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Aristides Kontogeorgis

Publications and source records attributed to Aristides Kontogeorgis.

At least 19 recordsLinked to original sources
arXiv2026

Equivariant Koszul Cohomology of Canonical Curves

This paper investigates the representation-theoretic structure of the Koszul cohomology of a smooth projective variety $X$ over an algebraically closed field $k$, admitting an action of a finite group $G$ of order coprime to ${\rm char}(k)$. Properties of $G$-equivariant functors are employed to show that the associated Koszul complex is a complex of $kG$-modules, and to generalize known dimension formulas to identities between virtual representations. In the case of canonical curves, explicit formulas are obtained by combining the theory of equivariant Euler characteristics and equivariant Riemann-Roch theorems with that of generating functions for Schur functors.

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arXiv2025

An Arithmetic Topology viewpoint on Descent theory and Equivariant Categories

We establish a unified group-theoretic framework bridging the arithmetic homotopy exact sequence of a variety and the Birman exact sequence of a surface. Within this framework, we reinterpret classical arithmetic notions - such as the descent of varieties and of covers - and construct their topological analogues. We formalize the parallel setting between closed subgroups of the absolute Galois group and subgroups of the Mapping Class Group of a base space and their actions on fundamental groups. This provides an analogy between arithmetic and topological invariants, allowing us to define the groups of moduli, definition, and invariance in both settings. Using this unified perspective, some purely group-theoretic proofs provide results in both settings simultaneously. Applications include a topological analogue of Weil's Descent Theorem for mapping class groups and an adaptation of Débes and Douai's cohomological obstructions regarding descent of algebraic covers to the topological setting. Finally, we elevate these results to the categorical level. We demonstrate that the classical Weil cocycle condition is equivalent to the existence of a linearization in the language of equivariant categories. Applying this perspective to the bounded derived category of coherent sheaves $\mathsf{D^b}(X)$, we show that the equivariant derived category $\mathsf{D^b}(X)^G$, under the action induced by a Weil descent datum, recovers the derived category of the descended variety.

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arXiv2025

On cyclic groups covers of the projective line

This article extends the study of cyclic ramified covers of the projective line defined by Kummer equations. We consider the most general case of such covers, allowing arbitrary orders in the roots of the generating radicant. The primary goal is the computation of the fundamental group of both the open and complete curve. We employ tools of combinatorial group theory utilizing the Smith Normal Form. This result is further visualized through the theory of foldings and $S$-graphs. Finally, we apply the theory of Alexander modules and the Crowell exact sequence to compute the abelianization of the fundamental group, $H_{1}(X, \mathbb{Z})$, and determine its Galois~module~structure over a field $k$ confirming the result using the Chevalley-Weil formula.

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arXiv2025

On the de Rham cohomology of cyclic covers

We compute explicit bases for the de Rham cohomology of cyclic covers of the projective line defined over an algebraically closed field of characteristic $p\geq 0$. For both Kummer and Artin-Schreier extensions, we describe precise $k$-bases for the cohomology groups $H^{1}(X,\mathcal{O}_{X})$ and $H^{0}(X,Ω_{X})$, and we use these to construct an explicit basis for the first de Rham cohomology group $H^{1}_{\mathrm{dR}}(X/k)$ via Čech cohomology. Our approach relies on detailed computations of divisors of functions and differentials, together with residue calculations and the duality pairing between $H^{0}(X,Ω_{X})$ and $H^{1}(X,\mathcal{O}_{X})$. The resulting expressions are given in closed form in terms of the defining equation of the cover, making the cohomology fully explicit and readily applicable to questions involving group actions, and the study of $p$-cyclic covers.

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arXiv2025

An obstruction to the local lifting problem

We are investigating the lifting problem for local actions involving semidirect products of a cyclic $p$-group with a cyclic group prime to $p$, where $p$ represents the characteristic of the special fiber. We establish a criterion based on the Harbater-Katz-Gabber compactification of local actions, enabling us to determine whether a given local action can be lifted or not. Specifically, in the case of the dihedral group, we present an example of a local dihedral action that cannot be lifted. This instance provides a more potent obstruction than the KGB obstruction.

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arXiv2025

The de Rham cohomology of covers with cyclic $p$-Sylow subgroup

Let $X$ be a smooth projective curve over a field $k$ with an action of a finite group $G$. A well-known result of Chevalley and Weil describes the $k[G]$-module structure of cohomologies of $X$ in the case when the characteristic of $k$ does not divide $\# G$. It is unlikely that such a formula can be derived in the general case, since the representation theory of groups with non-cyclic $p$-Sylow subgroups is wild in characteristic $p$. The goal of this article is to show that when $G$ has a cyclic $p$-Sylow subgroup, the $G$-structure of the de Rham cohomology of $X$ is completely determined by the ramification data. In principle, this leads to new formulas in the spirit of Chevalley and Weil for such curves. We provide such an explicit description of the de Rham cohomology in the cases when $G = \mathbb Z/p^n$ and when the $p$-Sylow subgroup of $G$ is normal of order $p$.

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arXiv2024

Galois Action on Homology of the Heisenberg Curve

The Heisenberg curve is defined topologically as a cover of the Fermat curve and corresponds to an extension of the projective line minus three points by the non-abelian Heisenberg group modulo n. We compute its fundamental group and investigate an action from Artin's Braid group to the curve itself and its homology. We also provide a description of the homology in terms of irreducible representations of the Heisenberg group over a field of characteristic $0$.

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arXiv2023

On the lifting problem of representations of a metacyclic group

We give a necessary and sufficient condition for a modular representation of a group $G=C_{p^h} \rtimes C_m$ in a field of characteristic zero to be lifted to a representation over local principal ideal domain of characteristic zero containing the $p^h$ roots of unity.

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arXiv2021

The equivariant Hilbert series of the canonical ring of Fermat curves

We consider a Fermat curve $F_n:x^n+y^n+z^n=1$ over an algebraically closed field $k$ of characteristic $p\geq0$ and study the action of the automorphism group $G=\left(\mathbb{Z}/n\mathbb{Z}\times\mathbb{Z}/n\mathbb{Z}\right)\rtimes S_3$ on the canonical ring $R=\bigoplus H^0(F_n,Ω_{F_n}^{\otimes m})$ when $p>3$, $p\nmid n$ and $n-1$ is not a power of $p$. In particular, we explicitly determine the classes $[H^0(F_n,Ω_{F_n}^{\otimes m})]$ in the Grothendieck group $K_0(G,k)$ of finitely generated $k[G]$-modules, describe the respective equivariant Hilbert series $H_{R,G}(t)$ as a rational function, and use our results to write a program in Sage that computes $H_{R,G}(t)$ for an arbitrary Fermat curve.

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arXiv2020

Galois action on homology of generalized Fermat Curves

The fundamental group of Fermat and generalized Fermat curves is computed. These curves are Galois ramified covers of the projective line with abelian Galois groups $H$. We provide a unified study of the action of both cover Galois group $H$ and the absolute Galois group $\mathrm{Gal}(\bar{\Q}/\Q)$ on the pro-$\ell$ homology of the curves in study. Also the relation to the pro-$\ell$ Burau representation is investigated.

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arXiv2020

Galois structure of the holomorphic differentials of curves

Let $X$ be a smooth projective geometrically irreducible curve over a perfect field $k$ of positive characteristic $p$. Suppose $G$ is a finite group acting faithfully on $X$ such that $G$ has non-trivial cyclic Sylow $p$-subgroups. We show that the decomposition of the space of holomorphic differentials of $X$ into a direct sum of indecomposable $k[G]$-modules is uniquely determined by the lower ramification groups and the fundamental characters of closed points of $X$ that are ramified in the cover $X\to X/G$. We apply our method to determine the $\mathrm{PSL}(2,\mathbb{F}_\ell)$-module structure of the space of holomorphic differentials of the reduction of the modular curve $\mathcal{X}(\ell)$ modulo $p$ when $p$ and $\ell$ are distinct odd primes and the action of $\mathrm{PSL}(2,\mathbb{F}_\ell)$ on this reduction is not tamely ramified. This provides some non-trivial congruences modulo appropriate maximal ideals containing $p$ between modular forms arising from isotypic components with respect to the action of $\mathrm{PSL}(2,\mathbb{F}_\ell)$ on $\mathcal{X}(\ell)$.

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arXiv2020

Arithmetic actions on cyclotomic function fields

We derive the group structure for cyclotomic function fields obtained by applying the Carlitz action for extensions of an initial constant field. The tame and wild structures are isolated to describe the Galois action on differentials. We show that the associated invariant rings are not polynomial.

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arXiv2020

Biduality and Reflexivity in Positive Characteristic

The biduality and reflexivity theorems are known to hold for projective varieties defined over fields of characteristic zero, and to fail in positive characteristic. In this article, we construct a notion of reflexivity and biduality in positive characteristic by generalizing the ordinary tangent space to the notion of $h$-tangent spaces. The ordinary reflexivity theory can be recovered as the special case $h=0$, of our theory. Several varieties that are not ordinary reflexive or bidual become reflexive in our extended theory.

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arXiv2019

A generalized skein relation for Khovanov homology and a categorification of the $θ$-invariant

The Jones polynomial is a famous link invariant that can be defined diagrammatically via a skein relation. Khovanov homology is a richer link invariant that categorifies the Jones polynomial. Using spectral sequences, we obtain a skein-type relation satisfied by the Khovanov homology. Thanks to this relation, we are able to generalize the Khovanov homology in order to obtain a categorification of the $θ$-invariant, which is itself a generalization of the Jones polynomial.

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