Victor Groth: Formality of Compact Kähler Manifolds and Bigraded Notions of Formality

In this thesis, we study the notion of formality from rational homotopy theory. We give a modern treatment, using the language of commutative bidifferential bigraded algebras (cbba's), of an influential theorem of Deligne-Griffiths-Morgan-Sullivan, which states that compact Kähler manifolds are formal. We show that if \((A^{*,*}, \partial , \bar{\partial})\) is a cbba with the \(\partial\bar{\partial}\)-property, then its total commutative differential graded algebra (cdga) \((A^* , d)\) is formal. The fact that compact Kähler manifolds are formal will then follow due to the \(\partial\bar{\partial}\)-Lemma, which we prove using the necessary material from complex geometry. We also discuss a further result: if the cohomology algebra of a cdga is of complete intersection type, then it is formal. This result can be generalized to cbba's: If a cbba A satisfies the \(\partial\bar{\partial}\)-property and has cohomology algebra of bigraded complete intersection type, then A is strongly formal. As a consequence, we prove that compact homogeneous Kähler manifolds are strongly formal. Lastly, we discuss obstructions to formality and weak formality, given by Massey products and their bigraded generalization ABC-Massey products. Examples are provided, mostly of nilmanifolds, both of computational importance and as counterexamples.