The Hodge-𝒟-conjecture for 𝒦3 and Abelian surfaces

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