On the geometric André–Oort conjecture for variations of Hodge structures

Abstract

Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝕍</m:mi> </m:math> {{\mathbb{V}}} be a polarized variation of integral Hodge structure on a smooth complex quasi-projective variety S . In this paper, we show that the union of the non-factor special subvarieties for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>S</m:mi> <m:mo>,</m:mo> <m:mi>𝕍</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(S,{\mathbb{V}})} , which are of Shimura type with dominant period maps, is a finite union of special subvarieties of S . This generalizes previous results of Clozel and Ullmo (2005) and Ullmo (2007) on the distribution of the non-factor (in particular, strongly) special subvarieties in a Shimura variety to the non-classical setting and also answers positively the geometric part of a conjecture of Klingler on the André–Oort conjecture for variations of Hodge structures.

Locations

• arXiv (Cornell University) - View - PDF
• Journal für die reine und angewandte Mathematik (Crelles Journal) - View