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In this paper, we introduce the notion of abi-$\overline{\mathbb{Q}}$-structure on the tangent space at a CM point on alocally Hermitian symmetric domain. We prove that thisbi-$\overline{\mathbb{Q}}$-structure decomposes into the direct sum of$1$-dimensional bi-$\overline{\mathbb{Q}}$-subspaces, and make thisdecomposition explicit for the moduli space of abelian varieties$\mathbb{A}_g$.  We propose an Analytic Subspace Conjecture, which is the analogue of theW\"{u}stholz's Analytic Subgroup Theorem in this context. We show that thisconjecture, applied to $\mathbb{A}_g$, implies that all quadratic$\overline{\mathbb{Q}}$-relations among the holomorphic periods of CM abelianvarieties arise from elementary ones.

Bi-$\overline{\mathbb{Q}}$-structures on Hermitian symmetric spaces and quadratic relations between CM periods