Nakano vanishing theorem

{{short description|Generalizes the Kodaira vanishing theorem}}

In mathematics, specifically in the study of vector bundles over complex Kähler manifolds, the Nakano vanishing theorem, sometimes called the Akizuki–Nakano vanishing theorem, generalizes the Kodaira vanishing theorem.{{Cite journal|last=Hitchin|first=N. J.|date=1981-07-01|title=Kählerian Twistor Spaces|url=http://www2.maths.ox.ac.uk/~hitchin/hitchinlist/Hitchin%20KAHLERIAN%20TWISTOR%20SPACES%20(PLMS%201981).pdf|journal=Proceedings of the London Mathematical Society|language=en|volume=s3-43|issue=1|pages=133–150|doi=10.1112/plms/s3-43.1.133|issn=1460-244X|s2cid=121623969}}{{cite arXiv|last=Raufi|first=Hossein|date=2012-12-18|title=The Nakano vanishing theorem and a vanishing theorem of Demailly-Nadel type for holomorphic vector bundles|eprint=1212.4417|class=math.CV}}{{Cite book|url=https://books.google.com/books?id=ePf_AwAAQBAJ|title=Differential Geometry of Complex Vector Bundles|last=Kobayashi|first=Shoshichi|date=2014-07-14|publisher=Princeton University Press|isbn=9781400858682|pages=68|language=en}} Given a compact complex manifold M with a holomorphic line bundle F over M, the Nakano vanishing theorem provides a condition on when the cohomology groups $H^q(M;\; \backslash Omega^p(F))$ equal zero. Here, $\backslash Omega^p(F)$ denotes the sheaf of holomorphic (p,0)-forms taking values on F. The theorem states that, if the first Chern class of F is negative,

Alternatively, if the first Chern class of F is positive,$$H^q(M;\; \backslash Omega^p(F))\; =\; 0\; \backslash text\{\; when\; \}\; q\; +\; p\; >\; n.$$