Stabilization of the cohomology of thickenings

Abstract

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.

arXiv comment24 pages, comments welcome