Questions tagged [log-geometry]

Log structures, semistable degenerations, log crystalline cohomology, log de Rham cohomology, log smoothness, log Gromov-Witten theory

Filter by
Sorted by
Tagged with
5
votes
0answers
79 views

algebraic de Rham cohomology of toric varieties (reference request)

I haven't been able to find anything workable yet, but I'm looking for a reference on the de Rham cohomology of toric varieties, where as many as possible of the following conditions are handled: ...
6
votes
1answer
223 views

An unpublished note by Bloch-Kato on p-divisible groups and Dieudonné crystals

I wonder if anyone could find the following unpublished paper of Bloch-Kato: Spencer Bloch and Kazuya Kato, $p$-divisible groups and Dieudonné crystals, unpublished. A similar question is here ...
3
votes
1answer
247 views

$p$-adic Kato--Nakayama space

Given a log scheme over $\mathbb{C}$ whose underlying scheme is locally of finite type, you can associate to it a ringed space called the Kato--Nakayama space. Is there a $p$-adic analogue of this ...
4
votes
1answer
363 views

Properties of log smooth schemes

Let $k$ be a field and $M$ be sharp monoid (with no invertible element) consider the log point $\eta_M=(\operatorname{Spec}(k), M)$. Let $X$ be a fine saturated scheme over $\eta_M$ such that the ...
6
votes
2answers
339 views

A log structure on the moduli space of curves

Let $M_{g, n}$ be the moduli space of curves of genus $g$ with $n$ marked points. Let $M_{g, \vec{n}}$ be the moduli space of marked curves with a choice of a (possibly zero) tangent vector at each ...
2
votes
1answer
132 views

Locally toric resolutions of compactifications

Suppose $U$ is a smooth, open $n$-dimensional variety over $\mathbb{C}.$ Say $X, X'$ are two proper normal-crossings compactifications of $U$. Call a map $m: X'\to X$ a modification if it is an ...
4
votes
0answers
324 views

An unpublished note by Spencer Bloch and Kazuya Kato

I am looking for an unpublished note by Spencer Bloch and Kazuya Kato, p-divisible groups and Dieudonné crystals. This note is always cited as Spencer Bloch and Kazuya Kato, p-divisible groups and ...
8
votes
1answer
788 views

When is a map from a logarithmic tangent bundle to a normal bundle surjective?

Suppose that $X$ is an algebraic variety with divisor $D$ such that the logarithmic tangent bundle $\mathcal{T}_X(-\log D)$ is locally free. Suppose moreover that $\iota\colon Y\to X$ is a regular ...
5
votes
0answers
465 views

Semistable reduction and log structures

I have been reading Hyodo-Kato's paper on log-crystalline cohomology, and there is one statement there that has been troubling me. To explain this, suppose we have a perfect field $k$ of ...
1
vote
1answer
372 views

trivialities on log-structures

I would like to understand some trivialities about log-structures. Given a log-scheme $(X,M_X)$ the log-structure $M_X$ is defined via push-out. Are there stupid examples in which this push-out is ...
2
votes
1answer
426 views

Minimal semistable model for K3-surfaces.

I wonder if a semistalbe K3 surface over a $p$-adic field has a minimal semistable model. I guess yes but I do not find any reference. Also, if we have a semistable K3 surface with a log structure, ...
1
vote
0answers
220 views

Moduli Space of an Algebraic K3 surface with singularities.

Suppose that $X$ is an algebraic K3 surface (say polarized). If the singular divisor of $X$ is normal crossing... Do we have a moduli space parametrizing such $K3$ surfaces? If yes do we have a ...
6
votes
3answers
1k views

relation between toric geometry and log geometry

Hello, I'm trying to understand the relation between the points of view of log geometry (monoids) and toric geometry (fans). Suppose that $k$ is a field and $P$ is a finitely generated monoid. Then $...
14
votes
4answers
4k views

References for logarithmic geometry

Hi everyone, I'm looking for a systematical introduction to (or treatment of) logarithmic structures on schemes. I am reading Kato's article ("Logarithmic structures of Fontaine-Illusie") at the ...
5
votes
3answers
1k views

What are Log Stacks

So, I've been running in both stacky circles and logarithmic circles and I've been wondering: is there a definition of log stack that is "useful"? I can imagine two such definitions: 1) A log stack ...
13
votes
0answers
791 views

Kato's log motives

What are they and what are their intended uses? Does anyone have notes/slides of this talk? I am curious about "log motives" because there seems to exist a "log motivic yoga" among experts in ...
8
votes
2answers
832 views

Logarithmic structures on moduli of elliptic curves over Z

I've heard it stated that if you take the moduli of elliptic curves with some level structure imposed (as a moduli scheme over Spec(Z)), there is a logarithmic structure that you can impose at the ...