MULTIPLICATIVE SUB-HODGE STRUCTURES OF CONJUGATE VARIETIES
Forum of Mathematics, Sigma, Tome 2 (2014)
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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.
@article{10_1017_fms_2013_7,
author = {STEFAN SCHREIEDER},
title = {MULTIPLICATIVE {SUB-HODGE} {STRUCTURES} {OF} {CONJUGATE} {VARIETIES}},
journal = {Forum of Mathematics, Sigma},
publisher = {mathdoc},
volume = {2},
year = {2014},
doi = {10.1017/fms.2013.7},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2013.7/}
}
STEFAN SCHREIEDER. MULTIPLICATIVE SUB-HODGE STRUCTURES OF CONJUGATE VARIETIES. Forum of Mathematics, Sigma, Tome 2 (2014). doi: 10.1017/fms.2013.7
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