Secondary Kodaira-Spencer classes and nonabelian Dolbeault cohomology

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If X is a smooth projective variety moving in a family, we define a secondary Kodaira-Spencer class for nonabelian Dolbeault cohomology Hom(XDol, T) of X with coefficients in the complexified 2-sphere T=S²⊗ (which is a 3-stack on Sch /). Let Z be a simply connected projective surface with h²,⁰≠ 0, and let X be the blow-up of Z at a point P. As P moves in Z, the blow-up X moves in a family and we show that the secondary Kodaira-Spencer class is nontrivial. This contrasts with the fact that the variations of mixed Hodge structures on the homotopy groups of X are constant. We discuss various surrounding notions, including two appendices where we give some details about the Breen calculations in characteristic zero and representability of simply connected complex shapes.