Multiplicative sub-Hodge structures of conjugate varieties

TL;DR

This paper constructs conjugate varieties with non-isomorphic K-rational (p,p)-class algebras, revealing that complex Hodge structures are not invariant under automorphisms of the complex numbers, especially outside imaginary quadratic fields.

Contribution

It provides explicit examples of conjugate varieties with differing algebraic structures, challenging assumptions about Hodge structure invariance under Galois actions.

Findings

Constructs conjugate varieties with non-isomorphic K-rational (p,p)-class algebras.

Shows complex Hodge structures are not invariant under Aut( ext{C}) actions.

Detects non-homeomorphic conjugate varieties across various fundamental groups and dimensions.

Abstract

For any subfield K of the complex numbers which is not contained in an imaginary quadratic number field, 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 number field; additionally, it shows that the complex Hodge structure on the complex cohomology algebra is not invariant under the Aut(\C)-action on varieties. In our proofs, we find simply connected conjugate varieties whose multilinear intersection forms on their second real cohomology groups are not (weakly) isomorphic. Using these, we detect non-homeomorphic conjugate varieties for any fundamental group and in any birational equivalence class of dimension at least 10.