Study of vector bundles on algebraic varieties

Lange-Pauly证明了线丛在亏格大于或等于2的非奇异射影曲线上的直像是半稳定的。我们将此结果扩展到非奇异曲线和非奇异投影曲面上的半稳定向量束。作为推论，我们得到了关于正特征的一般类型非奇异极小曲面地理学的新结果。此外，我们还得到了 4 维射影空间上的二阶向量丛分裂成线丛的新定理。

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.