Logarithmic Derivatives of Solutions to Linear Differential Equations

Logarithmic Combinatorial Differentials

Given a morphism $X \to S$ of fine log schemes, we develop a geometric description of the sheaves of higher-order differentials $Ω^n_{X/S}$ for $n > 1$, as well as a definition of the de Rham complex in terms of this description.

Decomposition of direct images of logarithmic differentials

The purpose of this note is to give a short proof of a theorem of Kollár that the derived direct image of the canonical sheaf splits into a sum of its cohomology sheaves. This is deduced from a stronger decomposition theorem for direct images of sheaves of logarithmic differentials, together with the weak semistable reduction theorem.

Logarithmic $p$-bases and arithmetical differential modules

We introduce the notion of log $p$-smoothness which weakens that of log-smoothness and that of having locally $p$-bases. We extend Berthelot's construction of arithmetic $D$-modules and some properties in this context.

Computing Regular Meromorphic Differential Forms via Saito's Logarithmic Residues

Logarithmic differential forms and logarithmic vector fields associated to a hypersurface with an isolated singularity are considered in the context of computational complex analysis. As applications, based on the concept of torsion differential forms due to A.G. Aleksandrov, regular meromorphic differential forms introduced by D. Barlet and M. Kersken, and Brieskorn formulae on Gauss-Manin connections are investigated. (i) A method is given to describe singular parts of regular meromorphic differential forms in terms of non-trivial logarithmic vector fields via Saito's logarithmic residues. T…

Stability of logarithmic differential one-forms

This article deals with the irreducible components of the space of codimension one foliations in a projective space defined by logarithmic forms of a certain degree. We study the geometry of the natural parametrization of the logarithmic components and we give a new proof of the stability of logarithmic foliations, obtaining also that these irreducible components are reduced.

Logarithmic forms and anti-invariant forms of reflection groups

Let W be a finite group generated by unitary reflections and A be the set of reflecting hyperplanes. We will give a characterization of the logarithmic differential forms with poles along A in terms of anti-invariant differential forms. If W is a Coxeter group defined over the real numbers, then the characterization provides a new method to find a basis for the module of logarithmic differential forms out of basic invariants.

Center conditions: Rigidity of logarithmic differential equations

In this paper we prove that any degree $d$ deformation of a generic logarithmic polynomial differential equation with a persistent center must be logarithmic again. This is a generalization of Ilyashenko's result on Hamiltonian differential equations. The main tools are Picard-Lefschetz theory of a polynomial with complex coefficients in two variables, specially the Gusein-Zade/A'Campo's theorem on calculating the Dynkin diagram of the polynomial, and the action of Gauss-Manin connection on the so called Brieskorn lattice/Petrov module of the polynomial. Some applications on the cyclicity of c…

超椭圆Gorenstein曲线与对数微分

Hyperelliptic Gorenstein curves and logarithmic differentials

本文提出一种局部且与基变换可交换的收缩对数超椭圆曲线子曲线的工具，并证明超椭圆多重尺度微分能决定底层节点曲线的Gorenstein收缩序列，为微分极限的更一般猜想提供了首个证据。

离散环式 adic 空间上的对数微分

Logarithmic differentials on discretely ringed adic spaces

本文在离散环式 adic 空间上借助 Kähler 半范数定义有界微分子层，并用对数微分刻画它，证明在光滑对数结构情形其整体截面同构于相应对数微分。

A criterion for the logarithmic differential operators to be generated by vector fields

We study divisors in a complex manifold in view of the property that the algebra of logarithmic differential operators along the divisor is generated by logarithmic vector fields. We give a sufficient criterion for the property, a simple proof of F.J. Calderon-Moreno's theorem that free divisors have the property, a proof that divisors in dimension 3 with only isolated quasi-homogeneous singularities have the property, an example of a non-free divisor with non-isolated singularity having the property, an example of a divisor not having the property, and an algorithm to compute the V-filtration…

Jet differentials on toroidal compactifications of ball quotients

We give explicit estimates for the volume of the Green-Griffiths jet differentials of any order on a toroidal compactification of a ball quotient. To this end, we first determine the growth of the logarithmic Green-Griffiths jet differentials on these objects, using a natural deformation of the logarithmic jet space of a given order, to a suitable weighted projective bundle. Then, we estimate the growth of the vanishing conditions that a logarithmic jet differential must satisfy over the boundary to be a standard one.

A residual duality over Gorenstein rings with application to logarithmic differential forms

Kyoji Saito's notion of a free divisor was generalized by the first author to reduced Gorenstein spaces and by Delphine Pol to reduced Cohen-Macaulay spaces. Starting point is the Aleksandrov-Terao theorem: A hypersurface is free if and only if its Jacobian ideal is maximal Cohen-Macaulay. Pol obtains a generalized Jacobian ideal as a cokernel by dualizing Aleksandrov's multi-logarithmic residue sequence. Notably it is essentially a suitably chosen complete intersection ideal that is used for dualizing. Pol shows that this generalized Jacobian ideal is maximal Cohen-Macaulay if and only if the…

Quasihomogeneity of isolated singularities and logarithmic cohomology

We characterize quasihomogeneity of isolated singularities by the injectivity of the map induced by the first differential of the logarithmic differential complex in the top local cohomology supported in the singular point.

On the existence of logarithmic and orbifold jet differentials

We introduce the concept of directed orbifold, namely triples (X, V, D) formed by a directed algebraic or analytic variety (X, V), and a ramification divisor D, where V is a coherent subsheaf of the tangent bundle TX. In this context, we introduce an algebra of orbifold jet differentials and their sections. These jet sections can be seen as algebraic differential operators acting on germs of curves, with meromorphic coefficients, whose poles are supported by D and multiplicities are bounded by the ramification indices of the components of D. We estimate precisely the curvature tensor of the co…

通过log Frobenius-Witt微分的准则来研究log正则性

A criterion for log regularity via log Frobenius-Witt differentials

本文研究了log Frobenius-Witt微分的性质，并基于log FW-微分模块提出了一种log正则性准则。

Complete sets of logarithmic vector fields for integration-by-parts identities of Feynman integrals

Integration-by-parts identities between loop integrals arise from the vanishing integration of total derivatives in dimensional regularization. Generic choices of total derivatives in the Baikov or parametric representations lead to identities which involve dimension shifts. These dimension shifts can be avoided by imposing a certain constraint on the total derivatives. The solutions of this constraint turn out to be a specific type of syzygies which correspond to logarithmic vector fields along the Gram determinant formed of the independent external and loop momenta. We present an explicit ge…