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Main discovery The authors give a geometric description of the formal local data of holonomic modules over the first Weyl algebra. Restricted to elementary objects, the exponential part is encoded by a truncated jet of an affine plane curve at infinity, and the regular part is encoded by a regular formal meromorphic connection; for elementary objects these data determine the corresponding elementary block bijectively (Theorem 4.4). (F1, F8, F3)
What the words mean The first Weyl algebra is the algebra of differential operators in one variable. A truncated jet records the initial terms of a curve’s Laurent expansion near the line at infinity. A formal meromorphic connection is a linear system described over formal Laurent series. (F1, F5)
How the result is built The authors construct a functorial passage from global Weyl-algebra modules to formal objects. They factor this construction as two steps. First they take a germ to a module over formal power series. Then they apply a formalization/extension functor Φ that lands in modules over formal Laurent series. (F4)
They formalize the exponential data using Kontsevich’s space J_∞, viewed concretely as truncated jets of affine plane curves at infinity. They introduce a decorated version eJ_∞ that pairs a truncated jet with a regular C((t))-connection. (F5)
The core correspondence and uniqueness Restricted to elementary objects, the paper proves a bijection between isomorphism classes of elementary blocks El(ρ,φ,R) and decorated jets (j, R ⊗ H_j) in eJ_∞. Here H_j is a one-dimensional regular formal connection constructed from a minimal triple corresponding to the truncated jet. This is stated as the precise Theorem 4.4 in the work. (F3, F6)
Key structural tools The formalization functors satisfy the adjunction chain Ψ ⊣ ι* ⊣ Φ ⊣ ι! ⊣ Ψ. The functor Φ admits a microlocal description: Φ(M_c) = (μD ⊗_{D_{t_c}} M_c)|_{C((t_c))}, where μD denotes the microlocalization ring used to pass from D_{t_c}-modules to their microlocal incarnations. These identities are used to track how local data transform under automorphisms and under integral transforms such as the Fourier transform. (F7)
Fourier transform and stationary phase Local Fourier transforms and a stationary phase formula are central. The paper gives explicit local Fourier-transform formulas for elementary blocks and uses stationary phase formulas to track how the geometric data transform. The resulting correspondence is equivariant under automorphisms of the first Weyl algebra, including the Fourier transform. The authors compare this Weyl-algebra action with the action of a subgroup of polynomial automorphisms of the affine plane and show they agree. (F9, F14, F2, F10)
Significance The work gives a concrete geometric encoding of the two parts of formal local data: an exponential factor coming from a truncated jet of a curve at infinity, and a regular factor coming from a formal connection. For elementary objects, that encoding determines the formal building blocks uniquely. The description is compatible with natural symmetries of the Weyl algebra, so it respects operations such as the Fourier transform. (F1, F3, F2)
Caveats and limits The bijection is proved only for elementary objects. In general, more detailed quiver-representation data of regular holonomic C[[t]]<∂>-module germs carry extra structure that eJ_∞ does not recover. (F3, F13)
Also, one formalization functor (the meromorphic localization Ψ) vanishes on modules supported at points. That means Ψ alone does not capture all behavior under Fourier transform. The paper follows Arinkin’s approach to associate a canonical W1-module to a C((t))-connection so the global Fourier transform can be applied; this step is needed because connections with nonzero horizontal sections can send part of their data to point-supported modules, which are lost if one passes only to formal meromorphic connections. (F11, F12)
Technical context The results sit inside the refined Turrittin–Levelt theory: every finite-dimensional formal meromorphic connection decomposes into elementary blocks, and with minimal presentations the decomposition is unique up to order and isomorphism. The paper identifies the geometric source of the exponential factors and shows how to pair them with regular factors to recover elementary blocks. (F8)