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Le Potier's vanishing theorem

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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[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 is Dolbeault cohomology group, where denotes the sheaf of holomorphic p-forms on X. If E is an ample, then

for .

from Dolbeault theorem,

for .

By Serre duality, the statements are equivalent to the assertions:

for .

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. Also, Schneider (1974) found another proof.

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

for .

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. If E is an ample, then

for is false for

See also

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Note

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References

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Further reading

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  • Schneider, Michael; Zintl, Jörg (1993). "The theorem of Barth-Lefschetz as a consequence of le Potier's vanishing theorem". Manuscripta Mathematica. 80: 259–263. doi:10.1007/BF03026551. S2CID 119887533.
  • Huang, Chunle; Liu, Kefeng; Wan, Xueyuan; Yang, Xiaokui (2022). "Vanishing Theorems for Sheaves of Logarithmic Differential Forms on Compact Kähler Manifolds". International Mathematics Research Notices. doi:10.1093/imrn/rnac204.
  • Bădescu, Lucian; Repetto, Flavia (2009). "A Barth–Lefschetz Theorem for Submanifolds of a Product of Projective Spaces". International Journal of Mathematics. 20: 77–96. arXiv:math/0701376. doi:10.1142/S0129167X09005182. S2CID 10539504.
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