Sommersemester 2020
Algebraic Geometry 4: Introduction to Log Geometry
Log geometry is a framework in which suitable mildly degenerate objects in algebraic geometry behave like non-degenerate objects.
Let $X$ be a scheme, a log structure on $X$ is a just a homomorphism $\alpha:M_X\rightarrow \mathcal{O}_X$ of sheaves of monoids, such that
$$\begin{array}[c]{ccc} M_X&\stackrel{\alpha}{\rightarrow} &\mathcal{O}_X \\ \downarrow &&\downarrow\\ \alpha^{-1}(\mathcal{O}_X^{\times})&\stackrel{\cong}{\rightarrow}&\mathcal{O}_X^{\times} \end{array}.$$
Here the structure sheaf $\mathcal{O}_X$ is regarded as a sheaf of monoids with respect the multiplication. A log scheme is just a scheme equipped with a log structure.
One benefit of putting the exact structure $M_X\rightarrow\mathcal{O}_X$ is that we gain more smooth (more precisely log smooth) morphisms. For example, the morphism
$$f:X:=\mathop{\mathrm{Spec}}\mathbb{Z}_p[X,Y]/(XY-p)\rightarrow \mathop{\mathrm{Spec}}\mathbb{Z}_p=:S$$
of schemes is clearly not smooth on the special fiber. Let $\eta$ be the generic point of $S$, and let $X_{\eta}:=X\times_S\eta$. The natural homomorphisms $M_X:=\mathcal{O}_{X_{\eta}}^{\times}\cap\mathcal{O}_X\hookrightarrow\mathcal{O}_X$ and $M_S:=\mathcal{O}_{\eta}^{\times}\cap\mathcal{O}_S\hookrightarrow\mathcal{O}_S$ are log structures on $X$ and $S$ respectively. The morphism $f:X\rightarrow S$ extends to a morphism $(X,M_X)\rightarrow (S,M_S)$ of log schemes which we still denote by $f$ by abuse of notation. A fascinating fact is that, for any commutative diagram
$$\begin{array} [a]{ccc} (T_0,M_{T_0}) &\rightarrow&(X,M_X) \\ \downarrow&&\downarrow\scriptstyle{f}\\ (T,M_T)&\rightarrow&(S,M_S) \end{array},$$
where $T_0\hookrightarrow T$ is a closed embedding defined by a square-zero sheaf of ideals and $M_{T_0}$ is the log structure induced from $T$, locally there always exists a morphism $(T,M_T)\rightarrow (X,M_X)$ such that the diagram remains commutative. Apparently such a lifting property does not hold if we do not take the log structures into account. In view of the infinitesimal criterion of smoothness, the morphism $(X,M_X)\rightarrow (S,M_S)$ should be considered to be smooth in the world of log schemes. This is indeed what we do, and we call it a log smooth morphism. In other words, the degenerate family $X\rightarrow S$ of schemes becomes a non-degenerate family $(X,M_X)\rightarrow (S,M_S)$ of log schemes, which justifies what we said in the beginning. The feature making mildly degenerate objects non-degenerate, makes log geometry extremely useful in compactifying classical moduli spaces.
Let us look at another example. Consider the tamely ramified cover $g:\mathbb{A}^1_k\rightarrow\mathbb{A}^1_k,a\mapsto a^n$, where $k$ is a field and $n$ is a natural number which is invertible in $k$. Let $U:=\mathbb{A}^1_k-\{0\}$. We endow $\mathbb{A}^1_k$ with the log structure $\mathcal{O}_{U}^{\times}\cap\mathcal{O}_{\mathbb{A}^1_k}\hookrightarrow \mathcal{O}_{\mathbb{A}^1_k}$. Similar as before, $g$ extends to a morphism of log schemes, which we still denote by $g$. Now $g$ even satisfies a stronger lifting property, namely similar as above there exists local liftings and they are unique (hence they glue to a unique global one). The morphism $g$ is an example of log etale morphisms. Analogous to the classical case, one has a theory of log etale topology, log etale fundamental group, and log etale cohomology.
Prerequisites:
Algebraic Geometry 2 (SS 2019) by Prof. Ulrich Görtz, see: AlgGeo2
Content:
In this course, we are going to introduce log schemes, charts, fs log schemes, compactifying log structures, log smoothness and log differentials, and log flatness.
Time and place:
Vorlesung: Monday 10-12 WSC-S-U-3.01
Vorlesung: Wednesday 14-16 WSC-S-U-3.01
References:
Kato, Kazuya. Logarithmic structures of Fontaine-Illusie, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988)." (1989): 191-224.
Comment: This is probably the first published source about log geometry. It could serve as a place where one can start to learn log geometry.
Ogus, Arthur. Lectures on logarithmic algebraic geometry. (Cambridge Studies in Advanced Mathematics 178). Cambridge University Press, 2018. See: due-library e-book
Comment: This is a comprehensive text book on log geometry, and also the only text book on log geometry. The first two chapters (270 pages) are about monoids and monoidal spaces, and log schemes only appear afterwards. For learning the essence of log geometry, one could follows the author's suggestion on page xvii to proceed as Chapter I 1.4, 4.1, 4.2, Chapter II 1.1, 2.1, Chapter III-V.
Gabber, Ofer and Ramero, Lorenzo. Foundations for almost ring theory--Release 7.5. arXiv preprint arXiv:math/0409584 (2004).
Comment: The chapters 6, 12, 13 of this book are about the theory of monoids and log schemes.
Further readings:
Kato, Kazuya. Logarithmic structures of fontaine-illusie. II, arXiv preprint arXiv:1905.10678 (2019).
Comment: In this paper, the Kummer etale topology and the Kummer flat topology are introduced. They are finer than the classical etale topology and the classical flat topology respectively.
Illusie, Luc. An overview of the work of K. Fujiwara, K. Kato, and C. Nakayama on logarithmic etale cohomology. Astérisque 279 (2002): 271-322.
Comment: This paper is a survey of the theory of log etale cohomology. As commented in the beginning by the author, the log techniques provide more natural proofs to known theorems as well as interesting generalizations and refinements.
Kajiwara, Takeshi, Kazuya Kato, and Chikara Nakayama. Logarithmic abelian varieties I-VI.
Comment: Degenerating abelian varieties cannot preserve properness, smoothness, and group structure at the same time. However the theory of log abelian varieties makes the impossible possible in the world of log geometry.
Abramovich, Dan, Qile Chen, Danny Gillam, Yuhao Huang, Martin Olsson, Matthew Satriano, and Shenghao Sun. Logarithmic geometry and moduli, arXiv preprint arXiv:1006.5870 (2010).
Comment: This paper discusses the role played by logarithmic structures in the theory of moduli.
Gross, Mark, and Bernd Siebert. "Mirror Symmetry via Logarithmic Degeneration Data I & II. arXiv:math/0309070 (2003) & arXiv:0709.2290 (2007).