Motivic Integration and the Cohomology of Non-Archimedean Analytic Spaces
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2026
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| _version_ | 1866901068447744000 |
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| author | Revista, Zen MFC, 10 |
| author_facet | Revista, Zen MFC, 10 |
| contents | We construct a canonical motivic measure on the category of smooth rigid analytic varieties over a discretely valued field K, taking values in a completion of the Grothendieck ring of varieties extended by rational powers of the Lefschetz motive. By bridging the model-theoretic integration of Hrushovski-Kazhdan with the geometric formalism of Berkovich spaces, we establish a comparison isomorphism between the motivic volume of a bounded semi-algebraic set and the weight-graded pieces of its compactly supported étale cohomology. Furthermore, we analyze the behavior of the motivic zeta function under the tropicalization map, proving a non-archimedean analogue of the Monodromy Conjecture for the analytic Milnor fiber. Our results provide a cohomological interpretation of the motivic volume in terms of the weight filtration on the vanishing cycles sheaf, generalizing the work of Denef and Loeser to the analytic setting. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18763366 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Motivic Integration and the Cohomology of Non-Archimedean Analytic Spaces Revista, Zen MFC, 10 Motivic Integration Berkovich Spaces Non-Archimedean Geometry Etale Cohomology Grothendieck Ring of Varieties We construct a canonical motivic measure on the category of smooth rigid analytic varieties over a discretely valued field K, taking values in a completion of the Grothendieck ring of varieties extended by rational powers of the Lefschetz motive. By bridging the model-theoretic integration of Hrushovski-Kazhdan with the geometric formalism of Berkovich spaces, we establish a comparison isomorphism between the motivic volume of a bounded semi-algebraic set and the weight-graded pieces of its compactly supported étale cohomology. Furthermore, we analyze the behavior of the motivic zeta function under the tropicalization map, proving a non-archimedean analogue of the Monodromy Conjecture for the analytic Milnor fiber. Our results provide a cohomological interpretation of the motivic volume in terms of the weight filtration on the vanishing cycles sheaf, generalizing the work of Denef and Loeser to the analytic setting. |
| title | Motivic Integration and the Cohomology of Non-Archimedean Analytic Spaces |
| topic | Motivic Integration Berkovich Spaces Non-Archimedean Geometry Etale Cohomology Grothendieck Ring of Varieties |
| url | https://doi.org/10.5281/zenodo.18763366 |