Vol. 3, No. 2, 2021

Download this article
Download this article For screen
For printing
Recent Issues
Volume 6, Issue 1
Volume 5, Issue 4
Volume 5, Issue 3
Volume 5, Issue 2
Volume 5, Issue 1
Volume 4, Issue 4
Volume 4, Issue 3
Volume 4, Issue 2
Volume 4, Issue 1
Volume 3, Issue 4
Volume 3, Issue 3
Volume 3, Issue 2
Volume 3, Issue 1
Volume 2, Issue 4
Volume 2, Issue 3
Volume 2, Issue 2
Volume 2, Issue 1
Volume 1, Issue 4
Volume 1, Issue 3
Volume 1, Issue 2
Volume 1, Issue 1
The Journal
About the journal
Ethics and policies
Peer-review process
Statement, 2023
 
Submission guidelines
Submission form
Editorial board
 
Subscriptions
 
ISSN (electronic): 2576-7666
ISSN (print): 2576-7658
Author index
To appear
 
Other MSP Journals
This article is available for purchase or by subscription. See below.
$\mathbb{P}^1$-localisation et une classe de Kodaira–Spencer arithmétique

Joseph Ayoub

Vol. 3 (2021), No. 2, 259–308
Abstract

Dans cet article, on introduit et on étudie le concept de 1-localisation qui est la variante du concept de 𝔸1-localisation où l’on remplace la droite affine par la droite projective. On démontre un théorème de 1-connexité en adaptant la preuve de Morel de son théorème de 𝔸1-connexité. On s’intéresse ensuite à la 1-localisation du faisceau des formes différentielles absolues et on montre que son 0-ième faisceau d’homologie est nul. Ceci nous amène naturellement à la définition d’une classe de Kodaira–Spencer arithmétique. Enfin, nous montrons un lien entre cette classe de Kodaira–Spencer arithmétique et les classes de Deligne–Illusie pour presque tout nombre premier, ce qui nous permet de prouver qu’elle est non identiquement nulle.

In this article, we introduce and study the concept of 1-localisation which is the variant of the concept of 𝔸1-localisation where the affine line is replaced by the projective line. We prove a 1-connectivity theorem following the proof of Morel of his 𝔸1-connectivity theorem. We then consider the 1-localisation of the sheaf of absolute differential forms and we show that its 0-th homology sheaf vanishes. This naturally brings us to the definition of an arithmetic Kodaira–Spencer class. Finally, we establish a link between this arithmetic Kodaira–Spencer class and the Deligne–Illusie classes for almost all prime numbers, which enables us to prove that the former is not identically zero.

PDF Access Denied

We have not been able to recognize your IP address 3.239.76.25 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recom­mendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

Keywords
$\mathbb{P}^1$-localisation, formes différentielles, classe de Kodaira–Spencer arithmétique
Mathematical Subject Classification 2010
Primary: 14F05, 14F10, 14F40, 14F42, 14G25
Milestones
Received: 27 February 2019
Revised: 6 May 2020
Accepted: 21 May 2020
Published: 5 December 2020
Authors
Joseph Ayoub
Institut für Mathematik
Universität Zürich
Switzerland
CNRS LAGA
Université Paris 13
France