Hodge and Frobenius colevels of algebraic varieties

TL;DR

The paper develops new, sharp lower bounds for Hodge and Frobenius colevels of algebraic varieties across all cohomological degrees, expressed in terms of dimension and the multidegree data of defining equations. It introduces refined invariants $ u_j^{(e)}(N;d_1, ldots,d_r)$ and proves bounds for affine and projective cases that extend and sharpen Ax–Katz–Deligne-type results, using a Lefschetz–type Gysin framework built from Deligne’s generic base change and weak Lefschetz, together with the del Angel–Esnault–Katz reduction method. These bounds imply stronger polar-divisibility statements for zeta functions and answer affirmatively a divisibility question for affine varieties, with concrete determinantal examples illustrating the improvements. The work blends $p$-adic insights with formal, base-change–driven arguments to achieve bounds for non-complete intersections and to develop projective consequences from affine results. Overall, it advances the coniveau–colevel philosophy by tying defining-equation data to cohomological divisibility across all degrees, with arithmetic ramifications via zeta-function poles.

Abstract

We provide new, improved lower bounds for the Hodge and Frobenius colevels of algebraic varieties (over $\mathbf{C}$ or over a finite field) in all cohomological degrees. These bounds are expressed in terms of the dimension of the variety and multi-degrees of its defining equations. Our results lead to an enhanced positive answer to a question raised by Esnault and the first author.

Theorems & Definitions (32)

• Definition 1.1: Frobenius colevel
• Definition 1.2: Hodge colevel
• Example 1.3
• Example 1.4
• Remark 1.5
• Remark 1.7
• Theorem 1.8: Deligne
• Definition 1.9
• Theorem 1.10
• Theorem 1.12: See Lemma \ref{['lemma:complete-intersection-bound']}