Bogomolov–Sommese vanishing theorem explained

Bogomolov–Sommese vanishing theorem should not be confused with Le Potier's vanishing theorem.

In algebraic geometry, the Bogomolov–Sommese vanishing theorem is a result related to the Kodaira–Itaka dimension. It is named after Fedor Bogomolov and Andrew Sommese. Its statement has differing versions:

This result is equivalent to the statement that:

H ^{0} \left(X,A ^{-} ⊗ \Omega ^{p} ^{X} ( log D) \right) = 0

for every complex projective snc pair

(X, D) and every invertible sheaf A \in Pic (X) with \kappa(A) > p .

Therefore, this theorem is called the vanishing theorem.

See also

• Bogomolov–Miyaoka–Yau inequality
• Vanishing theorem (disambiguation)

References

• Book: 10.1007/978-3-0348-8600-0_7 . . Differential forms and higher direct images . Lectures on Vanishing Theorems . 1992 . Esnault . Hélène . Hélène Esnault . Viehweg . Eckart . 54–64 . 978-3-7643-2822-1.
• 10.1515/crelle-2013-0031 . Bogomolov–Sommese vanishing on log canonical pairs . 2015 . Graf . Patrick . Journal für die reine und angewandte Mathematik (Crelle's Journal) . 2015 . 702 . 1210.0421 . 119627680.
• 10.1112/S0010437X09004321 . Extension theorems for differential forms and Bogomolov–Sommese vanishing on log canonical varieties . 2010 . Greb . Daniel . Kebekus . Stefan . Kovács . Sándor J. . Compositio Mathematica . 146 . 193–219 . 1474399 . 0808.3647 .
• 10.1007/s10240-011-0036-0 . Differential forms on log canonical spaces . 2011 . Greb . Daniel . Kebekus . Stefan . Kovács . Sándor J. . Peternell . Thomas . Publications Mathématiques de l'IHÉS . 114 . 87–169 . 1003.2913 . 115177340 .
• Book: Handbook of Moduli II . 2013 . International Press of Boston, Inc. . 9781571462589 . Advanced Lectures in Mathematics Volume 25 . Kebekus . Stefan . Differential forms on singular spaces, the minimal model program, and hyperbolicity of moduli stacks . 1107.4239 . 71–113.
• Book: 10.4171/114-1/14 . https://www.impan.pl/~pragacz/kebmich1.pdf . Notes on Kebekus' lectures on differential forms on singular spaces . Contributions to Algebraic Geometry . EMS Series of Congress Reports . 2012 . Michałek . Mateusz . 375–388 . 978-3-03719-114-9.

Further reading

• Holomorphic Tensors and Vector Bundles on Projective Varieties . 1979 . Bogomolov . F. A. . Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya . 42 . 6 . 1227–1287 . 10.1070/IM1979v013n03ABEH002076 . 1979IzMat..13..499B .
• Unstable vector bundles and curves on surfaces . Proceedings of the International Congress of Mathematicians. Helsinki, 1978 . 1980 . 517–524 . Bogomolov . Fedor .
• Une généralisation du théorème d'annulation de Kawamata-Viehweg . Demailly . Jean-Pierre . C. R. Acad. Sci. Paris Sér. I . 123–126 . 1989. 309. 1004954.
• Logarithmic de Rham complexes and vanishing theorems . Inventiones Mathematicae . 1986 . 86 . 161–194 . Esnault . H. . Viehweg . E. . 10.1007/BF01391499 . 1986InMat..86..161E . 123388645 .
• 10.1007/s00209-010-0758-6. Families over special base manifolds and a conjecture of Campana . 2011 . Jabbusch . Kelly . Kebekus . Stefan . Mathematische Zeitschrift . 269 . 3–4 . 847–878 . 0905.1746 . 17138847 .
• 10.1007/s00209-021-02740-8. Bogomolov–Sommese type vanishing for globally F-regular threefolds . 2021 . Kawakami . Tatsuro . Mathematische Zeitschrift . 299 . 3–4 . 1821–1835 . 1911.08240 . 215768942 .
• 10.1016/j.aim.2022.108640. Bogomolov-Sommese vanishing and liftability for surface pairs in positive characteristic . 2022 . Kawakami . Tatsuro . . 409 . 108640 . 2108.03768 . 236956885 .
• Book: Müller-Stach . Stefan J. . Global Aspects of Complex Geometry . 451–469. 10.1007/3-540-35480-8_12. Hodge Theory and Algebraic Cycles. .
• 10.1007/s00209-023-03252-3. Bogomolov–Sommese type vanishing theorem for holomorphic vector bundles equipped with positive singular Hermitian metrics . 2023 . Watanabe . Yuta . Mathematische Zeitschrift . 303 . 4 . 246823913 . 2202.06603 .
• Vanishing theorems . Journal für die Reine und Angewandte Mathematik . 1982 . 335 . 1–8 . Viehweg . Eckart . 10.1515/crll.1982.335.1.