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In this article, we give a necessary and sufficient condition for ampleness of semistable vector bundles with vanishing discriminant on a smooth projective variety X . As an application, we show ampleness of some special vector bundles on certain ruled surfaces. We prove similar results for parabolic ampleness.

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We investigate the relative logarithmic connections on a holomorphic vector bundle over a complex analytic family. We give a sufficient condition for the existence of a relative logarithmic connection on a holomorphic vector bundle singular over a relative simple normal crossing divisor. We define the relative residue of relative logarithmic connection and express relative Chern classes of a holomorphic vector bundle in terms of relative residues.

Serrano's conjecture asserts that if D is a strictly nef divisor on a projective variety X of dimension n, then K X + tD is ample for t > n + 1. In this article, we investigate this conjecture for strictly nef divisors on projective bundles over higher dimensional smooth projective varieties.