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!
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 $H^2(X)$.
This question is quite old, but some non-trivial things can be said. The case of K3 surfaces has already been treated, so let me restrict to the case of Calabi-Yau threefolds. By the Calabi-Bogomolov-Beauville decomposition theorem, a compact Kähler manifold with holomorphically torsion canonical bundle admits a finite étale cover that is the product of a complex torus, a product of simply connected compact Kähler manifolds of complex dimension at least 3, and hyperkähler manifolds (which have complex dimension divisible by 4). Hence, if $X$ is a compact Kähler threefold with holomorphically torsion canonical bundle, there is a finite étale cover of $X$ by a three-dimensional complex torus or a compact simply connected Kähler manifold with holomorphically trivial canonical bundle. Moreover, the algebra of holomorphic forms is non-zero only in the extreme degrees (i.e., $0$ and $3$). Hence, $h^{p,0} =0$ for $p=1$ and $p=2$. I will understand this to mean Calabi-Yau, here.
For a Calabi-Yau threefold, the intersection form allows us to view the second Chern class $c_2$ as an integral linear form on the closure of the Kähler cone. Miyaoka (Math. Ann. 1998) showed that this integral linear form $c_2 : \overline{\mathscr{K}} \to \mathbf{Z}$ is nonnnegative. Hence, there are three cases: If $c_2 \equiv 0$, then $X$ admits a finite étale cover by a complex torus. If $c_2$ is not strictly positive, but not identically zero, then Wilson (Contemp Math 162) showed that the Picard number $\rho \leq 5$. If $\rho=4$ or $\rho=5$, then the cubic hypersurface $W \subset \mathbf{RP}^{\rho-1}$ (associated to the cubic cone $W^{\ast} : = \{ D : D^3 \equiv 0 \}$) decomposes as $W = \Lambda \cup Q$, where $\Lambda$ is a hyperplane and $Q$ is a quadric of rank $\rho-1 \leq r \leq 4$. If $Q$ is a quadric cone, the vertex $p \not \in \Lambda$. If $\Lambda \neq \{ c_2 =0 \}$, then $X$ is an elliptic fiber space. If $\Lambda$ is the hyperplane and $Q$ is a quadric cone, then $X$ is a K3 fiber space over $\mathbf{P}^1$.
If $c_2>0$, then Wilson (Contemp Math 162) showed that the automorphism group of $X$ is finite. Oguiso and Peternell (CAG, 1998) showed that Calabi--Yau threefolds with positive second Chern class admit only finitely many proper algebraic fiber space structures. To my understanding, their results do not give any indication as to whether this finite number is non-zero.
