Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Coniveau filtrations with Z/2 coefficients

Masaki Kameko

TL;DR

This work examines two filtrations, the coniveau and strong coniveau, on mod $2$ cohomology of smooth projective complex varieties. It constructs explicit counterexamples by approximating classifying spaces $B(G\times S^1)$ with smooth projective varieties $X$ and analyzing the mod $2$ cohomology of $BPU(4)$ (with $G=PU(4)$) to produce a degree-3 generator whose coniveau is at least one but whose strong coniveau is strictly less than one. The method couples Steenrod and Milnor operations with Gysin pushforwards and leverages Ekedahl-type approximations to transport cohomology from the classifying space to $X$, yielding a concrete obstruction to strong coniveau. The result demonstrates that $\tilde{N}^1H^{3}(X;\mathbb{Z}/2)$ can be strictly contained in $N^1H^{3}(X;\mathbb{Z}/2)$, informing the nuanced relationship between coniveau and strong coniveau filtrations in mod $2$ settings and suggesting avenues for analogous constructions at odd primes.

Abstract

We show that two coniveau filtrations on the mod 2 cohomology group of a smooth projective complex variety differ.

Coniveau filtrations with Z/2 coefficients

TL;DR

This work examines two filtrations, the coniveau and strong coniveau, on mod cohomology of smooth projective complex varieties. It constructs explicit counterexamples by approximating classifying spaces with smooth projective varieties and analyzing the mod cohomology of (with ) to produce a degree-3 generator whose coniveau is at least one but whose strong coniveau is strictly less than one. The method couples Steenrod and Milnor operations with Gysin pushforwards and leverages Ekedahl-type approximations to transport cohomology from the classifying space to , yielding a concrete obstruction to strong coniveau. The result demonstrates that can be strictly contained in , informing the nuanced relationship between coniveau and strong coniveau filtrations in mod settings and suggesting avenues for analogous constructions at odd primes.

Abstract

We show that two coniveau filtrations on the mod 2 cohomology group of a smooth projective complex variety differ.
Paper Structure (15 sections, 10 theorems, 36 equations)

This paper contains 15 sections, 10 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. A sufficient condition for coniveau $\geq 1$
  3. The mod $2$ reduction map
  4. Definition of the coniveau filtration
  5. A sufficient condition for coniveau $\geq 1$
  6. A sufficient condition for strong coniveau $<1$
  7. Steenrod operations
  8. Milnor operations
  9. The Gysin pushforward map
  10. Definition of strong coniveau filtration
  11. A sufficient condition for strong coniveau $<1$
  12. Proof of Theorem \ref{['theorem:1.1']}
  13. Approximation of classifying space
  14. Cohomology of $BPU(4)$
  15. Proof of Theorem \ref{['theorem:1.1']}

Key Result

Theorem 1.1

There is a smooth projective complex variety $X$ such that the inclusion $\tilde{N}^1H^{3}(X;\mathbb{Z}/2)\subset N^1H^{3}(X;\mathbb{Z}/2)$ is strict.

Theorems & Definitions (18)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 4.1
  • ...and 8 more