Logarithmic Geometry Research Articles

Local log-regular rings vs. toric rings

Local log-regular rings are a certain class of Cohen-Macaulay local rings that are treated in logarithmic geometry. Our paper aims to provide purely commutative ring theoretic proof of several ring-theoretic properties of local log-regular rings such as an explicit description of a canonical module, and the finite generation of the divisor class group.

Logarithmic quasimaps

We construct a proper moduli space which is a Deligne–Mumford stack parametrising quasimaps relative to a simple normal crossings divisor in any genus using logarithmic geometry. We show this moduli space admits a virtual fundamental class of the expected dimension leading to numerical invariants which agree with the theory of Battistella–Nabijou where the latter is defined.

A smooth compactification of the space of genus two curves in projective space: via logarithmic geometry and Gorenstein curves

A smooth compactification of the space of genus two curves in projective space: via logarithmic geometry and Gorenstein curves

The monodromy pairing for logarithmic 1-motifs

We describe a 3-step filtration on all logarithmic abelian varieties with constant degeneration. The obstruction to descending this filtration, as a variegated extension, from logarithmic geometry to algebraic geometry is encoded in a bilinear pairing valued in the characteristic monoid of the base. This pairing is realized as the monodromy pairing in p-adic, l-adic, and Betti cohomolgies, and recovers the Picard-Lefschetz transformation in the case of Jacobians. The Hodge realization of the filtration is the monodromy weight filtration on the limit mixed Hodge structure.

A geographical study ofM‾2P2,4main $\overline{\mathcal{M}}_2\left(\mathbb{P}^2, 4\right)^{\text {main }}$

Abstract We discuss criteria for a stable map of genus two and degree 4 to the projective plane to be smoothable, as an application of our modular desingularisation of via logarithmic geometry and Gorenstein singularities.

Motives and homotopy theory in logarithmic geometry

This document is a short user’s guide to the theory of motives and homotopy theory in the setting of logarithmic geometry. We review some of the basic ideas and results in relation to other works on motives with modulus, motivic homotopy theory, and reciprocity sheaves.

A non-Archimedean analogue of Teichm\xfcller space and its tropicalization

In this article we use techniques from tropical and logarithmic geometry to construct a non-Archimedean analogue of Teichmüller spaceoverline{{{mathcal {T}}}}_g whose points are pairs consisting of a stable projective curve over a non-Archimedean field and a Teichmüller marking of the topological fundamental group of its Berkovich analytification. This construction is closely related to and inspired by the classical construction of a non-Archimedean Schottky space for Mumford curves by Gerritzen and Herrlich. We argue that the skeleton of non-Archimedean Teichmüller space is precisely the tropical Teichmüller space introduced by Chan–Melo–Viviani as a simplicial completion of Culler–Vogtmann Outer space. As a consequence, Outer space turns out to be a strong deformation retract of the locus of smooth Mumford curves in overline{{mathcal {T}}}_g.

Arithmetic surjectivity for zero-cycles

Let $f:X\to Y$ be a proper, dominant morphism of smooth varieties over a number field $k$. When is it true that for almost all places $v$ of $k$, the fibre $X_P$ over any point $P\in Y(k_v)$ contains a zero-cycle of degree $1$? We develop a necessary and sufficient condition to answer this question. The proof extends logarithmic geometry tools that have recently been developed by Denef and Loughran-Skorobogatov-Smeets to deal with analogous Ax-Kochen type statements for rational points.

Moduli of stable maps in genus one and logarithmic geometry, I

This is the first in a pair of papers developing a framework for the application of logarithmic structures in the study of singular curves of genus $1$. We construct a smooth and proper moduli space dominating the main component of Kontsevich's space of stable genus $1$ maps to projective space. A variation on this theme furnishes a modular interpretation for Vakil and Zinger's famous desingularization of the Kontsevich space of maps in genus $1$. Our methods also lead to smooth and proper moduli spaces of pointed genus $1$ quasimaps to projective space. Finally, we present an application to the log minimal model program for $\mathcal{M}_{1,n}$. We construct explicit factorizations of the rational maps among Smyth's modular compactifications of pointed elliptic curves.

Moduli of stable maps in genus one and logarithmic geometry, II

This is the second in a pair of papers developing a framework to apply logarithmic methods in the study of singular curves of genus $1$. This volume focuses on logarithmic Gromov--Witten theory and tropical geometry. We construct a logarithmically nonsingular moduli space of genus $1$ curves mapping to any toric variety. The space is a birational modification of the principal component of the Abramovich--Chen--Gross--Siebert space of logarithmic stable maps and produces an enumerative genus $1$ curve counting theory. We describe the non-archimedean analytic skeleton of this moduli space and, as a consequence, obtain a full resolution to the tropical realizability problem in genus $1$.

Clutching and gluing in tropical and logarithmic geometry

The classical clutching and gluing maps between the moduli stacks of stable marked algebraic curves are not logarithmic, i.e. they do not induce morphisms over the category of logarithmic schemes, since they factor through the boundary. Using insight from tropical geometry, we enrich the category of logarithmic schemes to include so-called sub-logarithmic morphisms and show that the clutching and gluing maps are naturally sub-logarithmic. Building on the recent framework developed by Cavalieri, Chan, Wise, and the third author, we further develop a stack-theoretic counterpart of these maps in the tropical world and show that the resulting maps naturally commute with the process of tropicalization.

Functorial tropicalization of logarithmic schemes: the case of constant coefficients

The purpose of this article is to develop foundational techniques from logarithmic geometry in order to define a functorial tropicalization map for fine and saturated logarithmic schemes in the case of constant coefficients. Our approach crucially uses the theory of fans in the sense of K. Kato and generalizes Thuillier's retraction map onto the non-Archimedean skeleton in the toroidal case. For the convenience of the reader many examples as well as an introductory treatment of the theory of Kato fans are included.

Logarithmic good reduction, monodromy and the rational volume

Let $R$ be a strictly local ring complete for a discrete valuation, with fraction field $K$ and residue field of characteristic $p > 0$. Let $X$ be a smooth, proper variety over $K$. Nicaise conjectured that the rational volume of $X$ is equal to the trace of the tame monodromy operator on $\ell$-adic cohomology if $X$ is cohomologically tame. He proved this equality if $X$ is a curve. We study his conjecture from the point of view of logarithmic geometry, and prove it for a class of varieties in any dimension: those having logarithmic good reduction.

Logarithmic geometry and the Milnor fibration

Inspired by a description of the logarithmic space of Kato and Nakayama in terms of real oriented blowups, we describe Milnor fibrations and related constructions used by A'Campo in the language of logarithmic geometry.

DERIVED LOGARITHMIC GEOMETRY I

In order to develop the foundations of derived logarithmic geometry, we introduce a model category of logarithmic simplicial rings and a notion of derived log-étale maps, and use them to define derived log stacks.

Parabolic sheaves on logarithmic schemes

We show how the natural context for the definition of parabolic sheaves on a scheme is that of logarithmic geometry. The key point is a reformulation of the concept of logarithmic structure in the language of symmetric monoidal categories, which might be of independent interest. Our main result states that parabolic sheaves can be interpreted as quasi-coherent sheaves on certain stacks of roots.

Relative rounding in toric and logarithmic geometry

We show that the introduction of polar coordinates in toric geometry smoothes a wide class of equivariant mappings, rendering them locally trivial in the topological category. As a consequence, we show that the Betti realization of a smooth proper and exact mapping of log analytic spaces is a topological fibration, whose fibers are orientable manifolds (possibly with boundary). This turns out to be true even for certain noncoherent log structures, including some families familiar from mirror symmetry. The moment mapping plays a key role in our proof. 14D06, 14M25, 14F45, 32S30; 53D20, 14T05

Logarithmic Geometry, Minimal Free Resolutions and Toric Algebraic Stacks

In this paper we will introduce a certain type of morphisms of log schemes (in the sense of Fontaine, Illusie and Kato) and study their moduli. Then by applying this we define the notion of toric algebraic stacks, which may be regarded as torus emebeddings in the framework of algebraic stacks and prove some fundamental properties. Also, we study the stack-theoretic analogue of toroidal embeddings.

The category of toric stacks

AbstractIn this paper, we show that there is an equivalence between the 2-category of smooth Deligne–Mumford stacks with torus embeddings and actions and the 1-category of stacky fans. To this end, we prove two main results. The first is related to a combinatorial aspect of the 2-category of toric algebraic stacks defined by I. Iwanari [Logarithmic geometry, minimal free resolutions and toric algebraic stacks, Preprint (2007)]; we establish an equivalence between the 2-category of toric algebraic stacks and the 1-category of stacky fans. The second result provides a geometric characterization of toric algebraic stacks. Logarithmic geometry in the sense of Fontaine–Illusie plays a central role in obtaining our results.

Komplexe Algebraische Geometrie

The Workshop Komplexe algebraische Geometrie , organized by Fabrizio Catanese (Bayreuth), Yujiro Kawamata (Tokyo), Gang Tian (Princeton), and Eckart Viehweg (Essen), drew together 50 participants. There were several young PhD students and other PostDocs in their 20's and early 30's, together with established leaders of the fields related to the thematic title of the workshop. There were 23 talks, each lasting 55 minutes or one hour, and each followed by a lively 10 minutes discussion. As usual at an Oberwolfach Meeting, the mathematical discussions continued outside the lecture room throughout the day and the night. The Conference fully fulfilled its purported aim, of setting in contact mathematicians with different specializations and non uniform background, of presenting new fashionable topics alongside with new insights on long standing classical open problems, and also cross-fertilizations with other research topics as arithmetic and physics. For the latter, cf. the talk by Bernd Siebert on the new approach to Mirror Symmetry through logarithmic geometry and toric affine Calabi Yau varieties and the one by van Straten on quantization of completely integrable Hamiltonian Systems. For the former, cf. the talks by Winkelmann on integral sets and by McQuillan on the Bloch principle. A central role was occupied by the new results around the Hodge conjecture presented by Voisin, and the new results by Hacon, McKernan et al which give an essential step towards the final solution of the Minimal Model Program, and were presented here by McKernan with a proposed approach to the Sarkisov program. There were many expositions dealing with several classical problems and classical and modern theories. It would take too long to dwell on each of the outstanding contributions presented at the Conference. For some topics there were several interrelated talks, for instance we could mention the following classical themes: There were also expositions on many other beautiful topics: The variety of striking results and the very interesting and challenging proposals made the participation in the workshop rather strenuous but certainly highly rewarding. We hope that the quality of the expositions in these abstracts will make them quite useful to the mathematical community.