Bounds for Chern classes of semistable vector bundles on complex projective spaces

This work concerns bounds for Chern classes of holomorphic semistable and stable vector bundles on ${\displaystyle {\mathbb{P}}^n}$. Non-negative polynomials in Chern classes are constructed for 4-vector bundles on ${\displaystyle {\mathbb{P}}^4}$ and a generalization of the presented method to r-bundles on ${\displaystyle {\mathbb{P}}^n}$ is given. At the end of this paper the construction of bundles from complete intersection is introduced to see how rough the estimates we obtain are.