A note on positivity of Einstein bundles (Q1190343)

The following result is obtained. Let \(X\) be a compact Kähler surface and \(E\) be a Hermitian bundle of rank 2 on \(X\) such that the first Chern form \(c_ 1(E)\) is equal to the Kähler form \(\omega\) up to a positive constant. Suppose that the first and second Chern forms satisfy \(c^ 2_ 1(E)-2c_ 2(E)=\varphi\centerdot\omega^ 2\), where \(\varphi\) is a positive function. Then \(E\) is positive. This is a differential algebraic version of an algebraic ampleness criterion of M. Schneider and A. Tancredi.