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MULTIPLICATIVE SUB-HODGE STRUCTURES OF CONJUGATE VARIETIES

Part of: (Co)homology theory Cycles and subschemes Real and complex geometry

Published online by Cambridge University Press:  18 February 2014

STEFAN SCHREIEDER
STEFAN SCHREIEDER*
Affiliation:
Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany; schreied@math.uni-bonn.de

Abstract

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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.

MSC classification

Secondary: 14C30: Transcendental methods, Hodge theory , Hodge conjecture 14F25: Classical real and complex (co)homology 51M15: Geometric constructions 14F35: Homotopy theory; fundamental groups

Information

Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 2 , 01 July 2014 , e1
Creative Commons
Creative Common License - CCCreative Common License - BY
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

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