We define and study infinite root stacks of fine and saturated logarithmic schemes, a limit version of the root stacks introduced by Niels Borne and the second author in Adv. Math. (231 (2012Math. (231 ( ) 1327Math. (231 ( -1363. We show in particular that the infinite root stack determines the logarithmic structure and recovers the Kummer-flat topos of the logarithmic scheme. We also extend the correspondence between parabolic sheaves and quasi-coherent sheaves on root stacks to this new setting. ContentsBoth authors were supported in part by the PRIN project 'Geometria delle varietà algebriche e dei loro spazi di moduli' from MIUR, and by research funds from the Scuola Normale SuperioreThe infinite root stack. In this paper we construct an algebraic version of the Kato-Nakayama space, the infinite root stack ∞ √ X of X, which is a proalgebraic stack over X. For this we build on the construction of the root stacks B √ X in [7] (denoted by X B/M X there), which in turn is based on several particular cases constructed by Olsson [29,34]. Denote by α X : M X → O X the logarithmic structure on X, a fine saturated logarithmic scheme, and set M X def = M X /O * X ; then M X is a sheaf of monoids on the smallétale site Xé t , whose geometric fibers are sharp fine saturated monoids (these are the monoids that appear in toric geometry, consisting of the integral points in a strictly convex rational polyhedral cone in some R n ). A Kummer extension B of M X is, roughly speaking, a sheaf of monoids containing M X , such that every section of B has locally a positive multiple in M X (see [7, Definition 4.2]); the typical example is the sheaf 1 d M X of fractions of sections of M X with some fixed denominator S ∞ √ X; this is a proalgebraic stack that can be described very explicitly, but is not an infinite root stack. It has some advantages over (X 0 , M X0 ); for example, the projection s 0 ×∞ √X → X 0 can be used as a substitute for (X 0 , M X0 ) in problems such as degenerations of Gromov-Witten invariants. We plan to go back to this in a later paper.This 'central fiber' of the infinite root stack, and much of the basic formalism developed in the present article, is crucial in the recent work [38] about a logarithmic version of the derived McKay correspondence, of S. Scherotzke, N. Sibilla and the first author.

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 [26]. Alternatively, one may consider the profinite homotopy type of the underlying topological stack of its infinite root stack [48]. Finally, for a log scheme not necessarily over C, another natural candidate is the profiniteétale 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 profiniteétale homotopy type of its infinite root stack.2010 Mathematics Subject Classification. MSC Primary: 14F35, 55P60; Secondary: 55U35. Key words and phrases. log scheme, Kato-Nakayama space, root stack, profinite spaces, infinity category, etale homotopy type, topological stack. 1 1.2. The comparison map and the equivalence of profinite completions. Our main result states:Theorem (see Theorem 6.4). Let X be a fine saturated log scheme locally of finite type over C. Then there is a canonical map of pro-topological stacksthat induces an equivalence upon profinite completion

We introduce the notion of a holonomic D-module on a smooth (idealized) logarithmic scheme and show that Verdier duality can be extended to this context. In contrast to the classical case, the pushforward of a holonomic module along an open immersion is in general not holonomic. We introduce a "perverse" t-structure on the category of coherent logarithmic D-modules which makes the dualizing functor t-exact on holonomic modules. Conversely this t-exactness characterizes holonomic modules among all coherent logarithmic D-modules. We also introduce logarithmic versions of the Gabber and Kashiwara-Malgrange filtrations.

We globalize the derived version of the McKay correspondence of Bridgeland-King-Reid, proven by Kawamata in the case of abelian quotient singularities, to certain log algebraic stacks with locally free log structure. The two sides of the correspondence are given respectively by the infinite root stack and by a certain version of the valuativization (the projective limit of every possible log blow-up). Our results imply, in particular, that in good cases the category of coherent parabolic sheaves with rational weights is invariant under log blow-up, up to Morita equivalence.