Weight decompositions on algebraic models for mapping spaces and homotopy automorphisms (Q6591064)

Positive weight decompositions on general commutative differential graded algebras were first considered by \textit{R. Body} and \textit{R. Douglas} [Pac. J. Math. 75, 331--338 (1978; Zbl 0403.55017); ibid. 90, 21--26 (1980; Zbl 0421.55007)]. They showed that such decompositions lead to unique factorizations of rational homotopy types. In this paper, the authors use weight-graded basic results in homotopy theory to better understand the homotopical consequences of having positive weights. Assume a differential graded algebra has a positive weight decomposition and finite-dimensional cohomology \(H\). The authors prove that there is a transferred structure on \(H\) and some finite number \(k\geq 3\) with the property that the higher \(A_\infty\)-operations \(H^{\otimes n}\to H\) vanish for \(n\geq k\). Similarly, they show that the \(n\)-fold Massey products on \(H\) are all trivial for sufficiently large \(n\).\N\NPositive weight decompositions can be defined for topological spaces via their algebra of cochains. One can show that a space that is either formal or coformal has positive weights. The authors' goal is to study the behaviour of weights on the rational homotopy types of mapping spaces and homotopy automorphisms. Let \(X\) and \(Y\) be compact Kähler manifolds, and assume \(Y\) is nilpotent. Assume \(f\colon X\to Y\) is holomorphic. The authors prove that the connected component of the mapping space \(\mathrm{map}(X,Y)\) containing \(f\) has a positive weight decomposition.