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MULTIPLICATIVE SUB-HODGE STRUCTURES OF CONJUGATE VARIETIES
Published online by Cambridge University Press: 18 February 2014
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.
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- Research Article
- Information
- Creative Commons
- The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution licence .
- Copyright
- © The Author 2014
References
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