Bogomolov–Sommese vanishing theorem

Not to 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:

Bogomolov–Sommese vanishing theorem for snc pair:^[1][2][3][4] Let X be a projective manifold (smooth projective variety), D a simple normal crossing divisor (snc divisor) and ${\displaystyle A\subseteq \Omega _{X}^{p}(\log D)}$ an invertible subsheaf. Then the Kodaira–Itaka dimension ${\displaystyle \kappa (A)}$ is not greater than p.

This result is equivalent to the statement that:^[5]

${\displaystyle H^{0}\left(X,A^{-1}\otimes \Omega _{X}^{p}(\log D)\right)=0}$

for every complex projective snc pair ${\displaystyle (X,D)}$ and every invertible sheaf ${\displaystyle A\in \mathrm {Pic} (X)}$ with ${\displaystyle \kappa (A)>p}$.

Therefore, this theorem is called the vanishing theorem.

Bogomolov–Sommese vanishing theorem for lc pair:^[6][7] Let (X,D) be a log canonical pair, where X is projective. If ${\displaystyle A\subseteq \Omega _{X}^{[p]}(\log \lfloor D\rfloor )}$ is a ${\displaystyle \mathbb {Q} }$-Cartier reflexive subsheaf of rank one,^[8] then ${\displaystyle \kappa (A)\leq p}$.

See also

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

Notes

1. (Michałek 2012)
2. (Greb, Kebekus & Kovács 2010)
3. (Esnault & Viehweg 1992, Corollary 6.9)
4. (Kebekus 2013, Theorem 2.17)
5. (Graf 2015)
6. (Greb et al. 2011, Theorem 7.2)
7. (Kebekus 2013, Corollary 4.14)
8. (Greb et al. 2011, Definition 2.20.)

References

• Esnault, Hélène; Viehweg, Eckart (1992). "Differential forms and higher direct images". Lectures on Vanishing Theorems. pp. 54–64. doi:10.1007/978-3-0348-8600-0_7. ISBN 978-3-7643-2822-1.
• Graf, Patrick (2015). "Bogomolov–Sommese vanishing on log canonical pairs". Journal für die reine und angewandte Mathematik (Crelle's Journal). 2015 (702). arXiv:1210.0421. doi:10.1515/crelle-2013-0031. S2CID 119627680.
• Greb, Daniel; Kebekus, Stefan; Kovács, Sándor J. (2010). "Extension theorems for differential forms and Bogomolov–Sommese vanishing on log canonical varieties". Compositio Mathematica. 146: 193–219. arXiv:0808.3647. doi:10.1112/S0010437X09004321. S2CID 1474399.
• Greb, Daniel; Kebekus, Stefan; Kovács, Sándor J.; Peternell, Thomas (2011). "Differential forms on log canonical spaces" (PDF). Publications Mathématiques de l'IHÉS. 114: 87–169. arXiv:1003.2913. doi:10.1007/s10240-011-0036-0. S2CID 115177340.
• Kebekus, Stefan (2013). "Differential forms on singular spaces, the minimal model program, and hyperbolicity of moduli stacks". Handbook of Moduli II. Advanced Lectures in Mathematics Volume 25. International Press of Boston, Inc. pp. 71–113. arXiv:1107.4239. ISBN 9781571462589.
• Michałek, Mateusz (2012). "Notes on Kebekus' lectures on differential forms on singular spaces" (PDF). Contributions to Algebraic Geometry. EMS Series of Congress Reports. pp. 375–388. doi:10.4171/114-1/14. ISBN 978-3-03719-114-9.

Further reading

• Bogomolov, F. A. (1979). "Holomorphic Tensors and Vector Bundles on Projective Varieties". Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya. 42 (6): 1227–1287. Bibcode:1979IzMat..13..499B. doi:10.1070/IM1979v013n03ABEH002076.
• Bogomolov, Fedor (1980). "Unstable vector bundles and curves on surfaces" (PDF). Proceedings of the International Congress of Mathematicians. Helsinki, 1978: 517–524.
• Demailly, Jean-Pierre (1989). "Une généralisation du théorème d'annulation de Kawamata-Viehweg". C. R. Acad. Sci. Paris Sér. I. 309: 123–126. MR 1004954.
• Esnault, H.; Viehweg, E. (1986). "Logarithmic de Rham complexes and vanishing theorems". Inventiones Mathematicae. 86: 161–194. Bibcode:1986InMat..86..161E. doi:10.1007/BF01391499. S2CID 123388645.
• Jabbusch, Kelly; Kebekus, Stefan (2011). "Families over special base manifolds and a conjecture of Campana". Mathematische Zeitschrift. 269 (3–4): 847–878. arXiv:0905.1746. doi:10.1007/s00209-010-0758-6. S2CID 17138847.
• Kawakami, Tatsuro (2021). "Bogomolov–Sommese type vanishing for globally F-regular threefolds". Mathematische Zeitschrift. 299 (3–4): 1821–1835. arXiv:1911.08240. doi:10.1007/s00209-021-02740-8. S2CID 215768942.
• Kawakami, Tatsuro (2022). "Bogomolov-Sommese vanishing and liftability for surface pairs in positive characteristic". Advances in Mathematics. 409: 108640. arXiv:2108.03768. doi:10.1016/j.aim.2022.108640. S2CID 236956885.
• Müller-Stach, Stefan J. "Hodge Theory and Algebraic Cycles". Global Aspects of Complex Geometry. pp. 451–469. doi:10.1007/3-540-35480-8_12.
• Watanabe, Yuta (2023). "Bogomolov–Sommese type vanishing theorem for holomorphic vector bundles equipped with positive singular Hermitian metrics". Mathematische Zeitschrift. 303 (4). arXiv:2202.06603. doi:10.1007/s00209-023-03252-3. S2CID 246823913.
• Viehweg, Eckart (1982). "Vanishing theorems". Journal für die Reine und Angewandte Mathematik. 335: 1–8. doi:10.1515/crll.1982.335.1.