在复代数曲线上的代数簇纤维化中，我们有 Fujita 半正定性定理。这一定理断言了这一类纤维化中相对典范丛正向层是一个具有相交理论等意义上半正定性的向量丛。这一定理被许多学者广泛地运用到复代数几何的研究中。尤其在代数曲面纤维化理论中，这一定理是基础性的。然而，这一定理在正特征情形的类比却并不成立。至今，我们对其失效的本质原因仍不清楚。该项目的目标就是研究正特征情形下，Fujita 半正定性定理失效的原因和成立的条件。同时，我们还将研究与之相关的其他问题，即一般型代数曲面结构层欧拉特征的正性以及 Miyaoka-丘成桐类型不等式在正特征情形下可能的类比。.我们希望通过对正特征曲面纤维化中这些问题的研究，产生一些新的研究方法。同时，希望这些方法能够帮助我们进一步去理解特征-p 的几何，以及促进复几何和数论相关问题的进一步发展。

Over the field of complex numbers, we have Fujita semipositivity theorem for algebraic varieties fibered over a curve. The theorem asserts the direct image of the relative canonical bundle of any such fibration is semipositive in the sense of intersection theory and some others . This theorem is widely applied in complex algebraic geometry and is especially fundamental to the theory of surface fibrations. In spite of its importance, the analogue of Fujita's theorem is known to fail in positive characteristics, for which the essential reason is not yet clear to us. This project aims to study explicitly how and when the Fujita's semipositive theorem could fail in positive characteristics. At the same time, we shall also study some closely related topics, the positivity of Euler characteristic of the structure sheaf of a surface of general type and the possible analogues of Miyaoka-Yau inequality in positive characteristics..We wish to create new methods in this project and hope these methods could in return give us a better understanding of characteristic-p geometry and contribute substantially to the study of related topics in complex geometry and number theory.