Abstract
Let \(k\) be an algebraically closed field of characteristic \(0\). For a log curve \(X/k^{\times }\) over the standard log point (Kato in Int J Math 11(2):215–232, 2000), we define (algebraically) a combinatorial monodromy operator on its log-de Rham cohomology group. The invariant part of this action has a cohomological description, it is the Du Bois cohomology of \(X\) (Du Bois in Bull Soc Math Fr 109(1):41–81, 1981). This can be seen as an analogue of the invariant cycles exact sequence for a semistable family (as in the complex, étale and \(p\)-adic settings). In the specific case in which \(k={\mathbb {C}}\) and \(X\) is the central fiber of a semistable degeneration over the complex disc, our construction recovers the topological monodromy and the classical local invariant cycles theorem. In particular, our description allows an explicit computation of the monodromy operator in this setting.
Résumé
Soit \(k\) un corps algébriquement clos de caractéristique \(0\). Pour une courbe logarithmique \(X/k^{\times }\) sur le point logarithmique standard ([16]), on définit (algébriquement) un opérateur de monodromie combinatoire sur sa cohomologie de de Rham logarithmique. La partie invariante de cette action possède une description cohomologique, elle est la cohomologie de Du Bois de \(X\) ([9]). Cela peut être vu comme un analogue de la suite exacte des cycles invariants pour une famille semi-stable (comme dans les cadres complexes, étale et \(p\)-adique). Dans le cas spécifique ou \(k={\mathbb {C}}\) et \(X\) est la fibre centrale d’une dégénération semi-stable sur le disque complexe, notre construction retrouve la monodromie topologique et le théorème des cycles invariants classique. En particulier, dans ce cadre, notre description fournis un calcul explicite de l’opérateur de monodromie.
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Acknowledgements
Pietro Gatti would like to thank Nicola Mazzari for the helpful discussions concerning this research. Bruno Chiarellotto is supported by the grant MIUR-PRIN 2017 “Geometric, algebraic and analytic methods in arithmetic”. Pietro Gatti was partially funded by Nero Budur’s research project G0B2115N from the Research Foundation of Flanders.
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Chiarellotto, B., Gatti, P. A combinatorial description of the monodromy of log curves. Ann. Math. Québec 45, 161–184 (2021). https://doi.org/10.1007/s40316-020-00133-7
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DOI: https://doi.org/10.1007/s40316-020-00133-7