Stabilization of the cohomology of thickenings

@article{Bhatt2019StabilizationOT,
  title={Stabilization of the cohomology of thickenings},
  author={Bhargav Bhatt and Manuel Blickle and Gennady Lyubeznik and Anurag Singh and Wenliang Zhang},
  journal={American Journal of Mathematics},
  year={2019},
  volume={141},
  pages={531 - 561}
}
Abstract: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… 

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