Let \(p\) be a prime number and \(k\) be a closed subfield of \(\mathbb C_{p}\). R. F. Coleman [Invent. Math. 69, 171–208 (1982; Zbl 0516.12017)] defined a well-behaved integration theory for one-forms on certain smooth \(k\)-analytic curves. In the book under review, V. G. Berkovich extends this theory to any smooth \(k\)-analytic space. Let’s mention that the analytic spaces considered here are Berkovich spaces in the sense of [Mathematical Surveys and Monographs, 33. Providence, RI: AMS (1990; Zbl 0715.14013) and Publ. Math., Inst. Hautes Étud. Sci. 78, 5–161 (1993; Zbl 0804.32019)].
In chapter \(1\), the author defines so-called naive analytic functions. Let \(X\) be a strictly \(k\)-analytic space. Let us recall that to any point \(x\) of \(X\) is associated a complete residue field \({\mathcal H}(x)\). We denote by \(s(x)\) the transcendence degree of \(\widetilde{\mathcal{H}(x)}\) over \(\tilde{k}\) and by \(t(x)\) the dimension of the \(\mathbb Q\)-vector space \(\sqrt{|\mathcal{H}(x)|^*}/\sqrt{|k|^*}\). Let \(X_{st}\) be the subset of \(X\) defined by \(s=t=0\) (it contains every \(\mathbb C_p\)-point of \(X\)). Naive analytic functions are germs of analytic functions in the neighborhood of \(X_{st}\). Any naive analytic function induces a locally analytic function \(X(k) \to k\) but the sheaf of naive analytic function is better behaved: in particular, it is stable by pullback to \(X \hat{\otimes}_{k} k'\), with \(k\subset k' \subset \mathbb C_p\).
In order to integrate one-forms on the space \(\mathbf G_{m}\) (with variable \(T\)), we need a logarithm function. Such a function \(\mathrm{Log}^\lambda(T)\) is defined by the usual series on the open unit disc \(D(1;1)\) and its value \(\lambda\) (in any \(k\)-algebra \(K^1\)) in \(p\). It is analytic on the open set in \({\mathbf G}_{m}\) (with infinitely many connected components) whose complement is the path joining \(0\) to \(\infty\), hence a naive analytic function on \({\mathbf G}_{m}\). In order to have a well-behaved integration theory, where two primitives differ from a constant, we need to choose such a branch of the logarithm.
Let’s now formulate the main theorem of the book. Let \(K = \bigcup_{i\geq 0} K^i\) be a filtered \(k\)-algebra and \(\lambda\) be an element of \(K^1\). On any smooth \(k\)-analytic space \(X\), there exists a filtered \(\mathcal{D}_{X}\)-subalgebra \(\mathcal{S}^{\lambda}_{X}\) of \(\mathfrak{N}^K_{X}\), the sheaf of naive analytic functions “with values in \(K\)”, which satisfies the following conditions: \(\mathcal{S}_{X}^{\lambda,0}\) is the sheaf \({\mathcal O}_{X} \otimes_{k} K^0\), for any \(i\geq 0\), \(\mathcal{S}_{X}^{\lambda,i+1}\) is generated by the local sections \(f\) for which \(\mathrm{d}f\) is a local section of \(\mathcal{S}_{X}^{\lambda,i}\) and \(\mathrm{Log}^\lambda(T) \in \mathcal{S}_{X}^{\lambda,1}({\mathbf G}_{m})\). Moreover, the de Rham complex is exact in degrees \(0\) and \(1\) : for any \(i\geq 0\), the kernel of \(\mathrm{d} : \mathcal{S}_{X}^{\lambda,i} \to \Omega^1_{\mathcal{S}_{X}^{\lambda,i},X}\) is the sheaf of locally constant functions with values in \(K^i\) and the kernel of \(\mathrm{d} : \Omega^1_{\mathcal{S}_{X}^{\lambda,i},X} \to \Omega^2_{\mathcal{S}_{X}^{\lambda,i},X}\) is contained in \(\mathrm{d}\mathcal{S}_{X}^{\lambda,i+1}\). Let’s mention that exactness in higher degrees is not known. The sheaf \(\mathcal{S}^{\lambda}_{X}\) also satisfies functoriality properties with regards to morphisms \(X' \to X\) and extensions of the base field to \(k' \subset\mathbb C_p\).
Chapter \(2\) and \(3\) are devoted to general facts on Berkovich spaces. Let \(X\) be a smooth analytic space, \(x \in X\) and \(n=\dim_x(X)\). If \(s(x)=n\), then, étale locally, the point looks like a “generic point” of the generic fiber of a formal scheme of some particular type. If \(s(x)<n\), then, \(x\) has a fundamental system of étale neighborhoods isomorphic to the direct product of a space of the previous type of dimension \(s(x)\), \(t(x)\) one-dimensional discs and \(n-s(x)-t(x)\) one-dimensional annuli. In chapter \(4\), the author studies the exactness of the de Rham complex in degree \(1\). He also proves that on a closed (i.e., with empty boundary) connected curve over a non-Archimedean field with non-trivial valuation, any two points can be connected by closed curves, in the spirit of a result of A. J. de Jong [Publ. Math., Inst. Hautes Étud. Sci. 82, 5–96 (1995; Zbl 0864.14009)] over discretely valued fields.
From then on, the author deals with spaces that are smooth germs of strictly affinoid spaces. They are related to the dagger spaces considered in [J. Reine Angew. Math. 519, 73–95 (2000; Zbl 0945.14013)] and [Duke Math. J. 113, No. 1, 57-91 (2002; Zbl 1057.14023)] by E. Grosse-Klönne, who gave a proof of the finite dimensionality of their de Rham cohomologies. Chapter \(5\) and \(6\) of the book are respectively devoted to isocrystals and \(F\)-isocrystals. The author associates to a germ an increasing sequence \((E^i)_{i\geq 0}\) of unipotent isocrystals, where \(E^0\) is the algebra of global sections of the germ, such that the morphism \(H^1_{\mathrm{dR}}(E^i) \to H^1_{\mathrm{dR}}(E^{i+1})\) is zero for any \(i\geq 0\). Next he endows these isocrystals with a Frobenius structure. This turns out to be very useful by means of the following: any morphism of \(F\)-isocrystals \(E^i \to M\) such that \(H^1_{\mathrm{dR}}(E^i) \to H^1_{\mathrm{dR}}(M)\) is zero uniquely extend to a morphism of \(F\)-isocrytal \(E^{i+1} \to M\) (the proof relies on the article [Ann. Sci. Éc. Norm. Supér. (4) 31, No. 5, 683–715 (1998; Zbl 0933.14008)] by B. Chiarellotto on the eigenvalues of a Frobenius lifting acting on the de Rham cohomology). This result is used to construct an injective map from \(\bigcup_{i\geq 0} E^i\) to the sheaf of naive analytic functions.
In chapter \(7\), the author constructs the sheaves \(\mathcal{S}_{X}^{\lambda,i}\) by some rather intricate induction process using the local structure of smooth analytic spaces and the image of the \(E^i\)’s into the sheaf of naive analytic functions. In chapter \(8\), he derives some properties of those sheaves. Let’s just mention that for germs with good reduction (i.e. isomorphic to the generic fiber of a smooth formal scheme), things globalise: there exists a filtered subalgebra \(\mathcal{E}^\lambda\) of the algebra of global sections of \(\mathcal{S}^\lambda\) on the germ such that global one-forms (in \(\Omega^1_{\mathcal{E}^{\lambda,i}}\)) admit global primitives (in \(\mathcal{E}^{\lambda,i+1}\)) and such that the algebra \(\mathcal{E}^{\lambda,i+1}\) is generated by those primitives.
Chapter \(9\) is devoted to applications. For any smooth \(k\)-analytic space \(X\) such that \[ H_{1}(X\hat{\otimes}_{k} \mathbb C_p, \mathbb Q) \simeq H_{1}(X,\mathbb Q), \] there is a way to integrate closed one-forms \(\omega\) along paths \(\gamma\) with ends in \(X(k)\). If \(\omega=\mathrm{d}f\), we find \(\int_{\gamma} \mathrm{d}f = f(\gamma(1))-f(\gamma(0))\). As we expect, the value of the integral depends on the path \(\gamma\) only up to homotopy. On the other hand, it depends non-trivially on the homotopy class, contrary to the integrals defined by Yu. Zarhin [J. Math. Sci., New York 81, No. 3, 2744–2750 (1996; Zbl 0884.14020)] and P. Colmez [Integration on \(p\)-adic varieties (in French). Astérisque. 248. Paris: Société Mathématique de France (1998; Zbl 0930.14013)], that only depend on the ends of \(\gamma\). Last, the author construct, for any locally unipotent \(\mathcal{D}\)-module, a parallel transport along a path or an étale path, which satisfies the usual properties.