated Mond conjecture, computable up to the theoretical bound.

math.AG↗
arXiv2025

Nakajima's creation operators and the Kirwan map

We consider the Hilbert scheme of points in the affine complex plane. We find explicit formulas for the Nakajima's creation operators and their K-theoretic counterparts in terms of the Kirwan map. We obtain a description of the action of Nakajima's creation operators on the Chern classes of the tautological bundle.

math.AG↗
arXiv2024

Hecke algebra action on twisted motivic Chern classes and K-theoretic stable envelopes

Let $G$ be a linear semisimple algebraic group and $B$ its Borel subgroup. Let $\mathbb{T}\subset B$ be the maximal torus. We study the inductive construction of Bott-Samelson varieties to obtain recursive formulas for the twisted motivic Chern classes of Schubert cells in $G/B$. To this end we introduce two families of operators acting on the equivariant K-theory $K_\mathbb{T}(G/B)[y]$, the right and left Demazure-Lusztig operators depending on a parameter. The twisted motivic Chern classes coincide (up to normalization) with the K-theoretic stable envelopes. Our results imply wall-crossing formulas for a change of the weight chamber and slope parameters. The right and left operators generate a twisted double Hecke algebra. We show that in the type $A$ this algebra acts on the Laurent polynomials. This action is a natural lift of the action on $K_\mathbb{T}(G/B)[y]$ with respect to the Kirwan map. We show that the left and right twisted Demazure-Lusztig operators provide a recursion for twisted motivic Chern classes of matrix Schubert varieties.

math.AG↗
arXiv2022

Twisted motivic Chern class and stable envelopes

We present a definition of {\em twisted motivic Chern classes} for singular pairs $(X,Δ)$ consisting of a singular space $X$ and a $\mathbb Q$-Cartier divisor containing the singularities of $X$. The definition is a mixture of the construction of motivic Chern classes previously defined by Brasselet-Sch{ü}rmann-Yokura with the construction of multiplier ideals. The twisted motivic Chern classes are the limits of the elliptic classes defined by Borisov-Libgober. We show that with a suitable choice of the divisor $Δ$ the twisted motivic Chern classes satisfy the axioms of the stable envelopes in the K-theory. Our construction is an extension of the results proven by the first author for the fundamental slope.

math.AG↗
arXiv2021

Comparison of motivic Chern classes and stable envelopes for cotangent bundles

We consider a complex smooth projective variety equipped with an action of an algebraic torus with a finite number of fixed points. We compare the motivic Chern classes of Białynicki-Birula cells with the $K$-theoretic stable envelopes of cotangent bundle. We prove that under certain geometric assumptions satisfied e.g. by homogenous spaces these two notions coincide up to normalization.

math.AG↗
arXiv2021

Motivic Chern classes of configuration spaces

We calculate the equivariant motivic Chern class for configuration space of a quasiprojective (maybe singular) variety and the space of vectors with different directions. We prove the formulas for generating series of these classes. We generalize the localization theorems results about Bialynicki-Birula decomposition to acquire some stability for the motivic Chern classes of configuration spaces.

math.AG↗