The second and third Chern classes of Calabi-Yau threefolds Let $X$ be a smooth projective Calabi-Yau threefold. Then the first Chern class vanishes: $$c_1(X)=c_1(T_X)=0.$$

Is anything known about $c_2(X)$ and $c_3(X)$? What about $c_2$ of a
  K$3$ surface?

(I am sorry if this is very well-known. This question is just a cultural curiosity: I started asking myself such questions while doing a Chern class computation on a threefold, and I realized I could not replace $c_i(X)$ by anything I knew...)
Thanks!
 A: In general the top Chern class is the Euler class of the real bundle underlying the holomorphic tangent bundle, so its degree is the Euler characteristic of the manifold.
So for $X$ a $K3$ surface, we always have $c_2(X)=24$.
Unfortunately for Calabi–Yau threefolds the Euler characteristic can vary wildly, and in fact it's not even known whether the Euler characteristics are bounded. See this MO question for some good information. According to one answer there, the smallest known value for $c_3$ of a CY threefold is -960. 
As for $c_2$ of a Calabi–Yau threefold, I find I have even less to say. Of course now it is a class in $H^4(X)$, rather than an integer; by cup-product, one can view it as a linear form on $H^2(X)$. This paper of Kanazawa and Wilson has some information on the properties of this linear form (and other things relevant to your question). For example, they quote the interesting fact (attributed to Miyaoka) that this form is nonnegative on nef divisor classes in $H^2(X)$.
