Algebraic cycles on the real \(K3\)-surfaces. (Q1284352)

For a real algebraic \(K3\)-surface \(X(\mathbb R)\), we give all possible values of the dimension \(h_{\text{alg}}^1 (X(\mathbb R))\) of the group \(H_{\text{alg}}^1 (X(\mathbb R),\mathbb Z/2)\) of algebraic cycles of \(X(\mathbb R)\). In particular, we prove that if \(X(\mathbb R)\) is not an \(M\)-surface, \(X(\mathbb R)\) can always be deformed over \(\mathbb R\) to some \(X'(\mathbb R)\) with \(h_{\text{alg}}^1 (X'(\mathbb R))= \dim H^1(X(\mathbb R),\mathbb Z/2))\). Furthermore, we obtain that in certain moduli space of real algebraic \(K3\)-surfaces, the collection of real isomorphism classes of \(K3\) surfaces such that \(h_{\text{alg}}^1(X(\mathbb R))\geq k\) is a countable union of subspaces of dimension \(20-k\).