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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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  • $\begingroup$ is the map $H^*(X, \mathbb{C})\to C^*(X, \mathbb{C})$ given by taking harmonic representatives? $\endgroup$
    – Nguyen
    Sep 9, 2020 at 14:18
  • $\begingroup$ @ranicky it is. $\endgroup$ Sep 9, 2020 at 15:23
  • $\begingroup$ Sullivan constructed a rational DGA via polynomial forms with respect to a triangulation (c.f. Griffiths, Morgan, Rational homotopy theory and differential forms). Would that work for you? $\endgroup$ Sep 9, 2020 at 16:10

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