ch (i, F)Given a locally-free sheaf $E$ of rank $r$ on a smooth variety such that its Chern class formally factor as chern $E = \prod_{j=1}^r (1 + \alpha_j)$, we define its Chern character to be ch $E := \sum_{j=1}^r exp(\alpha_j)$. The $i$-th graded piece of this power series is symmetric in the $\alpha_j$ and, hence, expressible as a polynomial in the elementary symmetric polynomials in the $\alpha_j$; we set ch $(i, E)$ to be this polynomial. Because the Chern character is additive on exact sequences of vector bundles and every coherent sheaf can be resolved by locally-free sheaves, we can extend this definition to all coherent sheaves.
The first few components of the Chern character are easily related to other invariants.
|
|
|
|
|
|
|
On a complete smooth normal toric variety, the Chern class of the cotangent bundle factors as a product over the irreducible torus-invariant divisors, so we can express the Chern character as a sum.
|
|
|
|
|
|