Secondary Kodaira-Spencer classes and nonabelian Dolbeault cohomology
Abstract
Description
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.
75 pages. Correction-existence and functoriality of decomposition of an infinite loop stack into product of Eilenberg-MacLane stacks don't hold in general. However, what we need for the calculation is still true
75 pages. Correction-existence and functoriality of decomposition of an infinite loop stack into product of Eilenberg-MacLane stacks don't hold in general. However, what we need for the calculation is still true