Let \(X\) be a scheme and \(E\) a vector bundle over \(X\) and denote by \(s_i(E)\) the \(i\)th Segre class of \(E\). Consider the formal power series \[s_t(E) := \sum_{i =0}^\infty s_i(E)t^i = 1 + s_1(E)t + s_2(E)t^2 + ...\] and note that it is invertible since it has nonzero constant term. Define the Chern polynomial to be the inverse of this power series: \[c_t(E) = \sum_{i=0}^\infty c_i(E)t^t = 1 + c_1(E)t + c_2(E)t^2 + ... := s_t(E)^{-1},\] so called because it will turn out to be a polynomial. The first few terms are given
- \(c_0(E) = 1\)
- \(c_1(E) = - s_1(E)\)
- \(c_2(E) = s_1(E)^2 - s_2(E)\),
and the \(n\)th term is
- \(c_n(E) = -c_{n-1}(E)s_{1}(E) - c_{n-2}(E)s_{2}(E) - ... - s_n(E)\).
Here the \(s_i(E)\) are regarded as endomorphisms of \(A_*(X)\), and since they all commute, there is no ambiguity. We
Definition
We call \(c_i(E)\) the \(i\)th Chern class of \(E\) over \(X\) and define \[c(E) = 1 + c_1(E) + ... + c_r(E)\] to be the total Chern class of \(E\), where \(r = \rank E\). We write \(c_i(E) \cap \alpha\) to mean “\(c_i(E)\) applied to \(\alpha\)”, and thus \[c(E) \cap \alpha = \sum_{i=0}^r c_i(E)\cap \alpha.\]