Bogomolov–Sommese 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:

{{blockquote|Bogomolov–Sommese vanishing theorem for snc pair:{{harv|Michałek|2012}}{{harv|Greb|Kebekus|Kovács|2010}}{{harv|Esnault|Viehweg|1992|loc=Corollary 6.9}}{{harv|Kebekus|2013|loc=Theorem 2.17}} Let X be a projective manifold (smooth projective variety), D a simple normal crossing divisor (snc divisor) and $A \subseteq \Omega ^{p} _ {X} (\log D)$ an invertible subsheaf. Then the Kodaira–Itaka dimension $\kappa(A)$ is not greater than p.

}}

This result is equivalent to the statement that:{{harv|Graf|2015}}

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

for every complex projective snc pair $(X, D)$ and every invertible sheaf $A \in \mathrm{Pic}(X)$

with $\kappa(A) > p$ .

Therefore, this theorem is called the vanishing theorem.

{{blockquote|Bogomolov–Sommese vanishing theorem for lc pair:{{harv|Greb|Kebekus|Kovács|Peternell|2011|loc=Theorem 7.2}}{{harv|Kebekus|2013|loc=Corollary 4.14 }} Let (X,D) be a log canonical pair, where X is projective. If $A \subseteq\Omega ^{[p]}_{X} (\log \lfloor D \rfloor)$ is a $\mathbb{Q}$ -Cartier reflexive subsheaf of rank one,{{harv|Greb|Kebekus|Kovács|Peternell|2011|loc=Definition 2.20.}} then $\kappa(A) \leq p$ .}}

Notes

References

{{cite book |doi=10.1007/978-3-0348-8600-0_7 |chapter-url={{Google books|Nmv0BwAAQBAJ|page=58|plainurl=yes}} |chapter=Differential forms and higher direct images |title=Lectures on Vanishing Theorems |year=1992 |last1=Esnault |first1=Hélène |author-link1=Hélène Esnault |last2=Viehweg |first2=Eckart |pages=54–64 |isbn=978-3-7643-2822-1}}
{{cite journal |doi=10.1515/crelle-2013-0031 |title=Bogomolov–Sommese vanishing on log canonical pairs |year=2015 |last1=Graf |first1=Patrick |journal=Journal für die reine und angewandte Mathematik (Crelle's Journal) |volume=2015 |issue=702 |arxiv=1210.0421 |s2cid=119627680}}
{{cite journal |doi=10.1112/S0010437X09004321 |title=Extension theorems for differential forms and Bogomolov–Sommese vanishing on log canonical varieties |year=2010 |last1=Greb |first1=Daniel |last2=Kebekus |first2=Stefan |last3=Kovács |first3=Sándor J. |journal=Compositio Mathematica |volume=146 |pages=193–219 |s2cid=1474399 |arxiv=0808.3647 }}
{{cite journal |doi=10.1007/s10240-011-0036-0 |title=Differential forms on log canonical spaces |year=2011 |last1=Greb |first1=Daniel |last2=Kebekus |first2=Stefan |last3=Kovács |first3=Sándor J. |last4=Peternell |first4=Thomas |journal=Publications Mathématiques de l'IHÉS |volume=114 |pages=87–169 |arxiv=1003.2913 |s2cid=115177340 |url=http://www.numdam.org/item/10.1007/s10240-011-0036-0.pdf}}
{{cite book |title=Handbook of Moduli II |year=2013 |publisher=International Press of Boston, Inc. |isbn=9781571462589 |series=Advanced Lectures in Mathematics Volume 25 |last1=Kebekus |first1=Stefan |chapter=Differential forms on singular spaces, the minimal model program, and hyperbolicity of moduli stacks |arxiv=1107.4239 |pages=71–113}}
{{cite book |doi=10.4171/114-1/14 |chapter-url=https://www.impan.pl/~pragacz/kebmich1.pdf |chapter=Notes on Kebekus' lectures on differential forms on singular spaces |title=Contributions to Algebraic Geometry |series=EMS Series of Congress Reports |year=2012 |last1=Michałek |first1=Mateusz |pages=375–388 |isbn=978-3-03719-114-9}}

Bogomolov–Sommese vanishing theorem

See also

Notes

References

Further reading