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 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 ${\displaystyle n=2r\geq 6.}$