Generic vanishing on homogeneous spaces in arbitrary characteristic

Technical Summary: Generic Vanishing on Homogeneous Spaces in Arbitrary Characteristic

Problem Statement
The paper addresses the problem of establishing a "generic vanishing" theorem for the intersection of locally closed smooth affine subvarieties within a proper homogeneous space $X$, acted upon by a connected algebraic group $G$. Specifically, the authors seek to prove that for a generic translate $gW$ of a subvariety $W$ by an element $g \in G$, the intersection $gW \cap Z$ satisfies a specific sign inequality regarding its Euler characteristic.

While Schürmann–Simpson–Wang [SSW25] proved this result in characteristic zero, the authors aim to extend this theorem to an arbitrary algebraically closed field, including positive characteristic. The core challenge lies in adapting the geometric and cohomological arguments used in characteristic zero to settings where standard tools (like Hodge theory or specific vanishing theorems relying on characteristic zero) may not apply directly.

Methodology
The authors employ the framework of $\ell$-adic constructible sheaves and the theory of perverse sheaves. Their approach relies on the following key methodological steps:

1. Radon Correspondence: Instead of viewing the family of intersections merely as a parameterized family, the authors model the situation as a Radon correspondence over the group $G$. They define a Radon transform functor $R_W! : D^b_c(X) \to D^b_c(G)$ associated with the incidence correspondence $\Gamma_W \subset G \times X$.
2. Base Change and Künneth Formula: A critical technical observation is that after a smooth pullback along an orbit map $\mu_x: G \to X$, the universal correspondence becomes a product. This reduction allows the authors to utilize the Künneth formula, which holds unconditionally in the derived category of constructible sheaves with finite tor-dimension ($D^b_c$).
3. $t$-Exactness: The authors prove that the composition of the Radon transform with the pushforward of a morphism $f: Z \to X$ is $t$-exact with respect to the perverse $t$-structure. This is achieved by analyzing the properties of the morphisms involved (smoothness, affineness, quasi-finiteness) and applying Artin vanishing and standard base-change theorems.
4. Support Bounds: By establishing the $t$-exactness, they deduce support bounds for the resulting perverse sheaf on $G$. These bounds imply that the cohomology groups of the intersections vanish generically outside a specific degree.
5. Arithmetic Descent: For the case where the base field is the algebraic closure of a finite field $\mathbb{F}_q$, the authors descend the geometric results to the finite field level. They utilize Deligne's generic base change theorem and the Lang–Weil estimate to analyze the trace functions and the size of the "bad" locus (where generic vanishing fails).

Key Contributions and Results

• Main Theorem (Generic Vanishing): The paper proves that for a perverse sheaf $K$ on a subvariety $Z$, there exists a dense open subset $U \subset G$ such that for all $g \in U(k)$, the compactly supported cohomology groups $H^i_c(gW, g f_Z^* g \phi_Z^* K)$ vanish for $i \neq \dim W - \dim X$.
• Sign Inequality: As a consequence, the authors establish the sign inequality:
  $$(-1)^{\dim X - \dim W + \dim Z} \chi(gW \cap Z) \geq 0$$
  for generic $g \in G$, where $\chi$ denotes the Euler characteristic. This result holds over any algebraically closed field, extending the characteristic-zero result of Schürmann–Simpson–Wang.
• Resolution of Conjectures: The result confirms Kumar's sign conjecture for triple intersections of Schubert cells in arbitrary characteristic, a conjecture previously known in characteristic zero only in specific cases or full generality via [SSW25].
• Arithmetic Consequences:
  • Trace-Function Identity: Over finite fields, the trace function of the Radon transform on a dense open subset of $G$ is identified with the Frobenius trace on a single middle cohomology group.
  • Lang–Weil Estimates: The paper provides an estimate for the number of points in the non-generic locus (the complement of the dense open set), showing it is bounded by $O(q^{n(\dim G - 1)})$.
  • Weil Bounds: Under additional assumptions (specifically, that the sheaf has no boundary contribution at infinity), the authors derive a Weil bound for the resulting "exponential sums" on the intersections.

Significance and Claims
The authors position this work as a natural extension of the generic vanishing tradition, which has historically treated families of characters or local systems on fixed supports (e.g., Green–Lazarsfeld, Gabber–Loeser, Schnell, Weissauer). In contrast, this paper treats a fixed sheaf with a support that varies by translation.

The paper claims significance in two main areas:

1. Characteristic Independence: It removes the restriction to characteristic zero for generic vanishing theorems on homogeneous spaces, utilizing the robustness of the Radon transform and perverse sheaf theory in arbitrary characteristic.
2. Arithmetic Applications: It demonstrates that the geometric $t$-exactness theorem yields concrete arithmetic consequences, including precise trace identities and bounds on exponential sums over finite fields, without requiring the data to be defined over characteristic zero.

The authors remain modest regarding the scope, noting that the methods rely on the specific structure of homogeneous spaces and the smoothness of orbit maps, and they do not propose new experimental applications or future directions beyond the immediate arithmetic refinements discussed in Section 4.