Stabilization of the cohomology of thickenings

For a local complete intersection subvariety $X=V({\cal I})$ in ${\Bbb P}^n$ 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~${\Bbb P}^n$ along $X$ can be effectively computed as the cohomology on any sufficiently high thickening~$X_t=V({\cal I}^t)$; 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.