Cohomologie non ramifi\'ee dans le produit avec une courbe elliptique

TL;DR

This paper explores unramified cohomology in products with elliptic curves, demonstrating how certain cohomology classes relate to the failure of the integral Hodge conjecture in specific algebraic varieties.

Contribution

It applies Gabber's method to produce unramified cohomology classes and connects these to the failure of the integral Hodge conjecture for products involving elliptic curves.

Findings

Identifies cases where the integral Hodge conjecture fails for certain products.

Uses Gabber's method to construct unramified cohomology classes.

Extends previous results to new classes of algebraic varieties.

Abstract

A method of Gabber (2002) produces unramified cohomology classes in the products of certain varieties with an elliptic curve. The connection between third unramified cohomology and integral Hodge conjecture for codimension 2 cycles (2012, which builds upon results from algebraic K-Theory, then gives many examples of such a product for which this conjecture fails. The special case of the product with an Enriques surface was established by Benoist and Ottem (2018).