Hodge classes on self-products of \(K3\) surfaces (Q2863508)

In this thesis, the Hodge conjecture for self-products of \(K3\) surfaces is studied.NEWLINENEWLINELet \(S\) be a \(K3\) surface. To prove the Hodge conjecture for \(S\times S\) it suffices to check the Hodge conjecture for \(H^4(S\times S)\). Let \(T(S)\) be the orthogonal complement of \(\mathrm{Pic}(S)\otimes \mathbb{Q}\) inside \(H^2(S,\mathbb{Q})\). Then the Hodge conjecture for \(H^4(S\times S)\) is equivalent to the statement that all endomorphisms of \(T(S)\), which respect the Hodge structure, on \(T(S)\) are induced by algebraic cycles.NEWLINENEWLINELet \(E(S)\) be the algebra of all these endomorphisms. Then \(E(S)\) is either a CM-field or a totally real field. A result by Mukai implies the Hodge conjecture for \(S\times S\) if \(E(S)\) is a CM-field. Therefore the author focuses mostly on the case where \(E(S)\) is a totally real number field.NEWLINENEWLINEIn the first chapter of the thesis the author proves a result on the deformation theory of \(S\), which shows that the Hodge conjecture for \(S\times S\) is implied by Grothendieck's invariant cycle conjecture. The same result was obtained before by André, but by using the Kuga-Satake correspondence.NEWLINENEWLINEIn the second chapter the Kuga-Sutake correspondence for \(K3\) surfaces is studied. The author proves that if \(S\) is a double cover of \(\mathbb{P}^2\) ramified along six lines then the Hodge conjecture holds true for \(S\times S\). This result seems to yield examples of fourfolds for which the Hodge conjecture was not known before.NEWLINENEWLINEIn the third chapter it is shown that the Hodge conjecture for \(S\times S\) holds true if and only if the Hodge conjecture holds true for \(\mathrm{Hilb}^{2}(S)\).NEWLINENEWLINEIn the fourth chapter two further results are proven. First it is shown that if \(E(S)\) is a CM-field then \(S\) is defined over a number field. This result was proven before by Piateski-Shapiro and Shafarevich and by Rizov using different techniques.NEWLINENEWLINEThe final result is that if \(Y\) is a moduli space of sheaves on a \(K3\)-surface \(S\) then the André motive of \(Y\) is contained in the smallest Tannakian subcategory of the category of motives containing \(\mathfrak{h}^2(Y)\).