Infinite root stacks and quasi-coherent sheaves on logarithmic schemes

Abstract

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.

Parabolic sheaves with real weights as sheaves on the Kato–Nakayama space

“…If we allow the index of the root to vary, these equivalences are compatible with the natural projections m √ X → n √ X for n | m, and in fact there is an analogous statement at the limit, on the infinite root stack ∞ √ X = lim ← −n n √ X [27,Theorem 7.3]. This "stacky" point of view allows to treat parabolic sheaves as "plain" quasi-coherent sheaves on a slightly more complicated object, and has been useful in several instances (see for example [9], [2] and [24]).…”

“…Remark 4.16. We refrain from using the term "coherent" for the sheaves that locally admit finite presentations, because already on the infinite root stack, the structure sheaf might not be coherent (see [27,Example 4.17]), so that "finitely presented" and "coherent" are not equivalent notions. We expect the same to happen in this context.…”

“…They also constructed an equivalence of abelian categories between parabolic sheaves with weights in a fixed Kummer extension, and quasi-coherent sheaves on the corresponding stack of roots. Versions of this correspondence were earlier investigated by Biswas [1] and Borne [4], and it was further generalized to arbitrary rational weights by the author and Vistoli in [27].…”

Gluing semi-orthogonal decompositions

“…is the infinite root stack [34] of (X, D). It is a pro-algebraic stack that embodies the "logarithmic geometry" of the pair (X, D) in its stacky structure, and it is an algebraic analogue of the so-called "Kato-Nakayama space" [10,35].…”

“…In recent years log techniques have become a mainstay of algebraic geometry and mirror symmetry: for instance, log geometry provides the language in which the Gross-Siebert program in mirror symmetry is formulated [14]. In [34] Talpo and Vistoli explain how to to associate to any log scheme an infinite root stack, which is a projective limit of root stacks. This assignment gives rise to a faithful functor from log schemes to stacks: log information is converted into stacky information without any data loss.…”