Ragnar-Olaf Buchweitz

Abstract:   In this joint work with H.Flenner (Bochum, Germany), we construct a general semiregularity map for algebraic cycles on complex spaces as asked for by S. Bloch in 1972. Existence of such a map was known before for curves on algebraic surfaces (Severi 1947), for divisors (Kodaira-Spencer 1959) and then locally complete intersections (Bloch 1972) on projective manifolds.

Our construction of the semiregularity map involves powers of the cotangent complex, Atiyah classes, and trace maps, but in contrast to the classically known cases avoids duality theory. The result extends not only to subspaces of manifolds but to perfect complexes on arbitrary complex spaces.

Existence of a semiregularity map has well known consequences for the structure of the Hilbert scheme and for the variational Hodge conjecture. We also give new applications to deformations of vector bundles or coherent sheaves that encompass, for example, results of Artamkin and Mukai.