Holonomic étale sheaves are constructible

Technical Summary: Holonomic Étale Sheaves Are Constructible

Problem Statement
The paper addresses the relationship between the notions of "constructibility" and "holonomicity" for étale sheaves on algebraic varieties. In the context of $\mathcal{D}_X$-modules on complex manifolds, Kashiwara's theorem establishes that holonomic modules are constructible. In the algebraic setting, Beilinson previously proved that every constructible étale sheaf possesses a singular support with irreducible components of dimension equal to the dimension of the underlying scheme, effectively characterizing constructible sheaves as "holonomic" (though he did not explicitly define holonomicity for non-constructible sheaves).

The central problem tackled here is the converse: Can one define a notion of holonomicity for étale sheaves without a priori assuming constructibility, and does this definition imply constructibility? The authors aim to establish that holonomic étale sheaves are indeed constructible, providing an étale analogue of Kashiwara's theorem.

Methodology
The authors develop a framework for "micro support" (singular support) for arbitrary étale sheaves, extending Beilinson's construction which originally assumed constructibility. The methodology proceeds through the following steps:

1. Geometric Foundations (Sections 1–3): The paper establishes the necessary geometric language regarding closed conical subsets of the cotangent bundle $T^*X$. It defines and analyzes two key properties of morphisms relative to a conical subset $C$:

   • $C$-transversality: A condition on a morphism $h: W \to X$ regarding the intersection of the pullback of $C$ with the kernel of the cotangent map.
   • $C$-acyclicity: A condition on a pair of morphisms $(h, f)$ involving the acyclicity of $f$ relative to the pushforward of $C$.
     These concepts are adapted from Beilinson's work but are rigorously defined without assuming the sheaves involved are constructible.

2. Micro Support Definition (Section 4): The authors define a sheaf $F$ to be micro supported on a closed conical subset $C \subset T^*X$ if, for every $C$-acyclic pair of morphisms $(h, f)$, the morphism $f$ is locally acyclic relative to the pullback $h^*F$. This definition generalizes the notion of micro support to non-constructible sheaves. Key lemmas establish the behavior of micro support under operations such as pullbacks, pushforwards, and distinguished triangles.

3. Existence of Singular Support (Section 5): The authors prove that for any sheaf $F$ on a smooth scheme $X$, a unique minimal closed conical subset $SSF$ (the singular support) exists such that $F$ is micro supported on it.

   • The proof relies on reducing the problem to the case where $X$ is a projective space $\mathbb{P}^n$.
   • The Radon transform and the Legendre transform are employed to analyze the micro support on projective spaces. Specifically, the authors utilize the geometry of the incidence variety $Q \subset \mathbb{P} \times \mathbb{P}^\vee$ and the properties of the Radon transform $R$ and its inverse $R^\vee$ to relate the micro support of a sheaf to that of its transform.
   • This section demonstrates that the singular support exists even without the constructibility assumption, a point raised by Tong Zhou.

4. Holonomicity and Constructibility (Section 6):

   • Definition: A sheaf $F$ is defined as holonomic if (1) its singular support $SSF$ satisfies $\dim SSF \leq \dim X$, and (2) the stalk $F_x$ is constructible for every geometric point $x$.
   • Main Theorem: The authors prove that if $F$ is holonomic, then $F$ is constructible.
   • Proof Strategy: The proof proceeds by induction on $\dim X$. It utilizes the fact that on a dense open subset, a holonomic sheaf is locally constant (micro supported on the zero section). By analyzing the distinguished triangle associated with the complement of this open set and using the inductive hypothesis, the authors show that the sheaf must be constructible everywhere. The argument is first established for perfect fields and then extended to general fields via base change to the perfect closure.

Key Contributions and Results

• Definition of Holonomicity: The paper introduces a rigorous definition of holonomicity for étale sheaves that does not presuppose constructibility.
• Existence of Singular Support: It establishes the existence of the singular support $SSF$ for arbitrary sheaves (not just constructible ones), resolving a question regarding the necessity of the constructibility assumption for the existence of $SSF$.
• Main Theorem (Theorem 6.3): The central result is the converse of Beilinson's theorem: Holonomic étale sheaves are constructible.
  • Specifically, if $\dim SSF \leq \dim X$ and all stalks are constructible, the sheaf itself is constructible.
• Extension to Singular Schemes: The notion of holonomicity is extended to sheaves on singular schemes via closed immersions into smooth schemes, proving that this definition is independent of the embedding.
• Stability: The paper confirms that the category of holonomic sheaves is stable under standard operations (direct and inverse images, proper direct images with compact support) for morphisms of finite type.

Significance
The paper claims its significance as an étale analogue of Kashiwara's theorem on holonomic $\mathcal{D}_X$-modules. Just as Kashiwara's theorem asserts that holonomic $\mathcal{D}$-modules have constructible cohomology (a key ingredient in the Riemann–Hilbert correspondence), this work asserts that holonomic étale sheaves are constructible.

The authors note that this result provides a robust substitute for the "deficient notion of constructibility" in contexts such as rigid varieties, where a similar notion of holonomicity is expected to be definable. The work builds directly on Beilinson's foundational article [3], refining the definitions to remove the a priori constructibility assumption and proving that the geometric condition of having a "small" singular support (dimension $\leq \dim X$) combined with pointwise constructibility is sufficient to guarantee global constructibility.