STABILIZATION OF THE COHOMOLOGY OF THICKENINGS

摘要

For a local complete intersection subvariety X = V (I) in P-n over a field of characteristic zero, we show that, in cohomological degrees smaller than the codimension of the singular locus of X, the cohomology of vector bundles on the formal completion of P-n along X can be effectively computed as the cohomology on any sufficiently high thickening X-t = V (I-t); the main ingredient here is a positivity result for the normal bundle of X. Furthermore, we show that the Kodaira vanishing theorem holds for all thickenings X-t in the same range of cohomological degrees; this extends the known version of Kodaira vanishing on X, and the main new ingredient is a version of the Kodaira-Akizuki-Nakano vanishing theorem for X, formulated in terms of the cotangent complex.