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
an invertible subsheaf.
is not greater than p. This result is equivalent to the statement that:[5] for every complex projective snc pair
and every invertible sheaf
Bogomolov–Sommese vanishing theorem for lc pair:[6][7] Let (X,D) be a log canonical pair, where X is projective.
-Cartier reflexive subsheaf of rank one,[8] then