1997/12/18 by Carlos Simpson, Simpson, Carlos
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #alg-geom #math.AG
paper · pdf · doi:10.48550/arxiv.alg-geom/9712020
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
openalex publication_date 1997/12/18 · arxiv created 1998/01/06 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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⊗ \cc (which is a 3-stack on Sch /\cc). 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.