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We obtain restrictions on the rational homotopy types of mapping spaces andof classifying spaces of homotopy automorphisms by means of the theory ofpositive weight decompositions. The theory applies, in particular, to connectedcomponents of holomorphic maps between compact K\"ahler manifolds as well ashomotopy automorphisms of K\"ahler manifolds.

Weight decompositions on algebraic models for mapping spaces and homotopy automorphisms