English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Looking at the finite \'etale congruence covers $X(p)$ of a complex algebraicvariety $X$ equipped with a variation of integral polarized Hodge structureswhose period map is quasi-finite, we show that both the minimal gonality amongall curves contained in $X(p)$ and the minimal volume among all subvarieties of$X(p)$ tend to infinity with $p$. This applies for example to Shimuravarieties, moduli spaces of curves, moduli spaces of abelian varieties, modulispaces of Calabi-Yau varieties, and can be made effective in many cases. Theproof goes roughly as follows. We first prove a generalization of the Arakelovinequalities valid for any variation of Hodge structures on higher-dimensionalalgebraic varieties, which implies that the hyperbolicity of the subvarietiesof $X$ is controlled by the positivity of a single line bundle. We then show ingeneral that a big line bundle on a normal proper algebraic variety $\bar X$can be made more and more positive by going to finite covers of $\bar X$defined using level structures of a local system defined on a Zariski-denseopen subset.

Increasing hyperbolicity of varieties supporting a variation of Hodge structures with level structures