On the Beilinson-Hodge conjecture for $H^2$ and rational varieties

TL;DR

This paper proves the Beilinson-Hodge conjecture for the case n=2 when the variety is rational, confirming a key aspect of the conjecture in this specific setting.

Contribution

It establishes the surjectivity of the cycle map for n=2 on rational varieties, advancing understanding of the conjecture in this particular case.

Findings

Proves the conjecture for n=2 on rational varieties

Confirms the surjectivity of the cycle map in this setting

Provides new evidence supporting the Beilinson-Hodge conjecture

Abstract

The Beilinson-Hodge conjecture asserts the surjectivity of the cycle map $$H^n_M(X,\Q(n)) \to {\rm Hom}_{MHS}(\Q(-n),H^n(X,\Q))$$ for all positive integers $n$ and every smooth complex algebraic variety $X$. For $n=2$, we prove the conjecture if $X$ is rational.