For a local complete intersection subvariety $X=V({mathcal I})$ in ${mathbbP}^n$ over a field of characteristic zero, we show that, in cohomologicaldegrees smaller than the codimension of the singular locus of $X$, thecohomology of vector bundles on the formal completion of ${mathbb P}^n$ along$X$ can be effectively computed as the cohomology on any sufficiently highthickening $X_t=V({mathcal I^t})$; the main ingredient here is a positivityresult for the normal bundle of $X$. Furthermore, we show that the Kodairavanishing theorem holds for all thickenings $X_t$ in the same range ofcohomological degrees; this extends the known version of Kodaira vanishing on$X$, and the main new ingredient is a version of the Kodaira-Akizuki-Nakanovanishing theorem for $X$, formulated in terms of the cotangent complex.
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机译：对于本地完整的交叉点数，$ x = v（{ mathcal i}）$ in $ { mathbbp} ^ n $超过一个特征零点，我们展示了，在CONOMOLOGYDEGREES中小于$的单数轨迹的CODIMING X $，在$ X $ x $的正式完成上的矢量包的载体捆绑层可以有效地计算在任何足够高的$ x_t = v（{ mathcal i ^ t}）$ ;这里的主要成分是$ x $的正常捆绑的正派事务。此外，我们表明Kodairavanishing定理为所有加厚$ x_t $ of Ihohomology degrology度;这将延长已知的Kodaira在$ x $上消失，而主要的新成分是Kodaira-akizuki-Nakanovananovananavanishing定理的版本，以X $，根据Citangent Complex制定。
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