In this paper we prove the Hodge conjecture for arbitrary products of surfaces, S 1 × · · · × S n such that q(S i ) = 2, p g (S i ) = 1. We also prove the Hodge conjecture for arbitrary self-products of a K3 surface X such that the field E = End hg T (X) is CM. Notation and preliminariesUnless otherwise stated, we use the terms curve and surface to denote smooth projective curves and surfaces, resp. The term p g (S) = h 2,0 (S) is called the geometric genus of S, and q(S) = h 1,0 (S) = dim Alb(S) is known as the irregularity of S. For any complex projective manifold X, H k (X) will denote the group H k (X, Q) regarded as a (rational) Hodge structure of (pure) weight k. All Hodge structures appearing in this paper are rational and pure [2]; as usual, a Hodge cycle (of codimension p) or Hodge class of a Hodge structure V is an element v ∈ V p,p C ∩ V . We denote the subspace of Hodge cycles of V by H(V ), and also H p (X) = H(H 2p (X)) for X a smooth projective variety; consequently, H(X) = ⊕ dim(X) i=0 H i (X) will denote the Hodge ring, or ring of Hodge classes of X.We define the (rational) transcendental lattice T (S) of a surface S by the following orthogonal decompositionwith respect to the cup-product. The cup-product induces, after a change of sign, a polarisation of the Hodge structure T (S) [2]. For V and W two (pure) Hodge structures of the same weight, we denote Hom hg (V, W ) to be the space of linear maps from V to W respecting the Hodge structures. For an introduction see [2], [5].For a Hodge structure V as above we define the Hodge group of V , Hg(V ) to be the minimal Q-defined algebraic subgroup of GL(V ) such that h(U (1)) ⊂ Hg(V ) R ; here h is the representation corresponding to the Hodge bigraduation as in [2]. The following is basic in this paper:

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We study Lagrangian points on smooth holomorphic curves in TP 1 equipped with a natural neutral Kähler structure, and prove that they must form real curves. By virtue of the identification of TP 1 with the space L(E 3 ) of oriented affine lines in Euclidean 3-space E 3 , these Lagrangian curves give rise to ruled surfaces in E 3 , which we prove have zero Gauss curvature.Each ruled surface is shown to be the tangent lines to a curve in E 3 , called the edge of regression of the ruled surface. We give an alternative characterization of these curves as the points in E 3 where the number of oriented lines in the complex curve Σ that pass through the point is less than the degree of Σ. We then apply these results to the spectral curves of certain monopoles and construct the ruled surfaces and edges of regression generated by the Lagrangian curves.