Logarithmic Geometry

Logarithmic Geometry was invented (or discovered) in the 1980's, with crucial ideas contributed by Deligne, Faltings, Fontaine, Illusie, and especially Kato. Its roots go back much further, at least to the work by Mumford and others on toroidal embeddings, as well as the theory of semistable reduction. The goal of all these theories is to understand, in a systmeatic way, the related phenomena of compactification and degeneration in algebraic geometry, including arithmetic geometry with applications to number theory.

I have been working on a manuscript which is intended to serve as a basic introduction to the main ideas and techniques of log geometry, for close to a decade now. Because of the long scope of the project, and the lack of a systematic treatment of these ideas and techniques in the published literature, I have freely shared, upon request, preliminary (and in many ways unsatisfactory) versions of the manuscript. As I warned recipients, these early versions were incomplete and included plans or sketches for proofs and statements that in some caseds did not pan out. Currently I am hoping to arrive at a more definitive manuscript that will be publishable and will, I hope, contain only relatively minor innaccuracies. It is quite likely that, once a final agreement with a publisher is reached, I will no longer be able to distribute the manuscript itself, but will try keep errata and other updates available here. I would urge those of you who have already received copies of the earlier versions of the manuscript to download newer versions before it is too late. Many of the problems with earlier versions have been and will be corrected in the current and forthcoming releases.

Here are some comments on the state of the manuscript, as of October, 24,2017.

Chapter I, which is quite long and deals with the category of monoids, is, I hope, mostly correct and nearly complete. However, it could be that some additional results will be added as I understand their relevance to log geometry. (For example, I just added a monoidal version of the local criterion for flatness that I found helpful in the study of the relationship between flatness and integrality, for example in the (new) theorem IV 4.3.5.)

Chapter II, on sheaves of monoids. It now includes a fair amount of material on monschemes and monoidal transformations, going a bit beyond what is trully essential for log geometry. I had hoped to use this material, as well as some adiditonal material that is not included, to work out a useful theory of frames (extending work of Abbes, Kato, and Saito), but was not able to develop a convincing set of results.

Chapter III contains the main exposition about logarithmic schemes, discussing both the language introduced by Deligne and Faltings and the more general and functorial notion due to Fontaine, Illusie,and Kato. It includes a discussion Kato's "regular'' log schemes and a related (and new) notion of "solid" log schemes. T There are now sections on inseparable morphisms, on blow-ups, and on exactification and saturation of morphisms.

Chapter IV, concerns the key concepts of differentials, smoothness, and flatness. Some improvements since the previous verison of Aug 29 are included, following suggestions of O. Gabber.

Chapter V, on cohomology, has now been extensively revised. It begins with a discussion of the Betti cohomology of the space X_log, and its computation by means of analytic de Rham cohomology. It then treats algebraic de Rham cohomology, including the Cartier isomorphism in positive characteristic and comparison theorems between analytic and algebraic de Rham cohomology in characteristic zero.

 

The version of October 24 replaces the previous October file. It is printed in smaller font and has some changes to the material on flatness, inpsired by suggestions of O. Gabber and discussions with M. Olsson. I expect that this will be the last revision of the manuscript before it is copyedited by Cambrdige University Press, who has agreed to publish it. I do not plan to add any new material beyond what appear here, but I still hope to correct any errors that I find or that are pointed out to me. Once the book goes into production, which should be soon, errors will have to be corrected here, rather than in the printed version.

 

 

 

 

The link to the current version of the manuscript is below. Please write to me to request the password, which should not be shared with anyone. And as before, please do communicate to me any errors or other issues that you find.

 

 

 

Lectures on Logarithmic Geometry

(version of October 24, 2017)