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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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Bott vanishing for Fano threefolds
Bott proved a strong vanishing theorem for sheaf cohomology on projective space, namely that the higher cohomology of every bundle of differential forms tensored with an ample line bundle is zero. This holds for toric varieties, but not for most other varieties. We classify the smooth Fano threefolds that satisfy Bott vanishing. There are many more than expected.
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- Award ID(s):
- 2054553
- PAR ID:
- 10519766
- Publisher / Repository:
- Springer Nature
- Date Published:
- Journal Name:
- Mathematische Zeitschrift
- Volume:
- 307
- Issue:
- 1
- ISSN:
- 0025-5874
- Page Range / eLocation ID:
- Paper No. 14, 31 pp.
- Subject(s) / Keyword(s):
- Bott vanishing vanishing theorems Fano threefold Fano variety
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s00209-024-03468-x
