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Andrzej Weber

Publications and source records attributed to Andrzej Weber.

At least 19 recordsLinked to original sources
arXiv2026

Link patterns and elliptic Hecke algebra

We compare three families of geometric objects: Schubert varieties in flag manifolds, matrix Schubert varieties, and Borel orbits of 2-nilpotent matrices. The first family is indexed by permutations, the second by partial permutations, and the third - the most general - by link patterns. Each of the geometric objects mentioned above carries a characteristic class in equivariant elliptic cohomology, defined in the framework provided by Borisov and Libgober. We introduce a Hecke-type algebra that gives inductive formulas for computing the equivariant elliptic classes of link patterns. This requires extending the action of the Hecke algebra to menage partial permutations and link patterns. In studying the extended action, an important role is played by the associated quadratic forms and by the action of reflections on forms. Also, we analyze the specialization of equivariant elliptic classes of link patterns to the corresponding classes of Schubert varieties.

math.CO↗
arXiv2025

Four paths from birational geometry to the elliptic genus

The article presents four reasons why the elliptic genus is the most general characteristic class that admits a generalization to singular spaces. We prove that the elliptic characteristic class (with an additional factor) is essentially the only characteristic class invariant under certain modifications, such as the Atiyah flop, Grassmannian flops, and modifications of Bott-Samelson resolutions.This result confirms and extends Totaro's result concerning the cobordism ring modulo classical flops. However, our approach is based on local calculus in equivariant cohomology.

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

Characteristic classes of Borel orbits of square-zero upper-triangular matrices

Anna Melnikov provided a parametrization of Borel orbits in the affine variety of square-zero $n \times n$ matrices by the set of involutions in the symmetric group. A related combinatorics leads to a construction a Bott-Samelson type resolution of the orbit closures. This allows to compute cohomological and K-theoretic invariants of the orbits: fundamental classes, Chern-Schwartz-MacPherson classes and motivic Chern classes in torus-equivariant theories. The formulas are given in terms of Demazure-Lusztig operations. The case of square-zero upper-triangular matrices is reach enough to include information about cohomological and K-theoretic classes of the double Borel orbits in $Hom(\mathbb C^k,\mathbb C^m)$ for $k+m=n$. We recall the relation with double Schubert polynomials and show analogous interpretation of Rimányi-Tarasov-Varchenko trigonometric weight function.

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↗
arXiv2020

Elliptic classes on Langlands dual flag varieties

Characteristic classes of Schubert varieties can be used to study the geometry and the combinatorics of homogeneous spaces. We prove a relation between elliptic classes of Schubert varieties on a generalized full flag variety and those on its Langlands dual. This new symmetry is only revealed if Schubert calculus is elevated from cohomology or K theory to the elliptic level.

math.AG↗
arXiv2020

Algebraic torus actions on contact manifolds

We prove the LeBrun-Salamon Conjecture in low dimensions. More precisely, we show that a contact Fano manifold X of dimension 2n+1 that has reductive automorphism group of rank at least n-2 is necessarily homogeneous. This implies that any positive quaternion-Kahler manifold of real dimension at most 16 is necessarily a symmetric space, one of the Wolf spaces. A similar result about contact Fano manifolds of dimension at most 9 with reductive automorphism group also holds. The main difficulty in approaching the conjecture is how to recognize a homogeneous space in an abstract variety. We contribute to such problem in general, by studying the action of algebraic torus on varieties and exploiting Bialynicki-Birula decomposition and equivariant Riemann-Roch theorems. From the point of view of T-varieties (that is, varieties with a torus action), our result is about high complexity T-manifolds. The complexity here is at most 1/2(dim X+5) with dim X arbitrarily high, but we require this special (contact) structure of X. Previous methods for studying T-varieties in general usually only apply for complexity at most 2 or 3.

math.AG↗
arXiv2020

Elliptic classes of Schubert varieties via Bott-Samelson resolution

Based on recent advances on the relation between geometry and representation theory, we propose a new approach to elliptic Schubert calculus. We study the equivariant elliptic characteristic classes of Schubert varieties of the generalized full flag variety $G/B$. For this first we need to twist the notion of elliptic characteristic class of Borisov-Libgober by a line bundle, and thus allow the elliptic classes to depend on extra variables. Using the Bott-Samelson resolution of Schubert varieties we prove a BGG-type recursion for the elliptic classes, and study the Hecke algebra of our elliptic BGG operators. For $G=GL_n(C)$ we find representatives of the elliptic classes of Schubert varieties in natural presentations of the K theory ring of $G/B$, and identify them with the Tarasov-Varchenko weight function. As a byproduct we find another recursion, different from the known R-matrix recursion for the fixed point restrictions of weight functions. On the other hand the R-matrix recursion generalizes for arbitrary reductive group $G$.

math.AG↗
arXiv2020

Elliptic classes, McKay correspondence and theta identities

We revisit the construction of elliptic class given by Borisov and Libgober for singular algebraic varieties. Assuming torus action we adjust the theory to equivariant local situation. We study theta function identities having geometric origin. In the case of quotient singularities $\mathbb C^n/G$, where $G$ is a finite group the theta identities arise from McKay correspondence. The symplectic singularities are of special interest. The Du Val surface singularity $A_n$ leads to a remarkable formula.

math.AG↗
arXiv2019

Characteristic classes of orbit stratifications, the axiomatic approach

Consider a complex algebraic group $G$ acting on a smooth variety $M$ with finitely many orbits, and let $Ω$ be an orbit. The following three invariants of $Ω\subset M$ can be characterized axiomatically: (1) the equivariant fundamental class $[\overlineΩ, M]\in H^*_G(M)$, (2) the equivariant Chern-Schwartz-MacPherson class $c(Ω, M)\in H^*_G(M)$, and (3) the equivariant motivic Chern class $mC(Ω, M) \in K_G(M)[y]$. The axioms for Chern-Schwartz-MacPherson and motivic Chern classes are motivated by the axioms for cohomological and K-theoretic stable envelopes of Okounkov and his coauthors. For $M$ a flag variety and $Ω$ a Schubert cell---an orbit of the Borel group acting---this implies that CSM and MC classes coincide with the weight functions studied by Rimanyi-Tarasov-Varchenko. In this paper we review the general theory and illustrate it with examples.

math.AG↗
arXiv2019

Elliptic classes of Schubert varieties

We introduce new notions in elliptic Schubert calculus: the (twisted) Borisov-Libgober classes of Schubert varieties in general homogeneous spaces G/P. While these classes do not depend on any choice, they depend on a set of new variables. For the definition of our classes we calculate multiplicities of some divisors in Schubert varieties, which were only known for full flag varieties before. Our approach leads to a simple recursions for the elliptic classes. Comparing this recursion with R-matrix recursions of the so-called elliptic weight functions of Rimanyi-Tarasov-Varchenko we prove that weight functions represent elliptic classes of Schubert varieties.

math.AG↗
arXiv2019

Motivic Chern classes and K-theoretic stable envelopes

We study a K-theoretic characteristic class of singular varieties, namely the equivariant motivic Chern class. We prove that the motivic Chern class is characterized by an axiom system inspired by that of "K-theoretic stable envelopes," recently defined by Okounkov and studied in relation with quantum group actions on the K-theory algebra of moduli spaces. We also give explicit formulas for the equivariant motivic Chern classes of Schubert cells and matrix Schubert cells. Lastly, we calculate the equivariant motivic Chern class of the orbits of the A2 quiver representation, which yields formulas for the motivic Chern classes of determinantal varieties and more general degeneracy loci.

math.AG↗
arXiv2018

Residues formulas for the push-forward in K-theory, the case of G2/P

We study residue formulas for push-forward in K-theory of homogeneous spaces. First we review formulas for classical groups, which we derive from a formula for the classical Grassmannian case. Next we consider the homogeneous spaces for G2. One of them embeds in the Grassmannian Gr(2,7). We find its fundamental class in the equivariant K-theory and obtain the residue formula for the push-forward. This formula is valid for G2/B as well.

math.RT↗
arXiv2017

Equivariant Hirzebruch classes and Molien series of quotient singularities

We study properties of the Hirzebruch class of quotient singularities $\mathbb{C}^n/G$, where $G$ is a finite matrix group. The main result states that the Hirzebruch class coincides with the Molien series of $G$ under suitable substitution of variables. The Hirzebruch class of a crepant resolution can be described specializing the orbifold elliptic genus constructed by Borisov and Libgober. It is equal to the combination of Molien series of centralizers of elements of $G$. This is an incarnation of the McKay correspondence. The results are illustrated with several examples, in particular of 4-dimensional symplectic quotient singularities.

math.AG↗
arXiv2016

On the Pelczynski conjecture on Auerbach Bases

We consider Auerbach bases in Banach spaces of dimension n>2. We show that there exists at least (n-1)n/2+1 such bases. This estimate follows from the calculation of the Lusternik-Schnirelmann category of the flag variety. A better estimate is obtained for generic Banach spaces by the Morse theory.

math.FA↗
arXiv2016

Paths of the directed suspension

We prove that the loop space of the directed suspension of a directed space is homotopy equivalent to the James construction. In particular, it does not depend on the directed structure of a given directed space.

math.AT↗
arXiv2016

On rigidity of flag varieties

We prove that the variety of complete flags for any semisimple algebraic group is rigid in any smooth family of Fano manifolds.

math.AG↗