Logarithmic spaces (according to K. Kato).

*(English)*Zbl 0832.14015
Cristante, Valentino (ed.) et al., Barsotti symposium in algebraic geometry. Memorial meeting in honor of Iacopo Barsotti, in Abano Terme, Italy, June 24-27, 1991. San Diego, CA: Academic Press. Perspect. Math. 15, 183-203 (1994).

The notion of logarithmic space was invented by J.-M. Fontaine and the author to come to a unified treatment of the various constructions of de Rham complexes with logarithmic poles, in particular to include the case of mixed characteristic. More specifically, let \(S\) be the spectrum of a complete discrete valuation ring \(A\), with closed (resp. generic) point \(s\) (resp. \(\eta)\) and let \(X/S\) be a scheme with semistable reduction, i.e. locally in the étale topology \(X\) is isomorphic to the closed subscheme of \(\mathbb{A}^n_S\) defined by \(x_1x_2 \ldots x_r = t\), where \(x_1, \ldots, x_n\) are coordinates of \(\mathbb{A}^n\) and \(t\) is a uniformizing parameter of \(A\), then \(X\) is regular, \(X_\eta\) is smooth, and \(Y = X_s\) is a normal crossings divisor. One is interested in a good definition of the relative de Rham complex of \(X\) over \(S\) with logarithmic poles along \(Y\), \(\omega^\bullet _{X/S} = \Omega^\bullet _{X/S} (\log Y)\), which coincides with the usual de Rham complex on the generic fibre, and induces on \(Y\) the complex \(\omega^\bullet _Y = {\mathcal O}_Y \otimes \omega^\bullet _{X/S}\). Over \(\mathbb{C}\), J. Steenbrink used the analogue of \(\omega^\bullet _Y\) in the study of limits of Hodge structures and vanishing cycles. In general, \(\omega^\bullet _Y\) will not only depend on \(Y\) but also on \(X\) and the question becomes to find out which extra structure on \(Y\) is needed to define \(\omega^\bullet _Y\). Here is where the theory of logarithmic structures as worked out by K. Kato (and O. Hyodo) may prove to be a successful enlargement of algebraic geometry. As a matter of fact, Hyodo defined the complex \(\omega^\bullet _{X/S}\) and he and Kato already found some striking applications of the theory.

Let \((X, {\mathcal O}_X)\) be a ringed space. A pre-logarithmic structure on \(X\) is a pair \((M, \alpha)\) of a sheaf of monoids \(M\) on \(X\) and a homomorphism \(\alpha : M \to {\mathcal O}_X \). The structure \((M, \alpha)\) is called logarithmic if \(\alpha\) induces an isomorphism \(\alpha^{- 1} ({\mathcal O}^\times_X) @>\sim>> {\mathcal O}^\times_X\). A ringed space with a (pre-)log structure is called a (pre-)log space. Morphisms between pre-log spaces are defined in an obvious way. To any pre-log structure \((P, \beta)\) on the ringed space \(X\) there exists a uniquely defined log structure \((M, \alpha)\), with \(M\) defined as the push-out of the diagram \({\mathcal O}^\times_X \leftarrow \beta^{- 1} ({\mathcal O}^\times_X) \to P\). Any ringed space \(X\) has the trivial log structure given by the inclusion \({\mathcal O}^\times_X \to {\mathcal O}_X\). A log structure \((M, \alpha)\) on a scheme \(X\) is called fine if locally (étale) \((M, \alpha)\) is associated to a pre-log structure \((P_X, \beta)\) with \(P_X\) a constant sheaf of monoids with value \(P\), where \(P\) is finitely generated and integral, i.e. the canonical map \(P \to P^{gr}\) of \(P\) to its group envelope is injective. For a fine log scheme \((X,M)\) one defines a chart of \(M\) as a homomorphism \(P_X \to M\) with \(P\) finitely generated and integral, inducing an isomorphism \((P_X)^a @>\sim>> M\), where the superscript a means the log structure associated to the pre-log structure. Basic examples of fine log structures are furnished by:

(i) monoid algebras;

(ii) normal crossings divisors;

(iii) log points, the standard log point \(s = \text{Spec} (k)\), \(k\) a field, being given by the chart \(\mathbb{N} \to k\), \(n \mapsto 0\) (resp. 1) if \(n \neq 0\) (resp. \(n = 1)\);

(iv) semistable reduction.

For a morphism \(\underline X = (X, M_X) @>f>> \underline Y = (Y,M_Y)\) of log schemes one may define an \({\mathcal O}_X\)-module of ‘Kähler differentials’ \(\omega^1_{X/Y} = \Omega^1_{\underline X/\underline{Y}}\) with a universal pair \(d : {\mathcal O}_X \to \Omega^1_{\underline X/ \underline Y}\), \(d \log : M_X \to \Omega^1_{\underline X/ \underline Y}\). \(d\) extends to a derivation of the exterior algebra \(\Omega^\bullet _{\underline X/ \underline Y} = \bigwedge \Omega^1_{ \underline X/ \underline Y}\) and the resulting complex is called the de Rham complex of \(\underline X/ \underline Y\). One defines log smoothness via an adapted notion of thickening for log schemes. Kato gave a nice criterion for log smoothness of a map between fine log schemes. However, log smooth maps are still far from being well understood.

One can transpose the basic definitions and results of crystalline cohomology to log schemes. One of the main applications is the construction of a monodromy operator \(N\) on the de Rham cohomology of schemes with semistable reduction in mixed characteristic. For the log scheme \(\underline Y/ \underline S\), where \(\underline S\) is the standard log point defined by \(S = \text{Spec} (k)\) for a perfect field \(k\) of characteristic \(p > 0\), one defines \(H^i (\underline Y/W_n (\underline S)) : =H^i ((\underline Y/W_n (\underline S)_{\text{crys}}, {\mathcal O})\), and one obtains a Frobenius operator \(\varphi : H^i (\underline Y/W_n (\underline S)) \to H^i (\underline Y/W_n (\underline S))\), where for \(n \geq 1\), \(W_n (\underline S)\) is the log scheme defined by \(\text{Spec} (W_n (k))\) with log structure associated to \(\mathbb{N} \to W_n (k)\), \(1 \mapsto 0\). Also, one has a monodromy operator \(N : H^i (\underline Y/W_n (\underline S)) \to H^i (\underline Y/W_n (\underline S))\). These operators are related by \(N \varphi = p \varphi N\). Restricting to so-called good log schemes \(\underline Y/ \underline S\) one obtains a cohomology theory \(H^\bullet (\underline Y/W (\underline S)) \otimes K\), where \(K\) is the fraction field of \(W\), with values in the (graded) \(F\)-isocrystals over \(k\). It has several nice properties (Poincaré duality, Künneth), but Chern classes, cycle classes and trace formulas are still missing. For the mixed characteristic case (A complete discrete valuation ring, \(T = \text{Spec} (A)\), \(K\) fraction field of characteristic zero, \(k\) perfect residue field of characteristic \(p > 0\), associated log schemes, etc.) with \(\underline X\) a smooth log scheme over \(\underline T\) such that \(X/T\) is proper and the special fiber \(\underline Y = \underline X \times_{\underline T} \underline S\) has suitable properties (is of Cartier type), let \(\underline X_K\) be the generic fiber, which is log smooth over \(\text{Spec} (K)\) and endowed with the trivial log structure. Fixing a uniformizing parameter \(t\), Hyodo and Kato constructed the following canonical comparison isomorphism: \(\rho_t : H^i (\underline Y/W (\underline S)) \otimes_WK @>\sim>> H^i_{DR} (\underline X_K/K)\) \((: = H^i (X_K, \Omega^\bullet _{\underline X_K/K}))\). For a unit \(u\) one has: \(\rho_t = \rho_{ut} \exp (\log (u)N)\), where the logarithm is the usual \(p\)-adic one. This may be applied to get information on the de Rham cohomology \(H^i_{DR} (\underline X_K/K)\), and in particular, to approach Fontaine’s (and Jannsen’s) \(C_{st}\)-conjecture (which was proved by Kato in the semistable reduction case with small relative dimension).

The notion of log smoothness sheds new light on classical problems of degenerations and compactifications. The Tate curve \(E^t\) with \(q\)- invariant \(t\) is revisited. Kato proved that in a suitable category of log spaces \(E^t\) acquires a group structure. One is led to the introduction of valuative and algebraic valuative log spaces. One obtains the Kato-Tate curve \(\underline E^t = (\underline E^t_{\underline S})^{\text{val}}\) over a suitable log scheme \(\underline S\) and one can study its points of order \(n\), \(_n \underline E^t\). Then \(_n \underline E^t\) fits in an exact sequence \(0 \to (\mathbb{Z}/n \mathbb{Z}) (1) \to_n \underline E^t \to \mathbb{Z}/n \mathbb{Z} \to 0\) in the category of algebraic fine and saturated log schemes over \(\underline S\). Over the generic fiber this sequence gives the well known one for \(_n E^t\). More generally, one can study log finite flat group schemes and (in the limit) log Barsotti-Tate groups. Some results of Kato are briefly discussed. The paper closes with a few words on log abelian schemes. Despite a plausible definition and some results, several basic questions (commutativity, rigidity, dual) remain open.

For the entire collection see [Zbl 0802.00020].

Let \((X, {\mathcal O}_X)\) be a ringed space. A pre-logarithmic structure on \(X\) is a pair \((M, \alpha)\) of a sheaf of monoids \(M\) on \(X\) and a homomorphism \(\alpha : M \to {\mathcal O}_X \). The structure \((M, \alpha)\) is called logarithmic if \(\alpha\) induces an isomorphism \(\alpha^{- 1} ({\mathcal O}^\times_X) @>\sim>> {\mathcal O}^\times_X\). A ringed space with a (pre-)log structure is called a (pre-)log space. Morphisms between pre-log spaces are defined in an obvious way. To any pre-log structure \((P, \beta)\) on the ringed space \(X\) there exists a uniquely defined log structure \((M, \alpha)\), with \(M\) defined as the push-out of the diagram \({\mathcal O}^\times_X \leftarrow \beta^{- 1} ({\mathcal O}^\times_X) \to P\). Any ringed space \(X\) has the trivial log structure given by the inclusion \({\mathcal O}^\times_X \to {\mathcal O}_X\). A log structure \((M, \alpha)\) on a scheme \(X\) is called fine if locally (étale) \((M, \alpha)\) is associated to a pre-log structure \((P_X, \beta)\) with \(P_X\) a constant sheaf of monoids with value \(P\), where \(P\) is finitely generated and integral, i.e. the canonical map \(P \to P^{gr}\) of \(P\) to its group envelope is injective. For a fine log scheme \((X,M)\) one defines a chart of \(M\) as a homomorphism \(P_X \to M\) with \(P\) finitely generated and integral, inducing an isomorphism \((P_X)^a @>\sim>> M\), where the superscript a means the log structure associated to the pre-log structure. Basic examples of fine log structures are furnished by:

(i) monoid algebras;

(ii) normal crossings divisors;

(iii) log points, the standard log point \(s = \text{Spec} (k)\), \(k\) a field, being given by the chart \(\mathbb{N} \to k\), \(n \mapsto 0\) (resp. 1) if \(n \neq 0\) (resp. \(n = 1)\);

(iv) semistable reduction.

For a morphism \(\underline X = (X, M_X) @>f>> \underline Y = (Y,M_Y)\) of log schemes one may define an \({\mathcal O}_X\)-module of ‘Kähler differentials’ \(\omega^1_{X/Y} = \Omega^1_{\underline X/\underline{Y}}\) with a universal pair \(d : {\mathcal O}_X \to \Omega^1_{\underline X/ \underline Y}\), \(d \log : M_X \to \Omega^1_{\underline X/ \underline Y}\). \(d\) extends to a derivation of the exterior algebra \(\Omega^\bullet _{\underline X/ \underline Y} = \bigwedge \Omega^1_{ \underline X/ \underline Y}\) and the resulting complex is called the de Rham complex of \(\underline X/ \underline Y\). One defines log smoothness via an adapted notion of thickening for log schemes. Kato gave a nice criterion for log smoothness of a map between fine log schemes. However, log smooth maps are still far from being well understood.

One can transpose the basic definitions and results of crystalline cohomology to log schemes. One of the main applications is the construction of a monodromy operator \(N\) on the de Rham cohomology of schemes with semistable reduction in mixed characteristic. For the log scheme \(\underline Y/ \underline S\), where \(\underline S\) is the standard log point defined by \(S = \text{Spec} (k)\) for a perfect field \(k\) of characteristic \(p > 0\), one defines \(H^i (\underline Y/W_n (\underline S)) : =H^i ((\underline Y/W_n (\underline S)_{\text{crys}}, {\mathcal O})\), and one obtains a Frobenius operator \(\varphi : H^i (\underline Y/W_n (\underline S)) \to H^i (\underline Y/W_n (\underline S))\), where for \(n \geq 1\), \(W_n (\underline S)\) is the log scheme defined by \(\text{Spec} (W_n (k))\) with log structure associated to \(\mathbb{N} \to W_n (k)\), \(1 \mapsto 0\). Also, one has a monodromy operator \(N : H^i (\underline Y/W_n (\underline S)) \to H^i (\underline Y/W_n (\underline S))\). These operators are related by \(N \varphi = p \varphi N\). Restricting to so-called good log schemes \(\underline Y/ \underline S\) one obtains a cohomology theory \(H^\bullet (\underline Y/W (\underline S)) \otimes K\), where \(K\) is the fraction field of \(W\), with values in the (graded) \(F\)-isocrystals over \(k\). It has several nice properties (Poincaré duality, Künneth), but Chern classes, cycle classes and trace formulas are still missing. For the mixed characteristic case (A complete discrete valuation ring, \(T = \text{Spec} (A)\), \(K\) fraction field of characteristic zero, \(k\) perfect residue field of characteristic \(p > 0\), associated log schemes, etc.) with \(\underline X\) a smooth log scheme over \(\underline T\) such that \(X/T\) is proper and the special fiber \(\underline Y = \underline X \times_{\underline T} \underline S\) has suitable properties (is of Cartier type), let \(\underline X_K\) be the generic fiber, which is log smooth over \(\text{Spec} (K)\) and endowed with the trivial log structure. Fixing a uniformizing parameter \(t\), Hyodo and Kato constructed the following canonical comparison isomorphism: \(\rho_t : H^i (\underline Y/W (\underline S)) \otimes_WK @>\sim>> H^i_{DR} (\underline X_K/K)\) \((: = H^i (X_K, \Omega^\bullet _{\underline X_K/K}))\). For a unit \(u\) one has: \(\rho_t = \rho_{ut} \exp (\log (u)N)\), where the logarithm is the usual \(p\)-adic one. This may be applied to get information on the de Rham cohomology \(H^i_{DR} (\underline X_K/K)\), and in particular, to approach Fontaine’s (and Jannsen’s) \(C_{st}\)-conjecture (which was proved by Kato in the semistable reduction case with small relative dimension).

The notion of log smoothness sheds new light on classical problems of degenerations and compactifications. The Tate curve \(E^t\) with \(q\)- invariant \(t\) is revisited. Kato proved that in a suitable category of log spaces \(E^t\) acquires a group structure. One is led to the introduction of valuative and algebraic valuative log spaces. One obtains the Kato-Tate curve \(\underline E^t = (\underline E^t_{\underline S})^{\text{val}}\) over a suitable log scheme \(\underline S\) and one can study its points of order \(n\), \(_n \underline E^t\). Then \(_n \underline E^t\) fits in an exact sequence \(0 \to (\mathbb{Z}/n \mathbb{Z}) (1) \to_n \underline E^t \to \mathbb{Z}/n \mathbb{Z} \to 0\) in the category of algebraic fine and saturated log schemes over \(\underline S\). Over the generic fiber this sequence gives the well known one for \(_n E^t\). More generally, one can study log finite flat group schemes and (in the limit) log Barsotti-Tate groups. Some results of Kato are briefly discussed. The paper closes with a few words on log abelian schemes. Despite a plausible definition and some results, several basic questions (commutativity, rigidity, dual) remain open.

For the entire collection see [Zbl 0802.00020].

Reviewer: W.W.J.Hulsbergen (Haarlem)

##### MSC:

14F40 | de Rham cohomology and algebraic geometry |

14F30 | \(p\)-adic cohomology, crystalline cohomology |