초록   한글번역

Abstract For any subfield$K\subseteq \mathbb{C}$, not contained in an imaginary quadratic extension of$\mathbb{Q}$, we construct conjugate varieties whose algebras of$K$-rational ($p,p$)-classes are not isomorphic. This compares to the Hodge conjecture which predicts isomorphisms when$K$is contained in an imaginary quadratic extension of$\mathbb{Q}$; additionally, it shows that the complex Hodge structure on the complex cohomology algebra is not invariant under the Aut($\mathbb{C}$)-action on varieties. In our proofs, we find simply connected conjugate varieties whose multilinear intersection forms on$H^{2}(-,\mathbb{R})$are not (weakly) isomorphic. Using these, we detect nonhomeomorphic conjugate varieties for any fundamental group and in any birational equivalence class of dimension$\geq $10.