Stabilization of the cohomology of thickenings

For a local complete intersection subvariety X = V (I) in ℙⁿ 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 ℙⁿ 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.