Author:

Abstract:

In rational homotopy theory, varieties are encoded by their algebraic models thanks to the work of Sullivan, Morgan, and Hain among others. Although often hard to detect, these algebraic objects are found to be formal in certain cases, that is, there is an equivalence between the algebraic object and its chain cohomology. Formality of an algebraic model has deep consequences in the topology of the associated space; for example, the rational homotopy type of a formal space is completely determined by its cohomology ring. In terms of deformation theory this implies that the deformation functors simplify considerably. In the first part of this thesis we provide formality criteria for homotopy $\cP$-algebras, which are a very broad class of algebraic objects. This unifies and generalizes previous results by Kaledin and Manetti. In particular, we construct operadic Kaledin classes and show that they are obstructions to formality. Moreover, we prove that degeneration at the $E_2$ page of the operadic cohomology spectral sequences implies formality. The second part of the thesis is concerned with deformation theory. Following Budur and Wang's theory of deformation, we develop a new deformation theory with cohomology constraints to accommodate $\Linf$ pairs, which are homotopy stable algebraic objects. As an application we prove a structure theorem for possibly singular varieties; in particular, we show that for complex algebraic varieties with vanishing weight-zero $1$-cohomology, the irreducible components of the cohomology jump loci of rank one local systems containing the constant sheaf are tori. This entails restrictions on the fundamental groups.