Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0706.3674

Conjugate varieties with distinct real cohomology algebras

François Charles
View original source ↗Read source document ↗Export metadata ↓

Abstract

Using constructions of Voisin, we exhibit a smooth projective variety defined over a number field k and two complex embeddings of k, such that the two complex manifolds induced by these embeddings have non isomorphic cohomology algebras with real coefficients. This contrasts with the fact that the cohomology algebras with l-adic coefficients are canonically isomorphic for any prime number l, and answers a question of Grothendieck.

Explore related subjects

math.AG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

François Charles. 2008-07-15. Conjugate varieties with distinct real cohomology algebras. https://arxiv.org/abs/0706.3674

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the fundamental groups of subelliptic varieties

We show that the fundamental group of any smooth subelliptic variety is finite. Moreover, it is also proved that every finite group can be realized as the fundamental group of a smooth subelliptic variety. As a consequence, it follows that there exists a smooth subelliptic variety homotopy equivalent to the $n$-sphere if and only if $n>1$. This result can be considered as a negative answer to the algebraic version of Gromov's problem on the homotopy types of Oka manifolds.

math.AG↗
arXiv2026

Minimal energy local systems on curves

Let $Σ_{g,n}$ be an orientable topological surface of genus $g$ with $n$ punctures. When $g = 0$, Deroin and Tholozan studied the class of supra-maximal representations $π_1(Σ_{0,n})\to \mathrm{PSL}_2(\mathbb{R})$, and they showed that the supra-maximal representations form a compact component of a real relative character variety. We study a collection of rank $r$ local systems on $Σ_{g,n}$ which we call of minimal energy. These are generalizations of supra-maximal representations, and underlie polarizable complex variations of Hodge structure for any choice of complex structure on $Σ_{g,n}$. Like the supra-maximal representations, the minimal energy local systems form compact components of relative character varieties of real forms of $\mathrm{GL}_r(\mathbb{C})$. We show that when the local monodromy data around the punctures is chosen to be unitary and generic, and the relative character variety is nonempty, these minimal energy local systems always exist. When $g > 0$, we show that the minimal energy local systems come from unitary representations of $π_1(Σ_{g,n})$. If $g = 0$ we show that they do not always come from unitary representations, and we study their structure in general.

math.AG↗
arXiv2026

Hypergeometric local systems and parabolic bundles

Beukers and Heckman gave necessary and sufficient conditions for a hypergeometric function $_n F_{n-1}$ to be algebraic. We give a new proof of this theorem by passing through the Mehta-Seshadri correspondence. In particular, we explicitly write down the parabolic bundle corresponding to a unitary hypergeometric local system.

math.AG↗