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