Le Potier's vanishing theorem

In algebraic geometry, Le Potier's vanishing theorem is an extension of the Kodaira vanishing theorem, on vector bundles. The theorem states the following ": Let X be a n-dimensional compact complex manifold and E a holomorphic vector bundle of rank r over X, here $H^{p,q}(X,E)$ is Dolbeault cohomology group, where $\Omega ^{p}_{X}$ denotes the sheaf of holomorphic p-forms on X. If E is an ample, then
 * $ H^{p,q}(X, E) = 0$ for $p + q \geq n + r$.

from Dolbeault theorem,


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

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


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

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, found another proof.

generalizes Le Potier's vanishing theorem to k-ample and the statement as follows:

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

gave a counterexample, which is as follows:

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