2018/12/21 by Fang, Hanlong
#Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.1812.08968
In this paper, we give a combinatorial formula for the Čech cocycles representing the power sums of the Chern roots of a holomorphic vector bundle over a complex manifold. By an observation motivation by author's previous paper, we also construct some new holomorphic invariants refining the Chern classes. Firstly, we define the refined first T invariants for all holomorphic vector bundles (or \mathcal Q-flat classes in the line bundle case) and give a criterion for determining whether a manifold has a line bundle whose \mathcal Q-flat class is strictly finer than its first Chern class in the Dolbeault cohomology. Then, we define the refined higher T invariants for holomorphic vector bundles with a full flag structure. At last, we generalize the notion of the T invariants (or equivalently the Chern classes) and the refined T invariants for the locally free sheaves of schemes over general fields.