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Tadej Starčič

Publications and source records attributed to Tadej Starčič.

12 recordsLinked to original sources
arXiv2026

Centralizers of elements in the orthogonal, symplectic and unitary groups

We explicitly compute and parametrize the centralizers of elements, encompassing dege\-ne\-rate and unipotent ones, in the orthogonal, symplectic, and unitary gro\-ups over the real and complex fields, including the indefinite cases. The same approach simultaneously yields the isotropy groups of $H$-skew-symme\-tric, Hamiltonian, and $H$-Hermitian matrices under the adjoint representations of the corresponding classical groups. The resulting groups are conjugate to subgroups of nonsingular block matrices whose blocks exhibit a rectangular block Toeplitz structure.

math.AG↗
arXiv2026

Centralizers of the complex orthogonal and symplectic group

We find a recursive algorithm for computing the precise centralizers of the complex orthogonal and symplectic groups, and hence the isotropy groups, with respect to the similarity transformation on the spaces of skew-symmetric and Hamiltonian matrices, respectively. These groups are conjugate to groups of certain nonsingular block matrices whose blocks are rectangular block Toeplitz.

math.AG↗
arXiv2023

Isotropy groups of the action of orthogonal *congruence on Hermitian matrices

We present a procedure which enables the computation and the description of structures of isotropy subgroups of the group of complex orthogonal matrices with respect to the action of *congruence on Hermitian matrices. A key ingredient in our proof is an algorithm giving solutions of a certain rectangular block (complex-alternating) upper triangular Toeplitz matrix equation.

math.DG↗
arXiv2023

Stein neighborhood bases of embedded strongly pseudoconvex domains and approximation of mappings

In this paper we construct a Stein neighborhood basis for any compact subvariety $A$ with strongly pseudoconvex boundary $bA$ and Stein interior $A\backslash bA$ in a complex space $X$. This is an extension of a well known theorem of Siu. When $A$ is a complex curve, our result coincides with the result proved by Drinovec-Drnovšek and Forstnerič. We shall adapt their proof to the higher dimensional case, using also some ideas of Demailly's proof of Siu's theorem. For embedded strongly pseudoconvex domain in a complex manifold we also find a basis of tubular Stein neighborhoods. These results are applied to the approximation problem for holomorphic mappings.

math.CV↗
arXiv2022

On structures of normal forms of complex points of small $\mathcal{C}^{2}$-perturbations of real $4$-manifolds embedded in a complex $3$-manifold

We extend our previous result on the behavior of the quadratic part of a complex points of a small $\mathcal{C}^{2}$-perturbation of a real $4$-manifold embedded in a complex $3$-manifold. We describe the change of the structure of a normal form of a complex point. It is an immediate consequence of a theorem clarifying how small perturbations can change the bundle of a pair of one arbitrary and one symmetric $2\times 2$ matrix with respect to an action of a certain linear group.

math.CV↗
arXiv2021

Isotropy groups of the action of orthogonal similarity on symmetric matrices

We find an algorithmic procedure that enables to compute and to describe the structure of the isotropy subgroups of the group of complex orthogonal matrices with respect to the action of similarity on complex symmetric matrices. A key step in our proof is to solve a certain rectangular block upper-triangular Toeplitz matrix equation.

math.DG↗
arXiv2020

The stabilizers for the action of orthogonal similarity on symmetric matrices and orthogonal $*$-conjugacy on Hermitian matrices

We describe the recursive algorithmic procedure to compute the stabilizers of the group of complex orthogonal matrices with respect to the action of similarity on the set of all symmetric matrices. Futhermore, lower bounds for dimensions of the stabilizers for the action of orthogonal $*$-conjugation on Hermitian matrices are obtained. We also prove a result that completes the classification of normal forms of Hermitian matrices under orthogonal $*$-conjugation. A key step in our proof is to solve a certain block matrix equation with Toeplitz blocks. These results are then applied to provide a theorem on normal forms of the quadratic parts of flat complex points in a real codimension $2$ submanifold in a complex manifold.

math.AG↗
arXiv2019

On normal forms of complex points of small $\mathcal{C}^{2}$-perturbations of real $4$-manifolds embedded in a complex $3$-manifold

The purpose of this paper is to give a better understanding of complex points up to quadratic terms of real codimension $2$ submanifolds embedded in a complex $3$-manifold. We answer the question how a normal form of a pair of one arbitrary and one symmetric $2\times 2$ matrix with respect to a certain linear group action changes under arbitrarily small perturbations. This result is then applied to describe the quadratic part of normal forms of complex points of small $\mathcal{C}^{2}$-perturbations of real $4$-manifolds embedded in a complex $3$-manifold.

math.CV↗
arXiv2018

On regular Stein neighborhoods of a union of two maximal totally real subspaces in $\mathbb{C}^n$

We present a construction of regular Stein neighborhoods of a union of maximally totally real subspaces $M=(A+iI)\mathbb{R}^n$ and $N=\mathbb{R}^n$ in $\mathbb{C}^n$, provided that the entries of a real $n \times n$ matrix $A$ are sufficiently small. Our proof is based on a local construction of a suitable plurisubharmonic function $ρ$ near the origin, such that the sublevel sets of $ρ$ are strongly pseudoconvex and admit strong deformation retraction to $M\cup N$. We also give the application of this result to totally real immersions of real $n$-manifolds in $\mathbb{C}^n$ with only finitely many double points, and such that the union of the tangent spaces at each intersection in some local coordinates coincides with $M\cup N$, described above.

math.CV↗
arXiv2018

On Normal Forms of Complex Points of codimension 2 submanifolds

In this paper we present some linear algebra behind quadratic parts of quadratically flat complex points of codimension two real submanifold in a complex manifold. Assuming some extra nondegenericity and using the result of Hong, complete normal form descriptions can be given, and in low dimensions, we obtain a complete classification without any extra assumptions.

math.CV↗
arXiv2016

On regular Stein neighborhoods of a union of two totally real planes in $\mathbb{C}^2$

In this paper we find regular Stein neighborhoods for a union of totally real planes $M=(A+iI)\mathbb{R}^2$ and $N=\mathbb{R}^2$ in $\mathbb{C}^2$ provided that the entries of a real $2 \times 2$ matrix $A$ are sufficiently small. A key step in our proof is a local construction of a suitable function $ρ$ near the origin. The sublevel sets of $ρ$ are strongly Levi pseudoconvex and admit strong deformation retraction to $M\cup N$.

math.CV↗