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20
Nonarchimedean amoebas and tropical varieties
, 2004
"... We study the nonarchimedean counterpart to the complex amoeba of an algebraic variety, and show that it coincides with a polyhedral set defined by Bieri and Groves using valuations. For hypersurfaces this set is also the tropical variety of the defining polynomial. Using nonarchimedean analysis a ..."
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Cited by 79 (0 self)
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We study the nonarchimedean counterpart to the complex amoeba of an algebraic variety, and show that it coincides with a polyhedral set defined by Bieri and Groves using valuations. For hypersurfaces this set is also the tropical variety of the defining polynomial. Using nonarchimedean analysis and a recent result of Conrad we prove that the amoeba of an irreducible variety is connected. We introduce the notion of an adelic amoeba for varieties over global fields, and establish a form of the localglobal principle for them. This principle is used to explain the calculation of the nonexpansive set for a related dynamical system.
Functorial desingularization of quasiexcellent schemes in characteristic zero: the nonembedded case
, 2009
"... For a Noetherian scheme X, let Xreg denote the regular locus of X. The scheme X is said to admit a resolution of singularities if there exists a blowup X ′ → X with center disjoint from Xreg and regular X ′. More generally, for a closed subscheme Z ⊂ X, let (X,Z)reg denote the set of points x ∈ Xre ..."
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Cited by 16 (1 self)
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For a Noetherian scheme X, let Xreg denote the regular locus of X. The scheme X is said to admit a resolution of singularities if there exists a blowup X ′ → X with center disjoint from Xreg and regular X ′. More generally, for a closed subscheme Z ⊂ X, let (X,Z)reg denote the set of points x ∈ Xreg
The motivic Serre invariant, ramification, and the analytic Milnor fiber
 Invent. Math
"... Let us recall the classical definition of a padic zeta function, as it was given by Igusa [29]. A survey of the theory of padic zeta functions can be found in Denef’s Bourbaki report [19]. ..."
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Cited by 8 (6 self)
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Let us recall the classical definition of a padic zeta function, as it was given by Igusa [29]. A survey of the theory of padic zeta functions can be found in Denef’s Bourbaki report [19].
A trace formula for rigid varieties, and motivic Weil generating series for formal schemes
, 2009
"... Abstract. We establish a trace formula for rigid varieties X over a complete discretely valued field, which relates the set of unramified points on X to the Galois action on its étale cohomology. We develop a theory of motivic integration for formal schemes of pseudofinite type over a complete disc ..."
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Cited by 8 (5 self)
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Abstract. We establish a trace formula for rigid varieties X over a complete discretely valued field, which relates the set of unramified points on X to the Galois action on its étale cohomology. We develop a theory of motivic integration for formal schemes of pseudofinite type over a complete discrete valuation ring R, and we introduce the Weil generating series of a regular formal Rscheme X of pseudofinite type, via the construction of a GelfandLeray form on its generic fiber. Our trace formula yields a cohomological interpretation of this Weil generating series. When Xis the formal completion of a morphism f from a smooth irreducible variety to the affine line, then its Weil generating series coincides (modulo normalization) with the motivic zeta function of f. When X is the formal completion of f at a closed point x of the special fiber f −1 (0), we obtain the local motivic zeta function of f at x. In the latter case, the generic fiber of X is the socalled analytic Milnor fiber of f at x; we show that it completely determines the formal germ of f at x. 1.
Nonarchimedean analytification of algebraic spaces
 JOURNAL OF ALGEBRAIC GEOMETRY 18
, 2009
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A trace formula for varieties over a discretely valued field
, 2008
"... We study the motivic Serre invariant of a smoothly bounded algebraic or rigid variety X over a complete discretely valued field K with perfect residue field k. If K has characteristic zero, we extend the definition to arbitrary Kvarieties using Bittner’s presentation of the Grothendieck ring and a ..."
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Cited by 4 (2 self)
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We study the motivic Serre invariant of a smoothly bounded algebraic or rigid variety X over a complete discretely valued field K with perfect residue field k. If K has characteristic zero, we extend the definition to arbitrary Kvarieties using Bittner’s presentation of the Grothendieck ring and a process of Néron smoothening of pairs of varieties. The motivic Serre invariant can be considered as a measure for the set of unramified points on X. Under certain tameness conditions, it admits a cohomological interpretation by means of a trace formula. In the curve case, we use T. Saito’s geometric criterion for cohomological tameness to obtain more detailed results. We discuss some applications to WeilChâtelet groups, Chow motives, and the structure of the Grothendieck ring of varieties.
Singular cohomology of the analytic Milnor fiber, and mixed Hodge structure on the nearby cohomology
 J. Algebraic Geom
"... Abstract. We describe the homotopy type of the analytic Milnor fiber in terms of a strictly semistable model, and we show that its singular cohomology coincides with the weight zero part of the mixed Hodge structure on the nearby cohomology. We give a similar expression for Denef and Loeser’s motiv ..."
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Cited by 3 (2 self)
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Abstract. We describe the homotopy type of the analytic Milnor fiber in terms of a strictly semistable model, and we show that its singular cohomology coincides with the weight zero part of the mixed Hodge structure on the nearby cohomology. We give a similar expression for Denef and Loeser’s motivic Milnor fiber in terms of a strictly semistable model. MSC 2000: 32S30, 32S55, 14D07, 14G22 1.