Let $X$ be a minimal smooth projective surface of general type (over $\mathbb{C}$). Let's call such a surface MSPGT. Such a surface has two chern numbers $c_1^2$ and $c_2$. It is known that they are both non-negative integers with $c_1^2 + c_2\equiv 0\mod 12$, and they must satisfy the BMY and Noether inequalities. The "geography of chern numbers" refers to the problem of determining which pairs of integers arise as chern numbers of such surfaces. From my cursory reading of a few articles and wikipedia, it seems like it is expected that the above relations are the only relations satisfied by the chern numbers of MSPGT surfaces (i.e., any pair of integers satisfying the above relations can be realized as the chern numbers of some MSPGT surface.)
Since $c_2$ is the topological Euler characteristic of $X$ and $c_1^2$ (together with $c_2$) determines the holomorphic Euler characteristic of $\mathcal{O}_X$ (via Noether's formula), knowledge of the chern numbers would give two linearly independent relations on the Hodge numbers of $X$, which of course is not enough to determine the Hodge numbers, since there are three independent Hodge numbers: $h^{0,1},h^{0,2},h^{1,1}$. Thus, given $c_1^2$ and $c_2$, knowledge of any of $h^{0,1},h^{0,2},h^{1,1}$ (or even the Betti numbers $b^1,b^2$) would be enough to determine the rest.
My naive questions are:
Are there any known relations that must be satisfied for the Hodge numbers $h^{0,1},h^{0,2},h^{1,1}$ of MSPGT surfaces which are not implied by the above relations on chern numbers together with Noether's formula and the Hirzebruch signature theorem?
Is there a conjectural picture of exactly which triples of integers can appear as the Hodge numbers of such surfaces?
