Nonabelian Spencer cohomology and deformation theory

Secondary Kodaira-Spencer classes and nonabelian Dol- beault cohomology

The Kodaira-Spencer map is a component of the connection ∇. In particular, this implies that if κs 6= 0 then the connection∇ is nontrivial with respect to the Hodge decomposition. Various Hodge-theory facts imply that the global monodromy must be nontrivial in this case. We can be a bit more precise: if u ∈ V p,q is a vector such that κs(v)(u) 6= 0 for some tangent vector v ∈ T (S)s, then u can...

Differential Equations, Spencer Cohomology, and Computing Resolutions

We propose a new point of view of the Spencer cohomology appearing in the formal theory of differential equations based on a dual approach via comodules. It allows us to relate the Spencer cohomology with standard constructions in homological algebra and, in particular, to express it as a Cotor. We discuss concrete methods for its construction based on homological perturbation theory. Appears i...

Nonabelian Cohomology and Obstructions, following Wojtkowiak [2]

Let D be a small category. We define a variety of cohomology objects. Each starts with a functor to a category of " coefficents, " and produces a different sort of object as output. The coefficient categories are: sets Set, groups Gp, abelian groups Ab, and " bands " HGp, that is, the category whose objects are groups and whose morphisms are conjugacy classes of homomorphisms. Let S : D → Set b...

The Hodge filtration on nonabelian cohomology

Whereas usual Hodge theory concerns mainly the usual or abelian cohomology of an algebraic variety—or eventually the rational homotopy theory or nilpotent completion of π1 which are in some sense obtained by extensions—nonabelian Hodge theory concerns the cohomology of a variety with nonabelian coefficients. Because of the basic fact that homotopy groups in higher dimensions are abelian, and si...

Spencer-Attix Cavity Theory