TITLE: How do Chern classes behave under connected sums?
QUESTION [8 upvotes]: 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!

REPLY [2 votes]: 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 
$$K_Y = \sigma^* K_X \otimes [E] \Rightarrow c_1(Y) = \sigma^* c_1(X) - c_1([E])$$