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

TL;DR

This paper proves that the local structure of the representation variety of the fundamental group of certain Sasakian manifolds is quadratic, using almost-formality of the de Rham complex and establishing related vanishing theorems.

Contribution

It introduces the concept of almost-formality for Sasakian manifolds' de Rham complexes and applies it to describe the local structure of representation varieties.

Findings

Representation variety germs are quadratic for n ≥ 2.

De Rham complex of Sasakian manifolds is almost-formal.

Vanishing theorem for cup products in cohomology.

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.