I often see people talking about Chern classes of manifolds like $\mathbb{CP}^2 \# \mathbb{CP}^2$, or $\mathbb{CP}^2\#\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$, where $\#$ denotes connected sum and the bar denotes the opposite orientation.
I know what Chern classes are, and I know that if a complex vector bundle $E$ splits as $E_1 \oplus E_2$, then $$c_k(E) = \sum_{i + j = k} c_i(E_1)\wedge c_j(E_2).$$ However, I've never seen a formula relating the Chern classes of (the tangent bundle of) a complex manifold $M \# N$ and those of $M$ and $N$. So my questions are:
1) Does such a formula exist, and where can I read about it?
2) How does one compute the Chern classes of connected sums of $\mathbb{CP}^2$ and $\overline{\mathbb{CP}^2}$?
For the time being, I'm actually more interested in 2), but any information is very welcome. Thanks!
For question 2, one can find the following in Daniel Huybrechts' complex geometry (p.98 - 102).
The connected sum $Y :=\mathbb {CP}^2 \# \overline{\mathbb{CP}^2}$ is diffeomorphic to the blowup of $X:=\mathbb {CP}^2$ at one point. Call $E$ the expectional divisor and $\sigma : Y\to X$ the blowup map, then
$$K_Y = \sigma^* K_X \otimes [E] \Rightarrow c_1(Y) = \sigma^* c_1(X) - c_1([E])$$