Construct holomorphic invariants in \v{C}ech cohomology by a combinatorial formula

In this paper, we give a combinatorial formula for the \v{C}ech 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.