Non-Archimedean tame topology and stably dominated types (Q2789579)

In non-archimedean analytic geometry (e.g. analytic geometry over \(F = \mathbb Q_p\) or over \(F = k((t))\)), the naive definition of ``analytic'' (i.e., locally given by a power series) does not work very well, due to \(V(F)\) being totally disconnected, for any variety \(V\). One modern approach to this problem consists in replacing \(V\) by the corresponding Berkovich space \(V_{\mathrm{Berk}}\), which is a topological space that can be thought of as being obtained from \(V(F)\) by ``adding missing points'', in a somewhat similar way as \(\mathbb Q\) can be made connected by adding all irrational numbers.NEWLINENEWLINEFrom a topological point of view, \(V_{\mathrm{Berk}}\) has many good properties. For example, \(V_{\mathrm{Berk}}\) is connected if \(V\) is connected in the Zariski topology and \(V_{\mathrm{Berk}}\) is compact if \(V\) is complete. However, the precise structure of \(V_{\mathrm{Berk}}\) is rather mysterious; already \(\mathbb A^1_{\mathrm{Berk}}\) is a kind of tree which has densely many branch points (under some conditions on \(F\)).NEWLINENEWLINEThe book under review introduces, using tools from model theory, a new space \(\widehat V\) associated to \(V\) which is very similar to \(V_{\mathrm{Berk}}\). In particular, if \(F\) is a maximally complete valued field of rank 1, then \(\widehat V(F)\) can be canonically identified with \(V_{\mathrm{Berk}}\). (Note that \(\widehat V\) itself is not a set, but a functor which associates, to each \(F\) in a suitable category of valued fields, a set \(\widehat V(F)\) of \(F\)-rational points.) One advantage over \(V_{\mathrm{Berk}}\) is that this \(\widehat V\) can be defined in bigger generality (namely for fields \(F\) of higher rank and for \(V\) which are not varieties but definable sets in the sense of model theory). More importantly, the deep knowledge of model theory of algebraically closed valued fields (in particular [\textit{D. Haskell} et al., Stable domination and independence in algebraically closed valued fields. Cambridge: Cambridge University Press (2008; Zbl 1149.03027)]) provides powerful tools to understand these spaces \(\widehat V\). The authors use this to answer several questions about the spaces \(\widehat V\) which were open for the spaces \(V_{\mathrm{Berk}}\). This then also solves the original problems in \(V_{\mathrm{Berk}}\). In particular, the following results are obtained: {\parindent=0.7cm\begin{itemize}\item[--] As already mentioned above, the precise shape of a Berkovich space tends to be very complicated. However, the authors prove that there always exists a strong deformation retraction to a finite simplicial complex. (The precise statement about the spaces \(\widehat V\) is Theorem 11.1.1, which can be considered as the main theorem of the book; Theorem 14.2.1 states the conclusion about \(V_{\mathrm{Berk}}\).) In particular, this yields (in Chapter 13) an equivalence of categories between a homotopy category of spaces \(\widehat V\) and the homotopy category of a certain kind of finite simplicial complexes. \item[--] If \(V\) is projective, then \(V_{\mathrm{Berk}}\) is even equal to a projective limit of finite simplicial complexes, to each of which \(V_{\mathrm{Berk}}\) has a strong deformation retraction in a compatible way (Theorem 14.2.4). \item[--] For any morphism \(f : V \times \mathbb P^n \to V\) over \(V\), the fibers ``\((f^{-1}(v))_{\mathrm{Berk}}\)'' admit only finitely many different homotopy types (Theorem 14.3.1 is a generalization of this). \item If \(V\) is quasi-projective, then \(V_{\mathrm{Berk}}\) is locally contractible. NEWLINENEWLINE\end{itemize}} Formally, \(\widehat V(F)\) is just the set of \textit{stably dominated types} in \(V\) defined over \(F\). This notion of types has already been introduced in the above-mentioned paper by Haskell-Hrushovski-Macpherson. The philosophy is that the model theory of algebraically closed valued fields should decompose into a ``stable part'' coming from the residue field and an ``o-minimal part'' coming from the value group; a stably dominated type is one coming from the stable part. In accordance with this philosophy, a type is stably dominated if and only if it is orthogonal to the value group.NEWLINENEWLINEAs an example, consider \(V = \mathbb A^1\). Then \(\widehat V(F)\) consists exactly of the generic types of the closed valuative balls in \(F\), including balls that are just points. (Those form a tree under inclusion, so this fits to the above-mentioned fact that \(\mathbb A^1_{\mathrm{Berk}}\) is a tree.)NEWLINENEWLINEIn the above definition, it is unclear how \(\widehat V(F)\) depends on \(F\). A key observation (Theorem~3.1.1) is that \(\widehat V\) can be considered as a pro-definable set, i.e., a projective limit of definable sets. This makes various tools from model theory applicable to \(\widehat V\); for example, \(\widehat V\) is already determined by \(\widehat V(\mathbb U)\) for some monster model \(\mathbb U\).