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The elements of $\\mathrm{Gr}^W_{2p}H^{2p}(X,\\mathbb{Q}) \\cap H^{p,p} \\mathrm{Gr}^W_{2p}H^{2p}(X,\\mathbb{C})$ will be called Hodge (p,p)-classes.The purpose of this article, is to study the Bloch-Gillet-Soul\\'{e} (BGS) cycle class map from the $p$-th operational Chow group $A^p(X)$ to the space of $(p,p)$-Hodge classes. We show that if $p=1$ and $X$ is a normal surface with at worst rational singularities, then the BGS cycle class map is surjective. 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