A Čech--Stokes Pushout Groupoid: a Log/Kummer Betti Presenter for Stokes Torsors

Abstract

Let $X$ be a complex manifold and let $D\subset X$ be a simple normal crossings divisor. Fix a Kummer level $n\ge 1$ and an irregular type $Φ$ along $D$, and consider the logarithmic Betti boundary $X^{\log}_n$ (the level-$n$ Kato--Nakayama space). We construct a strictly $1$-categorical, cover-based Betti presenter for Stokes torsors by building a small strict Čech--Stokes collar groupoid on a punctured logarithmic collar of $D$, together with a canonical projection $π$ to the underlying sector Čech groupoid. The central observation is that the appropriate collar Stokes moduli is the groupoid of strict functorial sections $\operatorname{Sec}(π)$ (equivalently, $\mathrm{St}_Φ$-torsors on the sector cover), yielding a fully explicit presenter that remains small and strictly functorial. We then glue the collar presenter to the ordinary Čech presenter of $U:=X\setminus D$ through a strict diagram of small groupoids and an explicit $2$-pushout presenter $G_{Φ,n}$. We prove that global Stokes objects, defined intrinsically as a $2$-fiber product, are computed by a strict torsorial gluing problem, i.e.\ by Stokes-corrected torsorial representations of $G_{Φ,n}$. In arbitrary SNC depth we solve the local corner presentation problem by an explicit chamber-complex model on the angular torus: Stokes torsors are nonabelian cocycles on the $1$-skeleton with relations read off from $2$-cells. The overall construction is canonical up to Morita equivalence with a contractible space of identifications (invariant under stratified subdivision), and it is compatible with Kummer descent along all normal crossings strata via $(μ_n)^k$-equivariant local models and descent to homotopy fixed points.

Figures (8)

• Figure 1: Cocycle data on the $1$-skeleton (edge labels) and relations from $2$-cells, modulo vertex gauge.
• Figure 2: A directed graph $E$ determines the free groupoid $F(E)$ by adding formal inverses and cancelling $aa^{-1}$; imposing relations $A_\rho=B_\rho$ is encoded by the coequalizer $Q=\mathrm{coeq}(F(R)\rightrightarrows F(E))$.
• Figure 3: At a puncture $p_i$, the level-$n$ circle of directions $S^1_{p_i,n}$ is cut by Stokes walls $\Sigma_{p_i}$ into finitely many arcs $I_{i,a}$. A punctured collar $N_i^\times$ over $\Delta_i^\times$ is then covered by sector boxes $B_{i,a}\simeq \Delta_i^\times\times I_{i,a}$ with adjacent overlaps.
• Figure 4: The interior Čech presenter $\mathscr G_U$ and the collar Čech--Stokes presenter $\mathscr G_C$ (with $\pi:\mathscr G_C\to\check{\mathrm C}(\mathcal{B})$) glued along the overlap presenter $\mathscr G_{\times}$. Their explicit $2$-pushout $\mathscr G_{\Phi,n}$ presents global Stokes objects via torsorial gluing.
• Figure 5: Boundary gluing: the peripheral class $\delta_i$ is identified with $(\theta_i,M_i,\theta_i)\in\mathrm{Aut}(\theta_i)\cong(\mathbb{C},+)$.

Theorems & Definitions (133)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3
• Corollary 1.4
• Remark 2.1
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Remark 3.4
• Proposition 3.5