Logarithmic morphisms, tangential basepoints, and little disks

TL;DR

The paper develops a comprehensive framework of virtual morphisms in logarithmic geometry to encode Deligne’s tangential basepoints and to achieve functorial Betti and de Rham cohomology in this extended setting. It then constructs an algebro-geometric model of the little disks operad as a moduli space of genus-zero curves with tangential basepoints, using virtual morphisms to transport tangential data along gluings; this yields a direct Beilinson-style proof of formality. The formalism unifies basepoint phenomena across topological, algebraic, and motivic contexts, and identifies a bridge to configuration-space operads via Kato–Nakayama spaces. The results illuminate the roles of virtual morphisms in cohomology, deformations, and operad theory, with explicit connections to Deligne–Goncharov tangential basepoints, Riemann–Hilbert-type splittings, and Bar-Nakayama–type groupoids, and point toward motivic refinements and extensions to other Weil cohomologies.

Abstract

We develop the theory of "virtual morphisms" in logarithmic algebraic geometry, introduced by Howell. It allows one to give algebro-geometric meaning to various useful maps of topological spaces that do not correspond to morphisms of (log) schemes in the classical sense, while retaining functoriality of key constructions. In particular, we explain how virtual morphisms provide a natural categorical home for Deligne's theory of tangential basepoints: the latter are simply the virtual morphisms from a point. We also extend Howell's results on the functoriality of Betti and de Rham cohomology. Using this framework, we lift the topological operad of little $2$-disks to an operad in log schemes over the integers, whose virtual points are isomorphism classes of stable marked curves of genus zero equipped with a tangential basepoint. The gluing of such curves along marked points is performed using virtual morphisms that transport tangential basepoints around the curves. This builds on Vaintrob's analogous construction for framed little disks, for which the classical notion of morphism in logarithmic geometry sufficed. In this way, we obtain a direct algebro-geometric proof of the formality of the little disks operad, following the strategy envisioned by Beilinson. Furthermore, Bar-Natan's parenthesized braids naturally appear as the fundamental groupoids of our moduli spaces, with all virtual basepoints defined over the integers.

Figures (4)

• Figure 1: Algebraic vs. $C^\infty$ tangential basepoints at the origin in $\mathbb{A}^1$.
• Figure 2: A curve defining a point of $D_{A',A"}$ (left) and its dual rooted tree (right).
• Figure 3: Degeneration of a smooth curve to a nodal curve with the marked points $p$ and $q$ separated by a node $\nu$, viewed inside the universal curve. The marked points other than $p$ and $q$ are omitted from the diagram.
• Figure 4: Virtual $\mathbb{Z}$-points of $\mathfrak{FM}_{A}$ are in bijection with planar rooted trees, or equivalently parenthesized monomials built from the elements of $A$; the correspondence is illustrated here with the set of labels $A = \{a,b,c\}$.

Theorems & Definitions (82)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• corollary 1
• definition 1
• definition 2
• lemma 1
• definition 3
• definition 4
• proposition 1