Stacky geometry and logarithmic topology of transversely affine foliations

Technical Summary: Stacky Geometry and Logarithmic Topology of Transversely Affine Foliations

Problem Statement
The paper addresses the geometric and topological structures underlying transversely affine foliations on complex manifolds. While Singer's classical theorem on Liouvillian first integrals establishes that such foliations are governed by quadratures, exponentials of primitives, and algebraic extensions, the authors seek to recast this rigidity in a modern geometric framework. Specifically, the work aims to provide a unified description of the "transverse developing coordinate" and its reparametrisations using two complementary perspectives:

1. Stacky Geometry: Describing the multiplicative developing coordinate and its monodromy via quotient stacks.
2. Logarithmic Topology: Analyzing the boundary behavior, residues, and induced dynamics via the Kato–Nakayama space.

The central challenge is to reconcile the algebraic nature of Singer's operations (reparametrisations and algebraic extensions) with the analytic and topological data of the foliation, particularly when the holonomy group is non-discrete or the action is non-proper.

Methodology
The authors employ a dual approach combining stack theory, logarithmic geometry, and dynamical systems:

1. Stack-Theoretic Formulation:

   • The authors associate a transversely affine foliation $(\omega, \eta)$ with a linear monodromy character $\rho: \pi_1(U) \to \mathbb{C}^*$.
   • They define the quotient stack $[\mathbb{C}^*/\Gamma]$, where $\Gamma = \text{im}(\rho)$, as the natural geometric base for the multiplicative developing coordinate.
   • Using the Morita/bibundle formalism, they characterize morphisms between these stacks. They distinguish between strict semi-equivariant maps on the atlas and intrinsic 2-morphisms of the stack.
   • They introduce Kummer (root) refinements $[\mathbb{C}^*/(\mu_N \times \Gamma)]$ to geometrically model Singer's algebraic extension steps.

2. Logarithmic-Topological Analysis:

   • The authors pass to the Kato–Nakayama space $X^{\log}$, the real oriented blow-up of the divisor $D$ with torus fibers over boundary strata.
   • They analyze the pull-back of the logarithmic form $\eta$ to $X^{\log}$, showing it induces a real codimension-two foliation.
   • They study the restriction of this foliation to the boundary tori, demonstrating that the dynamics are linear and governed by the arithmetic of the residue vector.

3. Synthesis:

   • The paper constructs a canonical logarithmic lift of the developing map, $F_{\log}: X^{\log} \to [\mathbb{C}^*/\Gamma]$, which unifies the interior stacky description with the boundary topological data.

Key Contributions and Results

• The Stacky Singer Theorem (Proposition 3.15):
  The authors prove that any holomorphic stacky first integral for a transversely affine foliation is obtained, up to 2-isomorphism, by post-composing the canonical developing morphism $F: U \to [\mathbb{C}^*/\Gamma]$ with an endomorphism of the stack $[\mathbb{C}^*/\Gamma]$. This establishes that the "reparametrisation freedom" is entirely captured by the endomorphism monoid of the quotient stack.

• Classification of Reparametrisations by Regime (Section 5):
  The paper provides a rigorous classification of the endomorphisms of $[\mathbb{C}^*/\Gamma]$ based on the topology of the holonomy group $\Gamma$:

  • Proper/Discrete Cases: If $\Gamma$ is finite (torsion) or infinite cyclic with $|q| \neq 1$ (elliptic regime), the quotient is a Deligne–Mumford curve or an elliptic curve. Here, analytic automorphisms coincide with algebraic ones (GAGA).
  • Wild/Non-Discrete Cases: If $\Gamma$ is not closed (e.g., dense in $\mathbb{C}^*$ or accumulating at 1), the coarse quotient is non-Hausdorff. The authors show that despite this, the stacky base remains well-defined. In these "accumulation" regimes, holomorphic and meromorphic endomorphisms are rigid: they are necessarily monomial maps ($z \mapsto az^n$) or their inverses.

• Kummer Refinements as Algebraic Extensions (Theorem 4.5):
  The paper identifies Kummer base changes (passage to root stacks) as the geometric realization of Singer's algebraic extension step. It proves that any stacky first integral can be factored through a Kummer refinement followed by a meromorphic endomorphism of the refined base.

• Logarithmic Dynamics on the Boundary (Section 7):
  On the Kato–Nakayama space $X^{\log}$, the residues of $\eta$ define canonical boundary characters. The induced real foliation on boundary tori is linear. The authors classify the topology of the leaves:

  • Leaves are translates of subtori.
  • Their closures are determined by the rational span of the residue vector (Kronecker-type dichotomy: closed, dense in the torus, or dense in a proper subtorus).
  • The radial behavior (attracting, repelling, or neutral) is determined by the real part of the residues.

• The Logarithmic Lift and Complementary Realisations (Theorem 8.15):
  The authors construct a canonical morphism $F_{\log}: X^{\log} \to [\mathbb{C}^*/\Gamma]$. They demonstrate that the same residue data admits two complementary boundary realisations:

  1. Kummer Realisation: Removes torsion in the meridional monodromy via root constructions (algebraic extension).
  2. Kato–Nakayama Realisation: Retains the angular boundary data and the induced linear dynamics, which are invisible in a purely radial compactification.

Significance and Claims
The paper claims to provide a "geometric counterpart" to Singer's classical theorem. Its primary significance lies in:

1. Unification: It bridges the gap between the algebraic operations of differential algebra (Singer's closure) and the analytic geometry of foliations by interpreting them as morphisms of quotient stacks.
2. Handling Non-Proper Actions: It demonstrates that quotient stacks are the correct geometric objects to study transversely affine foliations even when the holonomy group is non-discrete and the coarse quotient is pathological (non-Hausdorff). The stack retains the necessary monodromy and isotropy information.
3. Dynamical Insight: It reveals that the Kato–Nakayama space is not merely an auxiliary topological tool but the natural setting where the "logarithmic boundary" acquires concrete geometric and dynamical form. The residues govern not just the monodromy but the actual linear dynamics on the boundary tori.
4. Rigidity: It establishes strong rigidity results for reparametrisations in non-discrete regimes, showing that the "wild" nature of the holonomy forces the reparametrisations to be monomial, thereby simplifying the classification of first integrals in these cases.

The authors conclude that the quotient stack $[\mathbb{C}^*/\Gamma]$ controls the linear and reparametrisation-theoretic aspects of the theory, while the Kato–Nakayama space provides the natural topological and dynamical setting where the boundary geometry of transversely affine foliations becomes visible.