arXiv subjects

Carles Bivià-Ausina

Publications and source records attributed to Carles Bivià-Ausina.

12 recordsLinked to original sources

arXiv2026

Log canonical thresholds at infinity

The paper considers a global version of the notion of log canonical threshold for plurisubharmonic functions $u$ of logarithmic growth in $\mathbb{C}^n$, aiming at description of the range of all $p>0$ such that $e^{-u}\in L^p(\mathbb{C}^n)$. Explicit formulas are obtained in the toric case. By considering Bergman functions of corresponding weighted Hilbert spaces, a new polynomial approximation of plurisubharmonic functions of logarithmic growth with control over its singularities and behavior at infinity (a global version of Demailly's approximation theorem) is established. Some application to tame polynomial maps are given.

math.CV↗

arXiv2024

The log canonical threshold of products of ideals and mixed Łojasiewicz exponents

Given two ideals $I$ and $J$ of the ring $\mathcal O_n$ of analytic function germs $f:(\mathbb C^n,0)\to \mathbb C$, we show a sharp lower bound for the log canonical threshold of $IJ$ in terms of the sequences of mixed Łojasiewicz exponents of them. In particular, in the case where $J$ is the maximal ideal, the corresponding equality holds if and only if the integral closure of $I$ equals some power of the maximal ideal.

math.AC↗

arXiv2024

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

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 $X$.

math.AG↗

arXiv2022

Bruce-Roberts Numbers and Quasihomogeneous Functions on Analytic Varieties

Given a germ of an analytic variety $X$ and a germ of a holomorphic function $f$ with a stratified isolated singularity with respect to the logarithmic stratification of $X$, we show that under certain conditions on the singularity type of the pair $(f,X)$, the following relative analog of the well known K. Saito's theorem holds true: equality of the relative Milnor and Tjurina numbers of f with respect to X (also known as Bruce-Roberts numbers) is equivalent to the relative quasihomogeneity of the pair $(f,X)$, i.e. to the existence of a coordinate system such that both $f$ and $X$ are quasihomogeneous with respect to the same positive rational weights.

math.CV↗

arXiv2020

Analytic spread and integral closure of integrally decomposable modules

We relate the analytic spread of a module expressed as the direct sum of two submodules with the analytic spread of its components. We also study a class of submodules whose integral closure can be obtained by means of a simple computer algebra procedure once the integral closure of each row ideal is known. In particular, we analyze a class of modules, not necessarily of maximal rank, whose integral closure is determined by the family of Newton polyhedra of their row ideals.

math.AC↗

arXiv2019

The sequence of mixed Łojasiewicz exponents associated to pairs of ideals

We analyze the sequence $\mathcal L^*_J(I)$ of mixed Łojasiewicz exponents attached to any pair $I,J$ of monomial ideals of finite colength of the ring of analytic function germs $(\mathbb C^n,0)\to \mathbb C$. In particular, we obtain a combinatorial expression for this sequence when $J$ is diagonal. We also show several relations of $\mathcal L^*_J(I)$ with other numerical invariants associated to $I$ and $J$.

math.AC↗

arXiv2018

Mixed Bruce-Roberts numbers

We extend the notion of $μ^*$-sequence and Tjurina number of functions to the framework of Bruce-Roberts numbers, that is, to pairs formed by the germ at $0$ of a complex analytic variety $X\subseteq \mathbb C^n$ and a finitely $\mathcal R(X)$-determined analytic function germ $f:(\mathbb C^n,0)\to (\mathbb C,0)$. We analyze some fundamental properties of these numbers.

math.AG↗

arXiv2016

Log canonical threshold and diagonal ideals

We characterize the ideals $I$ of $\mathcal O_n$ of finite colength whose integral closure is equal to the integral closure of an ideal generated by pure monomials. This characterization, which is motivated by an inequality proven by Demailly and Pham, is given in terms of the log canonical threshold of $I$ and the sequence of mixed multiplicities of $I$.

math.AG↗

arXiv2014

Mixed Łojasiewicz exponents, log canonical thresholds of ideals and bi-Lipschitz equivalence

We study the Łojasiewicz exponent and the log canonical threshold of ideals of $\mathcal O_n$ when restricted to generic subspaces of $\mathbb C^n$ of different dimensions. We obtain effective formulas of the resulting numbers for ideals with monomial integral closure. An inequality relating these numbers is also proven. We also introduce the notion of bi-Lipschitz equivalence of ideals and we prove the bi-Lipschitz invariance of Łojasiewicz exponents and log canonical thresholds of ideals.

math.AG↗

arXiv2012

Lojasiewicz exponent of families of ideals, Rees mixed multiplicities and Newton filtrations

We give an expression for the Łojasiewicz exponent of a wide class of n-tuples of ideals $(I_1,..., I_n)$ in $Ø_n$ using the information given by a fixed Newton filtration. In order to obtain this expression we consider a reformulation of Łojasiewicz exponents in terms of Rees mixed multiplicities. As a consequence, we obtain a wide class of semi-weighted homogeneous functions $(\mathbb{C}^n,0)\to (\mathbb{C},0)$ for which the Łojasiewicz of its gradient map $\nabla f$ attains the maximum possible value.

math.AG↗

arXiv2009

The Łojasiewicz exponent of a set of weighted homogeneous ideals

We give an expression for the Łojasiewicz exponent of a set of ideals which are pieces of a weighted homogeneous filtration. We also study the application of this formula to the computation of the Łojasiewicz exponent of the gradient of a semi-weighted homogeneous function $(\C^n,0)\to (\C,0)$ with an isolated singularity at the origin.

math.AG↗

arXiv2009

Lojasiewicz exponents and resolution of singularities

We show an effective method to compute the Łojasiewicz exponent of an arbitrary sheaf of ideals of $\OO_X$, where $X$ is a non-singular scheme. This method is based on the algorithm of resolution of singularities.

math.AG↗