Atypical Hodge Loci
Phillip Griffiths
TL;DR
The paper reframes the Noether–Lefschetz problem for variations of Hodge structure in terms of period mappings into Mumford–Tate subdomains, and shows that for VHS with level at least three, positive-dimensional Hodge loci are atypical in the sense that the actual codimension is strictly smaller than naive counts due to integrability constraints from Pfaffian differential systems. It achieves this via a detailed Lie-theoretic analysis that aligns the Hodge and root space decompositions and exploits transversality of the period map, together with an infinitesimal tangent-space argument and a Cartan-type generation lemma to force contradictions unless atypical. The work connects to Ax–Schanuel type results and uses definable $o$-minimal structures to deduce algebraicity of period images, with implications for the arithmetic of period domains and moduli spaces (e.g., Calabi–Yau families and their Yukawa couplings). It clarifies how extra Hodge tensors arising in families correspond to integrability phenomena and provides a framework to understand when Noether–Lefschetz loci exhibit excess intersection beyond naive dimension counts.
Abstract
In the recent works of a number of people there has emerged a beautiful new perspective on the arithmetic properties of Hodge structures. A central result in that development appears in a paper by Baldi, Klingler, and Ullmo. In this expository work we will explain that result and give a proof. The main conceptual step is to formulate Noether-Lefschetz loci in terms of intersections of period images with Mumford-Tate subdomains of period domains. The main technical step is to use the alignment of the Hodge and root space decompositions of the Lie algebras of the associated groups and from it to use the integrability conditions associated to a Pfaffian PDE system. These integrability conditions explain the generally present excess intersection property associated to the integral varieties of a pair of Pfaffian exterior differential systems.