Logarithmic Geometry Research Articles

“ compositions”: A unified R package to analyze compositional data

This contribution presents a new R package, called “ compositions”. It provides tools to analyze amount or compositional data sets in four different geometries, each one associated with an R class: rplus (for amounts, or open compositions, in a real, classical geometry), aplus (for amounts in a logarithmic geometry), rcomp (for closed compositions in a real geometry) and acomp (for closed compositions in a logistic geometry, following a log-ratio approach). The package allows to compare results obtained with these four approaches, since an analogous analysis can be performed according to each geometry, with minimal and straightforward modifications of the instructions. Beside these grounding classes, the package also includes: the most-basic features such as data transformations (e.g. logarithm, or additive logistic transform), basic statistics (both the classical ones, and those developed in the log-ratio framework of compositional analysis), high-level graphics (like ternary diagram matrix and scatter-plots) and high-level analysis (e.g. principal components or cluster analysis). Results of these functions and analysis are also provided in a consistent way among the four geometries, to ease their comparison.

On the Ishida and Du Bois complexes

In [12] Ishida introduces a complex, denoted by $\tilde {\Omega}^{^.}_Y$, associated to a filtered semi-toroidal variety Y over Spec C and proves that it is quasi-isomorphic to the Du Bois complex $\underline{\underline{ \Omega}}^{^.}_Y$ ([5]). In this article we regard a filtered semi-toroidal variety Y as an ideally log smooth log scheme over Spec C, and we give an interpretation of the Ishida complex $\tilde {\Omega}^{^.}_Y$ in terms of logarithmic geometry. Therefore, given a log smooth log scheme X over Spec C, we use this logarithmic interpretation of the Ishida complex to construct the following distinguished triangle in the Du Bois derived category Ddiff(X): $I_M \omega^{^.}_X \longrightarrow \underline{\underline{\Omega}}^{^.}_X \longrightarrow \underline{\underline{\Omega}}^{^.}_D \longrightarrow .$, where D = X − Xtriv (Xtriv being the trivial locus for the log structure M on X). Since the complex $I_M \omega^{^.}_X$ calculates the De Rham cohomology with compact supports of the smooth analytic space $X_{triv}^{an}$ ([20, Corollary 1.6]), this triangle is useful to give an interpretation of $H^{^.}_{DR,c}$(Xtriv/C) as the hyper-cohomology of the simple complex $\underline{\underline{s}}[\underline{\underline{\Omega}}^{^.}_X \longrightarrow \underline{\underline{\Omega}}^{^.}_D]$.

Towards an Algebraic Proof of Deligne’s Regularity Criterion. An Informal Survey of Open Problems

We review the notion of regular singular point of a linear differential equation with meromorphic coefficients, from the viewpoint of algebraic geometry. We give several equivalent definitions of regularity along a divisor for a meromorphic connection on a complex algebraic manifold and discuss the global birational theory of fuchsian differential modules over a field of algebraic functions. We describe the generalized algebraic version of Deligne’s canonical extension, constructed in [1, I.4]. Our main interest lies in the algebraic form of Deligne’s regularity criterion [2, II.4.4 (iii)], asserting that, on a normal compactification, only one codimensional components of the locus at infinity need to be considered. If one considers the purely algebraic nature of the statement, it is surprising that the only existing proof of this criterion is the transcendental argument given by Deligne in his corrigendum to loc. cit. dated April 1971. The algebraic proof given in our book [1, I.5.4] is also incorrect, as J. Bernstein kindly indicated to us.We introduce some notions of logarithmic geometry to let the reader appreciate Bernstein’s (counter)examples to some statements in our book [1]. Standard methods of generic projection in projective spaces reduce the question to a two-dimensional puzzle. We report on ongoing correspondence with Y. Andre and N. Tsuzuki, leading to partial results and provide examples indicating the subtlety of the problem.

Semistable degenerations and period spaces for polarized K3 surfaces

Modular compactifications of moduli spaces for polarized K3 surfaces are constructed using the tools of logarithmic geometry in the sense of Fontaine and Illusie. The relationship between these new moduli spaces and the classical minimal and toroidal compactifications of period spaces is discussed, and it is explained how the techniques of this paper yield models for the latter spaces over number fields. The paper also contains a discussion of Picard functors for log schemes and a logarithmic version of Artin's method for proving representability by an algebraic stack.

A logarithmic view towards semistable reduction

A smooth, proper family of curves creates a monodromy action of the fundamental group of the base on the H of a fibre. The geometric condition of T. Saito for the action of the wild inertia of a boundary point to be trivial is transformed to the condition of logarithmic smooth reduction. The proof emphasizes methods and results from logarithmic geometry. It applies to quasi–projective smooth curves with etale boundary divisor.

Logarithmic geometry and algebraic stacks

We construct algebraic moduli stacks of log structures and give stack-theoretic interpretations of K. Kato's notions of log flat, log smooth, and log étale morphisms. In the last section we describe the local structure of these moduli stacks in terms of toric stacks.