In algebraic geometry, Le Potier's vanishing theorem is an extension of the Kodaira vanishing theorem, on vector bundles.
The theorem states the following[1][2][3][4][5][6][7][8][9] Le Potier (1975): Let X be a n-dimensional compact complex manifold and E a holomorphic vector bundle of rank r over X, here
denotes the sheaf of holomorphic p-forms on X.
If E is an ample, then from Dolbeault theorem, By Serre duality, the statements are equivalent to the assertions: In case of r = 1, and let E is an ample (or positive) line bundle on X, this theorem is equivalent to the Nakano vanishing theorem.
Sommese (1978) generalizes Le Potier's vanishing theorem to k-ample and the statement as follows:[2] Le Potier–Sommese vanishing theorem: Let X be a n-dimensional algebraic manifold and E is a k-ample holomorphic vector bundle of rank r over X, then Demailly (1988) gave a counterexample, which is as follows:[1][10] Conjecture of Sommese (1978): Let X be a n-dimensional compact complex manifold and E a holomorphic vector bundle of rank r over X.