The hodge de rham theory of relative malcev completion1

The pro-unipotent radical of the pro-algebraic fundamental group of a compact Kähler manifold

“…Since taking U = R u ( 1 (X, x)) pro-represents -148 -this functor when G = red 1 (X, x), the quadratic presentation for U gives quadratic presentations both for the hull of [GM88] and for the Lie algebra of [DGMS75]. This also generalises Hain's results ( [Hai98]) on relative Malcev completions of variations of Hodge structure, since here we are taking relative Malcev completions of arbitrary reductive representations.…”

“…-Note that Corollary 6.3 implies the results on the fundamental group of [DGMS75], of [GM88] and of [Hai98]. The pro-unipotent completion 1 (X, x) ⊗ R studied in [DGMS75] is just the maximal quotient of R u ( 1 (X, x)) on which π 1 (X, x) acts trivially.…”

“…In [Hai98] Theorem 13.14, this is proved only for those ρ which are polarised variations of Hodge structure, and no consequences are given.…”

Weight decompositions on étale fundamental groups

“…The structure result in the smooth and proper case is much the same as those established in [Hai2] and [Pri2] for fundamental groups of compact Kähler manifolds. Likewise, [Pri1] was the analogue in finite characteristic of Goldman and Millson's results on Kähler representations ( [GM]).…”

Homotopy Theory of Higher Categories

“…A particularly useful version is when COEF F is the n-category of certain n-stacks over a site Y , and Shape(X)(F ) = Hom(Π n (X) Y , F ) where Π n (X) Y denotes the constant stack on Y with values equal to Π n (X). This leads to subjects generalizing Malcev completions and rational homotopy theory [110] [111] [112], such as the schematization of homotopy types [210] [137] [174] [169], de Rham shapes and nonabelian Hodge theory.…”