Le Potier's vanishing theorem

In algebraic geometry, Le Potier's vanishing theorem is an extension of the Kodaira vanishing theorem, on vector bundle. 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 ${\displaystyle H^{p,q}(X,E)}$ is Dolbeault cohomology group, where ${\displaystyle \Omega _{X}^{p}}$ denotes the sheaf of holomorphic p-forms on X. If E is an ample, then

${\displaystyle H^{p,q}(X,E)=0}$ for ${\displaystyle p+q\geq n+r}$ .

from Dolbeault theorem,

${\displaystyle H^{q}(X,\Omega _{X}^{p}\otimes E)=0}$ for ${\displaystyle p+q\geq n+r}$ .

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

${\displaystyle H^{i}(X,\Omega _{X}^{j}\otimes E^{*})=0}$ for ${\displaystyle j+i\leq n-r}$ .

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 the 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

${\displaystyle H^{p,q}(X,E)=0}$ for ${\displaystyle p+q\geq n+r+k}$ .

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

${\displaystyle H^{p,q}(X,\Lambda ^{a}E)=0}$ for ${\displaystyle p+q\geq n+r-a+1}$ is false for ${\displaystyle n=2r\geq 6.}$