Almost-formality and deformations of representations of the fundamental groups of Sasakian manifolds

Abstract

For a \(2n+1\)-dimensional compact Sasakian manifold, if \(n\ge 2\), we prove that the analytic germ of the variety of representations of the fundamental group at every semi-simple representation is quadratic. To prove this result, we prove the almost-formality of de Rham complex of a Sasakian manifold with values in a semi-simple flat vector bundle. By the almost-formality, we also prove the vanishing theorem on the cup product of the cohomology of semi-simple flat vector bundles over a compact Sasakian manifold.