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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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This content will become publicly available on July 24, 2026
A uniform identification of stable sheaf cohomology
This paper considers generalizations of certain arithmetic complexes appearing in the work of Raicu and VandeBogert in connection with the study of stable sheaf cohomology on flag varieties. Defined over the ring of integer valued polynomials, we prove an isomorphism of these complexes as conjectured by Gao, Raicu, and VandeBogert. In particular, this shows that a previously made identification between the stable sheaf cohomology of hook and two column partition Schur functors applied to the cotangent sheaf of projective space can be made to be uniform with respect to these complexes. These results are extended to the projective space defined over the integers.
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- Award ID(s):
- 2101225
- PAR ID:
- 10632117
- Publisher / Repository:
- American Mathematical Society
- Date Published:
- Journal Name:
- Proceedings of the American Mathematical Society
- ISSN:
- 0002-9939
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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