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Integration in valued fields. (English) Zbl 1136.03025
Ginzburg, Victor (ed.), Algebraic geometry and number theory. In Honor of Vladimir Drinfeld’s 50th birthday. Basel: Birkhäuser (ISBN 978-0-8176-4471-0/hbk). Progress in Mathematics 253, 261-405 (2006).
This fascinating and important paper develops model-theoretically a theory of motivic integration over valued fields of residue characteristic 0. The authors work with the geometry of algebraically closed valued fields, and to some extent with ‘V-minimal’ expansions of such fields, such as the rigid analytic expansions of Lipshitz and Robinson. The work has applications for other Henselian fields.
Let \(L\) be a valued field with valuation ring \({\mathcal O}\), value group \(\Gamma\), maximal ideal \({\mathcal M}\), and residue field \(k\). Define \(\text{RV}:= L^*/(1+{\mathcal M})\). There is an exact sequence \[ 0\rightarrow k^*\rightarrow\text{RV}\rightarrow\Gamma\rightarrow0, \] so RV combines the structure of the value group and residue field.
For any complete theory \(T\), there is a naturally defined Grothendieck semiring of definable sets, considered up to definable bijections (in the paper and this review, ‘definable’ means ‘definable without parameters’). The semiring is denoted \(K_+(T)\), or \(K_+(M)\), where \(M\models T\). Mostly the authors work with semigroups and semirings rather than groups and rings, to avoid collapse. The authors work over a base field \(L_0\), but in the model-theoretic setting of algebraically closed valued fields, with VF denoting the field sort. There is a natural map \({\mathbb L}\) taking definable sets in RV to definable sets in VF, which induces a surjective homomorphism of filtered semirings \(K_+\text{RES}[*]\otimes K_+\Gamma[*]\rightarrow K_+\text{VF}[*]\); the \(*\) indicates the filtration, which is via a notion of dimension, and RES denotes a residue field structure, basically the residue field equipped with a family of vector spaces on it, with the induced structure of ‘generalised algebraic varieties’. The kernel of \({\mathbb L}\) is identified as a certain congruence \(I_{\text{sp}}\). The inverse map is regarded as an Euler characteristic. The content of surjectivity is that, up to definable bijections, definable sets in VF come from definable sets in the value group and the residue field.
There are notions of ‘definable set with volume form’ in the VF and RV categories, and corresponding Grothendieck rings, denoted \(K_+\mu\text{VF}\) and \(K_+\mu\text{RV}\), and \({\mathbb L}\) induces an isomorphism of semirings \(K_+\mu\text{RV}[*]\rightarrow K_+\mu\text{VF}[*]\), again with a precisely described kernel. Here \(K_+\mu\text{RV}[*]\) is a tensor product of corresponding semirings for \(\Gamma\) and the residue field structure RES. The inverse isomorphism \(K_+\mu\text{VF}[*]\rightarrow K_+\mu\text{RV}[*]/I_{\text{sp}}^{\mu}\) is viewed as a motivic integral.
The initial theory is developed in the setting of C-minimality, a slight specialisation of the theory developed in [D. Haskell and D. Macpherson, Ann. Pure Appl. Logic 66, No. 2, 113–162 (1994; Zbl 0790.03039)]. Essentially, a C-minimal structure is a set with a nested sequence of equivalence relations, indexed by a definable dense linear ordering, such that any 1-variable definable set is a Boolean combination of equivalence classes (‘balls’). Algebraically closed valued fields, with the equivalence relation \(E_{\gamma}\) (\(\gamma\in\Gamma\)) defined by \(E_{\gamma} xy\Leftrightarrow v(x-y)\geq\gamma\) and with other equivalence relations corresponding to open balls, provide the motivating example. The authors develop the geometry of C-minimality (for example the structure of stable definable sets, non-interaction between open and closed balls). They work with a further specialisation, V-minimality, for which one requires that any definable chain of balls has non-empty intersection, plus other conditions. Under this hypothesis they prove piecewise continuity and differentiability results for definable functions (part of the content of Sections 3–5). The map \({\mathbb L}\) lifts definable sets from RV to VF, and there is a corresponding lift of definable functions, respected by \({\mathbb L}\), described in Section 6. Via a theory of ‘RV-blow-ups’, the kernel \(I_{\text{sp}}\) is identified in Section 7. The main theorems about \({\mathbb L}\) are proved in Section 8, with the differentiability theory mediating in the volume form case.
The paper includes an analysis of Grothendieck semirings and rings for divisible ordered abelian groups, in work overlapping with that of J. Maříková. It is shown that for \(\text{DOAG}_A\) (the theory of divisible ordered abelian groups with constants for a fixed submodel \(A\)) the Grothendieck ring is \({\mathbb Z}^2\). In particular, this gives two independent Euler characteristics on \(\text{DOAG}_A\).
A theory of integration of definable functions into certain Grothendieck rings is developed in Section 11. Here the authors work with rings rather than semirings, avoiding collapse and loss of information by working with bounded sets. Some of the results here were obtained independently by Cluckers and Loeser. Among other results, it is shown that for sufficiently large \(p\), the \(p\)-adic Fourier transform of a rational polynomial is locally constant away from an exceptional subvariety.
As indicated above, the bulk of the work is in the setting of algebraically closed valued fields, but in Section 12 the authors work with more general Henselian fields, with a richer class of definable sets, under an assumption that quantifier elimination is obtained by adding relation symbols in the RV sort. Much of the theory extends to this setting. From this, for example, for the theory RCVF of real closed valued fields, they derive two Euler characteristics from \(K(\text{RCVF})\) to \({\mathbb Z}[t]\), extending a result of T. Mellor. Also, they re-prove rapidly the theorem of [R. Cluckers and D. Haskell, Bull. Symb. Log. 7, No. 2, 262–269 (2001; Zbl 0988.03058)] that the Grothendieck ring of each \(p\)-adic field is trivial. Some of the model theory of Henselian fields of residue characteristic 0, in particular QE in the Denef-Pas language, is also derived from these methods, thereby bypassing the machinery of pseudo-convergent sequences.
In the final section, the authors use the theory developed to answer a question of Kontsevich and Gromov (also answered by different methods by M. Larsen and V. A. Lunts). They show that if \(X,Y\) are smooth \(d\)-dimensional subvarieties of a smooth projective \(n\)-dimensional variety \(V\), with \(V\setminus X\) and \(V\setminus Y\) birationally isomorphic, then \(X\times{\mathbb A}^{n-d}\) and \(Y\times{\mathbb A}^{n-d}\) are birationally equivalent (and \(X\) and \(Y\) are birationally equivalent if they contain no rational curves). In particular, two elliptic curves with isomorphic complements in projective space are isomorphic.
For the entire collection see [Zbl 1113.00007].

03C60 Model-theoretic algebra
11S85 Other nonanalytic theory
12L12 Model theory of fields
14C99 Cycles and subschemes
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