The Attractor Conjecture

Abstract

This thesis studies the Attractor Conjecture due to Moore, which aims to produce arithmetic Calabi-Yau varieties using the attractor mechanism studied in string theory. The first part of this thesis gives counterexamples to the Attractor Conjecture in all odd dimensions except for a few small exceptions, assuming a standard conjecture in unlikely intersection theory. Our counterexamples come from a family of Calabi-Yau varieties first studied by Dolgachev, and we use crucially a transcendence result of Shiga-Wolfart. For this family of Dolgachev varieties, the conjecture holds if and only if the moduli space is a Shimura variety. The second part of this thesis proves the Attractor Conjecture in many cases of Calabi-Yau variations of Hodge structures (CYVHS) on Shimura varieties. More precisely, we study the canonical CYVHS on Shimura varieties constructed by Gross, and prove that attractor points are CM points.