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 cannot be preserved by the global monodromy. Thus a local calculation (which actually only depends on the first-order deformation of Xs) implies a global fact. In particular this global fact would hold for any family of varieties X ′ over any base S , such that the new family osculates to order 1 with the original one (say as a map from S ′ into the moduli stack of the fibers). A particularly nice aspect of this situation is that the Kodaira-Spencer map is defined on the Dolbeault cohomology H(Xs,Ω ) and in particular it is obtained involving only algebraic-geometric calculations (just a cup-product with the deformation class)—no analytic considerations are needed. The goal of this paper is to calculate an example showing a similar type of behavior with a secondary Kodaira-Spencer class coming from nonabelian cohomology with coefficients in the complexified 2-sphere T = S ⊗C. For such at T (or any other coefficient stack similar in nature) we define the nonabelian Dolbeault cohomology of X with coefficients in T , denoted Hom(XDol, T ). When X varies in a family parametrized by a base