Kato–Nakayama spaces, infinite root stacks and the profinite homotopy type of log schemes

Carchedi, David¹ ; Scherotzke, Sarah² ; Sibilla, Nicolò³ ; Talpo, Mattia⁴

¹ Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, United States
² Mathematisches Institut, University of Bonn, Endenicher Allee 60, D-53115 Bonn, Germany
³ School of Mathematics, Statistics and Actuarial Sciences, University of Kent, Canterbury, CT2 7FS, United Kingdom
⁴ Department of Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6, Canada

Geometry & topology, Tome 21 (2017) no. 5, pp. 3093-3158.

Résumé

For a log scheme locally of finite type over $ℂ$, a natural candidate for its profinite homotopy type is the profinite completion of its Kato–Nakayama space. Alternatively, one may consider the profinite homotopy type of the underlying topological stack of its infinite root stack. Finally, for a log scheme not necessarily over $ℂ$, another natural candidate is the profinite étale homotopy type of its infinite root stack. We prove that, for a fine saturated log scheme locally of finite type over $ℂ$, these three notions agree. In particular, we construct a comparison map from the Kato–Nakayama space to the underlying topological stack of the infinite root stack, and prove that it induces an equivalence on profinite completions. In light of these results, we define the profinite homotopy type of a general fine saturated log scheme as the profinite étale homotopy type of its infinite root stack.

Détail

DOI : 10.2140/gt.2017.21.3093

Classification : 14F35, 55U35, 55P60
Keywords: log scheme, Kato–Nakayama space, root stack, profinite spaces, infinity category, étale homotopy type, topological stack

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