The Hodge-${\mathcal D}$-conjecture for $\text \textrm {K3}$ and Abelian surfaces

Let $X$ be a projective algebraic manifold, and $\text {CH}^{k}(X,1)$ the higher Chow group, with corresponding real regulator $\text {r}_{k,1}\otimes {{\mathbb R}}: \text {CH}^k(X, 1)\otimes {{\mathbb R}} \to H_{\mathcal D}^{2k-1}(X,{{\mathbb R}}(k))$. If $X$ is a general K3 surface or Abelian surface, and $k=2$, we prove the Hodge-${\mathcal D}$-conjecture, i.e. the surjectivity of $\text {r}_{2,1}\otimes {{\mathbb R}}$. Since the Hodge-${\mathcal D}$-conjecture is not true for general surfaces in $\mathbb {P}^{3}$ of degree $\geq 5$, the results in this paper provide an effective bound for when this conjecture is true.