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Title: Stabilization of the cohomology of thickenings
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.
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American journal of mathematics
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National Science Foundation
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