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Bogomolov–Sommese vanishing theorem

Theorem in algebraic geometry From Wikipedia, the free encyclopedia

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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. Then the Kodaira–Itaka dimension is not greater than p.

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

for every complex projective snc pair and every invertible sheaf with .

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 is a -Cartier reflexive subsheaf of rank one,[8] then .

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