Stabilization of the cohomology of thickenings
journal contributionposted on 2022-04-15, 15:49 authored by Bhargav Bhatt, Manuel Blickle, Gennady Lyubeznik, Anurag K Singh, Wenliang ZhangWenliang Zhang
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 Xt = V (It); 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 Xt 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.
CitationBhatt, B., Blickle, M., Lyubeznik, G., Singh, A. K.Zhang, W. (2019). Stabilization of the cohomology of thickenings. American Journal of Mathematics, 141(2), 531-561. https://doi.org/10.1353/ajm.2019.0013