Uniform stratified vanishing and equidistribution on $\mathbb{G}_m^d$

Technical Summary: Uniform Stratified Vanishing and Equidistribution on $\mathbb{G}_m^d$

Problem Statement
The paper addresses the problem of establishing uniform bounds on the "bad" loci of character sums associated with perverse sheaves on the split $d$-dimensional torus $\mathbb{G}_m^d$ over finite fields. Specifically, it seeks to prove a uniform stratified generic vanishing theorem where the bounds depend only on the dimension $d$ and the complexity of the sheaf (in the sense of Sawin), rather than on the specific field characteristic or the prime $\ell$.

These vanishing theorems describe the locus of characters $\chi$ for which the forget-supports morphism $R\Gamma_c(\mathbb{G}_m^d, M_\chi) \to R\Gamma(\mathbb{G}_m^d, M_\chi)$ fails to be an isomorphism. While generic vanishing was previously established by Gabber–Loeser (conditional on resolution of singularities) and later unconditionally by Forey–Fresán–Kowalski (FFK), those results did not provide uniform quantitative bounds on the number of finite-field characters in these bad loci as the field cardinality grows.

The authors aim to extend equidistribution theorems for these character sums to arbitrary sequences of finite fields with growing cardinality, allowing the characteristic to vary. Previous results by Katz (for $d=1$) and FFK (for connected commutative groups) were either restricted to fixed objects over varying extensions or conditional on uniform bounds for higher-dimensional tori ($d \ge 2$).

Methodology
The proof strategy relies on three main components: a uniform one-coordinate Mellin estimate, induction on the dimension of the torus, and a Tannakian deduction for equidistribution.

1. Uniform One-Coordinate Mellin Estimate:
   The core technical innovation is a quantitative bound on the set of "exceptional" characters for a single multiplicative variable. By decomposing $\mathbb{G}_m^d = \mathbb{G}_m \times \mathbb{G}_m^{d-1}$, the authors analyze the relative direct images of a perverse sheaf twisted by a character of the first factor.

   • They utilize the localization triangle to relate the failure of the forget-supports morphism to the boundary stalks at $0$ and $\infty$.
   • These boundary stalks are identified with continuous inertia cohomology of relative nearby-cycle complexes.
   • Using the prime-to-$\ell$ vanishing of inertia cohomology (Appendix B) and the injectivity of the map $\lambda \mapsto \lambda_x(\tau_x)$ (where $\tau_x$ is a topological generator of tame inertia), they show that exceptional characters correspond to monodromy eigenvalues on these nearby-cycle complexes.
   • Crucially, they bound the number of such eigenvalues solely in terms of the complexity $cu(A)$ and dimension $d$, avoiding the use of de Jong's alterations or resolution of singularities, which would otherwise introduce non-uniform constants.

2. Induction on Dimension:
   The authors prove the main theorems by induction on $d$.

   • Theorem 1.1 (Uniform Point Count): For a complex $K$, the size of the support $\Delta(K)$ of the cone of the Mellin comparison morphism is bounded by $O(q^{n(d-1)})$. The induction step uses the relative Mellin transform to reduce the problem from dimension $d$ to $d-1$. For non-exceptional characters, the problem reduces to the lower-dimensional case; for exceptional characters, the bound is trivial but controlled by the complexity.
   • Theorem 1.2 (Stratified Point Count): This refines the result to specific cohomological degrees $i$. The authors use the perverse spectral sequence to relate cohomology jumps in degree $i$ on $\mathbb{G}_m^d$ to jumps in degrees $i$ and $i-1$ on $\mathbb{G}_m^{d-1}$ for the perverse cohomology sheaves of the relative direct images. This allows the inductive bound $O(q^{n(d-i)})$ to be preserved.

3. Tannakian Deduction:
   The equidistribution result (Corollary 1.4) is derived using the Tannakian formalism of FFK.

   • The authors establish that the unramified locus $X(M_0)$ is generic (Theorem 1.3) by relating it to the support $\Delta(L)$ of a specific object $L = M \oplus M^\vee$.
   • Using the Weyl criterion, they show that the Frobenius conjugacy classes equidistribute in the maximal compact subgroup of the Tannakian group. The uniform bounds from Theorems 1.2 and 1.3 ensure that the "bad" characters (where cohomology is non-zero in non-zero degrees or where ramification occurs) are negligible in number compared to the total number of characters as the field size tends to infinity.

Key Contributions and Results

• Theorem 1.1 (Uniform Stratified Generic Vanishing): Proves that for any $d \ge 1$, there exists a function $\Phi_d^\Delta$ such that for any finite field $k$, any prime $\ell \neq \text{char}(k)$, and any complex $K$ of bounded complexity $C$, the number of characters $\chi \in \widehat{\mathbb{G}_m^d}(k_n)$ where the forget-supports morphism is not an isomorphism is bounded by $\Phi_d^\Delta(C) q^{n(d-1)}$.
• Theorem 1.2 (Bounded-Complexity Stratified Point Count): Proves that for a perverse sheaf $A$ of complexity $C$, the number of characters where the cohomology $H^i_c$ or $H^i$ is non-zero is bounded by $\Psi_d(C) q^{n(d-i)}$. This provides the necessary uniform bounds for the stratified vanishing loci.
• Theorem 1.3 (Unramified Point Count): Establishes that the complement of the unramified locus for an arithmetically semisimple object is bounded by $O(q^{n(d-1)})$, uniformly in the characteristic and $\ell$.
• Corollary 1.4 (Equidistribution): Proves an equidistribution theorem for sequences of perverse sheaves of bounded complexity with a common Tannakian monodromy group. As the cardinality of the finite fields tends to infinity, the Frobenius conjugacy classes attached to the multiplicative characters equidistribute in the space of conjugacy classes of a maximal compact subgroup of the Tannakian group.

Significance
The paper claims to resolve the open case of horizontal equidistribution for tori of dimension $d \ge 2$ left by Forey–Fresán–Kowalski. While FFK established the structure of the bad loci unconditionally, their equidistribution theorem for $d \ge 2$ remained conditional on uniform bounds for these loci. This paper provides those bounds, making the equidistribution theorem unconditional for $\mathbb{G}_m^d$.

Furthermore, the results generalize Katz's equidistribution theorem (originally for $d=1$) to arbitrary sequences of finite fields, including those where the characteristic varies. The constants in the bounds depend only on the dimension and the complexity of the sheaf (in the sense of Sawin), ensuring uniformity across varying characteristics and primes $\ell$. This uniformity is achieved without relying on resolution of singularities or de Jong's alterations, which typically yield constants dependent on the specific geometry of the variety rather than just its complexity.

The work thus unifies the theory of generic vanishing and equidistribution for tori, providing a robust framework for analyzing exponential sums and character sums in families over varying finite fields.