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In algebraic geometry, Le Potier's vanishing theorem is an extension of the Kodaira vanishing theorem, on vector bundles. The theorem states the following
Le Potier (1975): Let X be a n-dimensional compact complex manifold and E a holomorphic vector bundle of rank r over X, here is Dolbeault cohomology group, where 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 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: Let X be a projective manifold, D a simple normal crossing divisor and an invertible subsheaf. Then the Kodaira–Itaka dimension is not greater than p.
References
- 1 2 3 "Bernard Shiffman, C.V." (PDF). Mathematics Department, Johns Hopkins University.
- ↑ Bernard Shiffman at the Mathematics Genealogy Project
- ↑ Lax, Peter D. (December 2003). "Max Shiffman (1914–2000)" (PDF). Notices of the AMS: 1401.
- ↑ Schneider, Michael (1987). "Review of Vanishing theorems on complex manifolds by Bernard Shiffman and Andrew John Sommese". Bull. Amer. Math. Soc. (N.S.). 17: 180–183. doi: 10.1090/S0273-0979-1987-15555-8 .
