Secondary Kodaira-Spencer classes and nonabelian Dolbeault cohomology

Abstract

If $X$ is a smooth projective variety moving in a family, we define a secondary Kodaira-Spencer class for nonabelian Dolbeault cohomology $Hom(X_{Dol}, T)$ of $X$ with coefficients in the complexified 2-sphere $T=S^2\otimes \cc$ (which is a 3-stack on $Sch /\cc$). Let $Z$ be a simply connected projective surface with $h^{2,0}\neq 0$, 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.