The Chern classes of a complex vector bundle are characteristic classes living in the cohomology of the base. They are natural under pullback, so , and additive on short exact sequences of bundles via the Whitney sum formula where is the total Chern class.
A line bundle is a complex vector bundle of rank one. The first Chern class is the sole nontrivial Chern class of a line bundle and classifies it up to isomorphism.
For a complex manifold one sets , the first Chern class of the holomorphic tangent bundle.
The key fact used elsewhere is , where is the canonical bundle. Hence is equivalent to being trivial, which is the defining condition of a Calabi–Yau manifold.
For a chosen Hermitian metric, is represented in de Rham cohomology by the curvature of the induced connection, and on a Kähler manifold this representative is proportional to the Ricci form.