Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2407.02947

Modules of derivations, logarithmic ideals and singularities of maps on analytic varieties

Carles Bivià-Ausina·Konstantinos Kourliouros·Maria Aparecida Soares Ruas
View original source ↗Read source document ↗Export metadata ↓

Abstract

We introduce the module of derivations $Θ_{h,M}$ attached to a given analytic map $h:(\mathbb C^n,0)\to (\mathbb C^p,0)$ and a submodule $M\subseteq \mathcal O_n^p$ and analyse several exact sequences related to $Θ_{h,M}$. Moreover, we obtain formulas for several numerical invariants associated to the pair $(h,M)$ and a given analytic map germ $f:(\mathbb C^n,0)\to (\mathbb C^q,0)$. In particular, if $X$ is an analytic subvariety of $\mathbb C^n$, we derive expressions for analytic invariants defined in terms of the module $Θ_X$ of logarithmic vector fields of $X$.

Explore related subjects

math.AGmath.ACmath.CV

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Carles Bivià-Ausina, Konstantinos Kourliouros, Maria Aparecida Soares Ruas. 2024-07-03. Modules of derivations, logarithmic ideals and singularities of maps on analytic varieties. https://arxiv.org/abs/2407.02947

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

G3-Criteria and Applications

The G3-property of a subvariety was introduced by Hironaka-Matsumura, and plays an important role for deducing connectedness and extension results. Unfortunately, it's a rather elusive notion, which is not always easy to establish. Most of the existing work is concentrated on subvarieties of homogeneous varieties. The first goal of this article is to show that mobility assumptions on the subvariety, considered in works of Badescu, Chow, Debarre, Voisin, yield a certain partial positivity property, slightly stronger than G3, previously introduced by the author. Second, we apply the result to prove that, in numerous situations, the splitting of the normal bundle of a smooth two-codimensional subvariety implies that it is a complete intersection.

math.AG↗
arXiv2026

Moduli Stacks of $G$-Curves in Homotopy Theory at Height $p-1$

Let $p$ be odd and $G' = \mathbb{Z}/p \rtimes \mathbb{Z}/(p-1)^2$ the maximal finite subgroup of the Morava stabilizer group at height $p-1$. Inverse Galois theory produces from $G'$ alone a curve $X$, the unique curve of minimal genus with $\operatorname{Aut}(X) \simeq G'$; its ramification, its field of definition and its equation are consequences of the group, not choices. We prove a $G'$-equivariant equivalence between the deformations of $X$ and Lubin--Tate space, so that the Lubin--Tate action of $G'$ is the action of $\operatorname{Aut}(X)$ on deformations of the curve. The proof is a coordinate-free Kodaira--Spencer argument reducing to a single character count. The action becomes explicit: $G'$ acts through $\mathbb{F}_p \rtimes \mathbb{F}_p^\times$ shifting and scaling $p+1$ points on $\mathbb{P}^1$. From this we compute $H^*(G', π_* E_{p-1})$ and its Tate cohomology. One identity, $π^{p-1} = -p$, runs through every section.

math.AG↗