Strictly nef vector bundles and characterizations of ℙ n

In this note, we give a brief exposition on the differences and similarities between strictly nef and ample vector bundles, with particular focus on the circle of problems surrounding the geometry of projective manifolds with strictly nef bundles.

Chern Class Inequalities on Polarized Manifolds and Nef Vector Bundles

This article is concerned with Chern class and Chern number inequalities on polarized manifolds and nef vector bundles. For a polarized pair $(M,L)$ with $L$ very ample, our 1st main result is a family of sharp Chern class inequalities. Among them the 1st one is a variant of a classical result and the equality case of the 2nd one is a characterization of hypersurfaces. The 2nd main result is a Chern number inequality on it, which includes a reverse Miyaoka–Yau-type inequality. The 3rd main result is that the Chern numbers of a nef vector bundle over a compact Kähler manifold are bounded below by the Euler number. As an application, we classify compact Kähler manifolds with nonnegative bisectional curvature whose Chern numbers are all positive. A conjecture related to the Euler number of compact Kähler manifolds with nonpositive bisectional curvature is proposed, which can be regarded as a complex analogue to the Hopf conjecture.

Nef vector bundles on a quadric surface with the first Chern class (2, 1)

We classify nef vector bundles on a smooth quadric surface with the first Chern class (2, 1) over an algebraically closed field of characteristic zero; we see in particular that such nef bundles are globally generated.