Bogomolov–Sommese vanishing theorem

Short description: Theorem in algebraic geometry

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: Let X be a projective manifold (smooth projective variety), D a simple normal crossing divisor (snc divisor) and

$A\subseteq {\mathrm{\Omega }}_X^p(\log D)$

an invertible subsheaf. Then the Kodaira–Itaka dimension

$\kappa (A)$

is not greater than p.

This result is equivalent to the statement that:

$H^0(X,A^{-1}\otimes {\mathrm{\Omega }}_X^p(\log D))=0$

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

Therefore, this theorem is called the vanishing theorem.

Bogomolov–Sommese vanishing theorem for lc pair: Let (X,D) be a log canonical pair, where X is projective. If

$A\subseteq {\mathrm{\Omega }}_X^{[p]}(\log \lfloor D\rfloor )$

is a

$\mathbb{\mathbb{Q}}$

-Cartier reflexive subsheaf of rank one, then

$\kappa (A)\le p$

.

See also

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

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. https://books.google.com/books?id=Nmv0BwAAQBAJ&pg=PA58. 
• Graf, Patrick (2015). "Bogomolov–Sommese vanishing on log canonical pairs". Journal für die reine und angewandte Mathematik (Crelle's Journal) 2015 (702): 109–142. doi:10.1515/crelle-2013-0031. 
• 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. doi:10.1112/S0010437X09004321. 
• Greb, Daniel; Kebekus, Stefan; Kovács, Sándor J.; Peternell, Thomas (2011). "Differential forms on log canonical spaces". Publications Mathématiques de l'IHÉS 114: 87–169. doi:10.1007/s10240-011-0036-0. http://www.numdam.org/item/10.1007/s10240-011-0036-0.pdf. 
• 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. ISBN 9781571462589. 
• Michałek, Mateusz (2012). "Notes on Kebekus' lectures on differential forms on singular spaces". Contributions to Algebraic Geometry. EMS Series of Congress Reports. 6. pp. 375–388. doi:10.4171/114-1/14. ISBN 978-3-03719-114-9. https://www.impan.pl/~pragacz/kebmich1.pdf. 

Further reading

• Bogomolov, F. A. (1979). "Holomorphic Tensors and Vector Bundles on Projective Varieties". Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 42 (6): 1227–1287. doi:10.1070/IM1979v013n03ABEH002076. Bibcode: 1979IzMat..13..499B. https://www.mathnet.ru/php/getFT.phtml?jrnid=im&paperid=1965&what=fullt&option_lang=eng. 
• Bogomolov, Fedor (1980). "Unstable vector bundles and curves on surfaces". Proceedings of the International Congress of Mathematicians. Helsinki, 1978: 517–524. https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1978.2/ICM1978.2.ocr.pdf. 
• Demailly, Jean-Pierre (1989). "Une généralisation du théorème d'annulation de Kawamata-Viehweg". C. R. Acad. Sci. Paris Sér. I 309: 123–126. https://gallica.bnf.fr/ark:/12148/bpt6k56752189/f132.item. 
• Esnault, H.; Viehweg, E. (1986). "Logarithmic de Rham complexes and vanishing theorems". Inventiones Mathematicae 86: 161–194. doi:10.1007/BF01391499. Bibcode: 1986InMat..86..161E. http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002103303. 
• Jabbusch, Kelly; Kebekus, Stefan (2011). "Families over special base manifolds and a conjecture of Campana". Mathematische Zeitschrift 269 (3–4): 847–878. doi:10.1007/s00209-010-0758-6. 
• Kawakami, Tatsuro (2021). "Bogomolov–Sommese type vanishing for globally F-regular threefolds". Mathematische Zeitschrift 299 (3–4): 1821–1835. doi:10.1007/s00209-021-02740-8. 
• Kawakami, Tatsuro (2022). "Bogomolov-Sommese vanishing and liftability for surface pairs in positive characteristic". Advances in Mathematics 409. doi:10.1016/j.aim.2022.108640. 
• Müller-Stach, Stefan J. (2006). "Hodge Theory and Algebraic Cycles". Global Aspects of Complex Geometry. pp. 451–469. doi:10.1007/3-540-35480-8_12. ISBN 978-3-540-35479-6. https://books.google.com/books?id=d_0ob8js1O8C&pg=466. 
• Watanabe, Yuta (2023). "Bogomolov–Sommese type vanishing theorem for holomorphic vector bundles equipped with positive singular Hermitian metrics". Mathematische Zeitschrift 303 (4). doi:10.1007/s00209-023-03252-3. 
• Viehweg, Eckart (1982). "Vanishing theorems". Journal für die Reine und Angewandte Mathematik 335: 1–8. doi:10.1515/crll.1982.335.1. http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002199688.