Mixed Hodge structure conjecture for the Sullivan differential group of a Kähler manifold

Let $M$ be a simply connected compact Kähler manifold. Let $G_M$ be the group in Theorem~, let $D_M\to G_M$ be the homomorphism from the differential graded automorphism group, and let

$$G_M\to \operatorname{Aut}\pi_{\bullet}(M,x_o)_{\mathbb Q}$$

be the natural homomorphism. Write ${\mathfrak u}_M$ for the Lie algebra of the unipotent radical of $M$.

Mixed Hodge structure conjecture. The coordinate ring and Lie algebra of $G_M$ should have natural mixed Hodge structures such that the homomorphisms

$$D_M\to G_M\qquad\text{and}\qquad G_M\to \operatorname{Aut}\pi_{\bullet}(M,x_o)_{\mathbb Q}$$

are morphisms of mixed Hodge structures. Equivalently, the induced map on coordinate rings should be a morphism of mixed Hodge structures. When $M$ is algebraic, these structures should form admissible variations of mixed Hodge structures over the moduli space ${\mathscr M}$ parameterizing algebraic structures on the manifold underlying $M$. The weights on ${\mathfrak u}_M$ should be strictly negative.

This conjecture proposes that the algebraic groups controlling the rational homotopy automorphisms of a simply connected compact Kähler manifold carry Hodge-theoretic structures compatible with the natural homomorphisms. The algebraic case further predicts a variation over the moduli space, while the strict negativity condition reflects the expected behavior of the unipotent part.