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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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Displacement techniques in bounded cohomology
Abstract Several algebraic criteria, reflecting displacement properties of transformation groups, have been used in the past years to prove vanishing of bounded cohomology and stable commutator length. Recently, the authors introduced the property ofcommuting cyclic conjugates, a new displacement technique that is widely applicable and provides vanishing of the bounded cohomology in all positive degrees and all dual separable coefficients. In this note we consider the most recent along with the by now classical displacement techniques and we study implications among them as well as counterexamples.
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- Award ID(s):
- 2240136
- PAR ID:
- 10595739
- Publisher / Repository:
- Springer Nature
- Date Published:
- Journal Name:
- manuscripta mathematica
- Volume:
- 176
- Issue:
- 2
- ISSN:
- 0025-2611
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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