Quick question: Second chern class of a tensor product of vector bundles

Question

Let $G$ and $F$ be vector bundles of rank $r$ and $3$ respectively over a complex projective variety $X$. Here is what I find for the second Chern class:

$c_2(G\otimes F)=3(c_1(G)^2+c_2(G))+(3r-1)c_1(G)c_1(F)+rc_2(F)+{{r}\choose 2}c_1(F)^2$

Is this correct? I really need your reply because I'm in the middle of some calculation. Thank you very much.

Answer 1

First we have, for tensoring with a line bundle,

$$ c(G \otimes L) = 1 + x(c_1(G) + r c_1(L)) + x^2(c_2(G) + (r-1)c_1(G)c_1(L) + {r \choose 2}c_1(L)^2) + \cdots. $$

Now with the splitting principle, we pretend $F = L \oplus M \oplus N$, multiply out and take the $x^2$ term.

Summing the three separate $x^2$ terms gives $3c_2(G) + (r-1)c_1(G)c_1(F) + {r \choose 2}(c_1(F)^2 - 2c_2(F))$.

Summing the pairwise products of the $x$ terms gives $3c_1(G)^2 + 2r c_1(G) c_1(F) + r^2 c_2(F)$.

So my total is:

$$3(c_2(G) + c_1(G)^2) + (3r-1)c_1(G)c_1(F) + r c_2(F) + {r \choose 2}c_1(F)^2.$$

This seems to match with yours!