Generic Integral Manifolds for Weight Two Period Domains

TL;DR

The paper studies the Griffiths distribution on weight-two period domains $D = G/V$ with $G = SO(2p,q)$, focusing on generic integral elements and their potential to generate infinite-dimensional families of integral manifolds. It introduces a matrix-valued local model $U$ in a unipotent group with Maurer–Cartan form $\omega$ and shows that integral elements correspond to abelian subspaces, enabling a canonical construction of integral manifolds via holomorphic functions $f_2,\dots,f_p$. The key finding is that the generating functions must satisfy Hessian-commutator relations $[H_{f_i},H_{f_j}]=0$, yielding an infinite-dimensional family of solutions; in particular, $p=2$ recovers the classical contact case, while $p>2$ introduces new constraints. The work demonstrates how to realize these manifolds using quadratic generators with commuting symmetric matrices and proves existence results for prescribed initial Hessians, revealing a spectrum of behaviors depending on the parameter $q$ (wave-type versus overdetermined).

Abstract

We define the notion of a generic integral element for the Griffiths distribution on a weight two period domain, draw the analogy with the classical contact distribution, and then show how to explicitly construct an infinite-dimensional family of integral manifolds tangent to a given element.

Theorems & Definitions (7)

• Theorem 1
• Lemma 1
• Definition 1
• Proposition 1
• Proposition 2
• Proposition 3
• Theorem 2