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:

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 ${\displaystyle \kappa (A)\leq p}$.