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Motivic integration on smooth rigid varieties and invariants of degenerations. (English) Zbl 1078.14029

In the last years, the theory of motivic integration has emerged as a new powerful tool in producing birational invariants of algebraic varieties over a field \(k\) of characteristic zero. These developments have been propelled, in particular, by the pioneering work of J. Denef and F. Loeser on the geometry of arc spaces of singular algebraic varieties and their motivic measures. In the present paper, the authors turn to the somewhat dual situation of degenerating families of schemes over a discrete valuation ring \(R\) with perfect residue field \(K\), for which the framework of rigid geometry appears to be the most appropriate one. Actually, they construct an analogous theory of motivic integration for smooth, separated and quasi-compact rigid \(K\)-spaces, which is essentially based on the use of associated formal schemes, the classical Greenberg functor, and the theory of weak Néron models in formal and rigid geometry. Another crucial ingredient is the general theory of motivic integration on formal schemes, which has recently been established by the second author of the present paper [J. Sebag, Bull. Soc. Math. Fr. 132, No. 1, 1–54 (2004; Zbl 1084.14012)]. As the authors show, there are several important applications of their theory of motivic integration on smooth rigid varieties, ranging from the construction of new birational invariants of degenerations of algebraic varieties up to the formulation of an analogue of the famous Nash problem on the arc structure of singularities.

MSC:

14E18 Arcs and motivic integration
14G22 Rigid analytic geometry
14D06 Fibrations, degenerations in algebraic geometry
14J17 Singularities of surfaces or higher-dimensional varieties
14E30 Minimal model program (Mori theory, extremal rays)
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
14J32 Calabi-Yau manifolds (algebro-geometric aspects)

Citations:

Zbl 1084.14012
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References:

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