Study of vector bundles on algebraic varieties

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Lange-Pauly proved that the direct images of line bundles on non-singular projective curve with genus greater than or equal 2 are semi-stable. We extended this result to semi-stable vector bundles on non-singular curves and non-singular projective surfaces. As a corollary, we get a new result on geography of non-singular minimal surfaces of general type in positive characteristics. Further, We obtained a new theorem for rank two vector bundles on 4-dimensional projective space to split into line bundles.