# Hodge structure and rational coefficients

MathOverflow Asked by Dmitry Vaintrob on November 19, 2020

Suppose $$X$$ is a complex projective variety with a model $$X_mathbb{Q}$$ defined over the rational numbers. Then there is a rational de Rham lattice $$H^k_{dR}(X_mathbb{Q}, mathbb{Q})subset H^k(X, mathbb{C})$$ coming from the hypercohomology of the de Rham complex with rational coefficients. Hodge theory implies a formality quasiisomorphism $$H^*(X, mathbb{C})to C^*(X, mathbb{C})$$ which is compatible with multiplication, and even (see this paper of Guillen, Navarro, Pascual and Roig) functorial in a dg sense with respect to maps between projective varieties. Now I want to understand the map from the sub-lattice $$H^*_{dR}(X_{mathbb{Q}}, mathbb{Q})to C^*(X, mathbb{C}).$$ I have two questions.

1. Is there a way to characterize the forms in the image of $$H^*(X_mathbb{Q}, mathbb{Q})$$ in terms of Dolbeault forms? For example are they in some sense local linear combinations of rational holomorphic forms wedged with rational antiholomorphic forms?

2. Is there a canonical way to relate $$H^*_{dR}(X_mathbb{Q}, mathbb{Q})$$ with the hypercohomology complex $$C^*_{dR}(X_{mathbb{Q}}. mathbb{Q})$$ via a canonical chain of quasiisomorphisms? By canonical I mean at least compatible with rational closed immersions, but hopefully more generally with all maps of rational projective varieties?

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