Positivity of Schur forms for Griffiths positive vector bundles of rank three over complex threefolds

Abstract: In this paper, we prove the positivity of the double mixed discriminant associated with a positive linear map between spaces of (3\times 3) complex matrices, thereby settling the three-dimensional case of Finski's open problem. As an application, we show that all Schur forms are weakly positive for Griffiths positive Hermitian holomorphic vector bundles of rank three over complex threefolds. This yields a complete affirmative answer, in the case where both the rank and the dimension are three, to the question posed by Griffiths in 1969.