The pro-unipotent radical of the pro-algebraic fundamental group of a compact Kähler manifold (Q2475507)

The pro-algebraic completion \(\varpi_1(X)\) of the fundamental group \(\pi_1(X)\) of a compact connected Kähler manifold \(X\) has a Levi decomposition \[ \varpi_1(X)=R_u\big(\varpi_1(X)\big)\rtimes \varpi_1^{\text{ red}}(X). \] Simpson defined a Hodge structure on \(\varpi_1(X)_{\mathbb C}\) and used it to gain some control over \(\varpi_1^{ \text{ red}}(X)\). In this paper the author uses the Hodge theory to control \(R_u\big(\varpi_1(X)\big)\). The result is that \(R_u\big(\varpi_1(X)\big)\) is quadratically presented as a pro-unipotent group, meaning that its Lie algebra is defined by equations of bracket length~\(2\). This gives some new examples of groups that cannot be Kähler groups. The idea is that the results of \textit{W. M. Goldman} and \textit{J. J. Millson} [Bull. Am. Math. Soc., New Ser. 18, No.~2, 153--158 (1988; Zbl 0663.53056)] and of \textit{P. Deligne, P. Griffiths, J. Morgan} and \textit{D. Sullivan} [Invent. Math. 29, 245--274 (1975; Zbl 0312.55011] on the De Rham fundamental group both arise by deforming a group representation \(\rho_0\colon \pi_1(X)\to G\) to give \(\rho\colon \pi_1(X)\to U\rtimes G\). Here we take \(U=R_u\varpi_1(X)\). The theory of these deformations is developed in \S{2}, after a convenient summary in \S{1} of the main facts about pro-algebraic groups. \S{3} is devoted to twisted differential graded algebras, analogous to the DGAs used by Deligne et al [loc. cit] in the case of trivial~\(G\). The author points out that his twisted DGAs are equivalent to the \(G\)-equivariant DGAs of \textit{L. Katzarkov, T. Pantev} and \textit{B. Toën} [Schematic homotopy types and non-abelian Hodge theory. Preprint 2001, \url{arXiv: math.AG/0107129}]. \S{4} collects some further facts about pro-algebraic groups and these are used in \S{5}, in which it is shown that the twisted DGA of \(C^\infty\)-sections (sections of the sheaf of real \(C^\infty\) forms on \(X\) twisted by the local system corresponding to a real semisimple representation of \(\pi_1(X)\)) is enough to recover \(R_u\varpi_1(X)\). The main result is proved in \S{6}: it comes, as in Deligne et al [loc. cit.] from the fact that the twisted DGA is formal, i.e.\ quasi-isomorphic to its cohomology DGA.