arXiv subjects

András Cristian Lőrincz

Publications and source records attributed to András Cristian Lőrincz.

7 recordsLinked to original sources

arXiv2023

Singularities of orthogonal and symplectic determinantal varieties

Let either $GL(E)\times SO(F)$ or $GL(E)\times Sp(F)$ act naturally on the space of matrices $E\otimes F$. There are only finitely many orbits, and the orbit closures are orthogonal and symplectic generalizations of determinantal varieties, which can be described similarly using rank conditions. In this paper, we study the singularities of these varieties and describe their defining equations. We prove that in the symplectic case, the orbit closures are normal with good filtrations, and in characteristic $0$ have rational singularities. In the orthogonal case we show that most orbit closures will have the same properties, and determine precisely the exceptions to this.

math.AG↗

arXiv2021

On the collapsing of homogeneous bundles in arbitrary characteristic

We study the geometry of equivariant, proper maps from homogeneous bundles $G\times_P V$ over flag varieties $G/P$ to representations of $G$, called collapsing maps. Kempf showed that, provided the bundle is completely reducible, the image $G\cdot V$ of a collapsing map has rational singularities in characteristic zero. We extend this result to positive characteristic and show that for the analogous bundles the saturation $G\cdot V$ is strongly $F$-regular if its coordinate ring has a good filtration. We further show that in this case the images of collapsing maps of homogeneous bundles restricted to Schubert varieties are $F$-rational in positive characteristic, and have rational singularities in characteristic zero. We provide results on the singularities and defining equations of saturations $G\cdot X$ for $P$-stable closed subvarieties $X\subset V$. We give criteria for the existence of good filtrations for the coordinate ring of $G\cdot X$. Our results give a uniform, characteristic-free approach for the study of the geometry of a number of important varieties: multicones over Schubert varieties, determinantal varieties in the space of matrices, symmetric matrices, skew-symmetric matrices, and certain matrix Schubert varieties therein, representation varieties of radical square zero algebras (e.g. varieties of complexes), subspace varieties, higher rank varieties, etc.

math.AG↗

arXiv2021

Holonomic functions and prehomogeneous spaces

A function that is analytic on a domain of $\mathbb{C}^n$ is holonomic if it is the solution to a holonomic system of linear homogeneous differential equations with polynomial coefficients. We define and study the Bernstein-Sato polynomial of a holonomic function on a smooth algebraic variety. We analyze the structure of certain sheaves of holonomic functions, such as the algebraic functions along a hypersurface, determining their direct sum decompositions into indecomposables, that further respect decompositions of Bernstein-Sato polynomials. When the space is endowed with the action of a linear algebraic group $G$, we study the class of $G$-finite analytic functions, i.e. functions that under the action of the Lie algebra of $G$ generate a finite dimensional rational $G$-module. These are automatically algebraic functions on a variety with a dense orbit. When $G$ is reductive, we give several representation-theoretic techniques toward the determination of Bernstein-Sato polynomials of $G$-finite functions. We classify the $G$-finite functions on all but one of the irreducible reduced prehomogeneous vector spaces, and compute the Bernstein-Sato polynomials for distinguished $G$-finite functions. The results can be used to construct explicitly equivariant $\mathcal{D}$-modules.

math.AG↗

arXiv2020

Minimal free resolutions of ideals of minors associated to pairs of matrices

Consider the affine space consisting of pairs of matrices $(A,B)$ of fixed size, and its closed subvariety given by the rank conditions $\operatorname{rank} A \leq a$, $\operatorname{rank} B \leq b$ and $\operatorname{rank} (A\cdot B) \leq c$, for three non-negative integers $a,b,c$. These varieties are precisely the orbit closures of representations for the equioriented $A_3$ quiver. In this paper we construct the (equivariant) minimal free resolutions of the defining ideals of such varieties. We show how this problem is equivalent to determining the cohomology groups of the tensor product of two Schur functors of tautological bundles on a 2-step flag variety. We provide several techniques for the determination of these groups, which is of independent interest.

math.AG↗

arXiv2018

The b-functions of semi-invariants of quivers

In this paper we compute b-functions (or Bernstein-Sato polynomials) of various semi-invariants of quivers. The main tool is an explicit relation for the b-functions between semi-invariants that correspond to each other under reflection functors (or castling transforms). This enables us to compute recursively the b-functions for all Dynkin quivers, and extended Dynkin quivers with prehomogeneous dimension vectors.

math.RT↗

arXiv2018

Decompositions of Bernstein-Sato polynomials and slices

Let $G$ be a linearly reductive group acting on a vector space $V$, and $f$ a (semi-)invariant polynomial on $V$. In this paper we study systematically decompositions of the Bernstein-Sato polynomial of $f$ in parallel with some representation-theoretic properties of the action of $G$ on $V$. We provide a technique based on a multiplicity one property, that we use to compute the Bernstein-Sato polynomials of several classical invariants in an elementary fashion. Furthermore, we derive a "slice method" which shows that the decomposition of $V$ as a representation of $G$ can induce a decomposition of the Bernstein-Sato polynomial of $f$ into a product of two Bernstein-Sato polynomials - that of an ideal and that of a semi-invariant of smaller degree. Using the slice method, we compute Bernstein-Sato polynomials for a large class of semi-invariants of quivers.

math.RT↗

arXiv2017

Singularities of zero sets of semi-invariants for quivers

Let $Q$ be a quiver with dimension vector $α$ prehomogeneous under the action of the product of general linear groups $\operatorname{GL}(α)$ on the representation variety $\operatorname{Rep}(Q,α)$. We study geometric properties of zero sets of semi-invariants of this space. It is known that for large numbers $N$, the nullcone in $\operatorname{Rep}(Q,N\cdot α)$ becomes a complete intersection. First, we show that it also becomes reduced. Then, using Bernstein-Sato polynomials, we discuss some criteria for zero sets to have rational singularities. In particular, we show that for Dynkin quivers codimension $1$ orbit closures have rational singularities.

math.RT↗