Chern classes and sums of line bundles

Question

Let $E$ be a complex vector bundle of rank $r$ and suppose we can write $E = \oplus_{i=1}^r L_i$ where $L_i$ are line bundles. I have read here (and think I more or less understand why) that the total chern class of $E$ can be written in this case as:

$$c(E) = \prod_{i=1}^r(1+c_1(L_i))$$

My question: is there any similar simple expression for $c_1(E)$? In particular, is it true that in this case I can write $c_1(E) = \sum_{i=1}^r c_1(L_i)$? Thanks in advance!

Answer 1

Expanding the product, we have

$$c(E) = \prod_{i=1}^r(1+c_1(L_i)) = \sum_{k=0}^re_k(c_1(L_1), \dots, c_1(L_r))$$

where $e_k$ is the $k^{\text{th}}$ elementary symmetric polynomial. Note that $\deg e_k(c_1(L_1), \dots, c_1(L_r)) = 2k$, so equating terms of the same degree, we see that $c_k(E) = e_k(c_1(L_1), \dots, c_1(L_r))$. In particular, we have

$$c_1(E) = e_1(c_1(L_1), \dots, c_1(L_r)) = c_1(L_1) + \dots + c_1(L_r).$$