Well, it is of the "straightforward" questions one may ask. I propose it here to see if someone could tell me more on the recent status of this quite long-standing problem.

To initiate, let me give a brief description of the "classical" invariants. Essentially they are provided by either the symplectic structure or the Hodge structure, most of which relating to vanishing theorems and integral theorems. As far as a compact manifold $M$ of complex dimension $n$ is concerned, we have the following:

$b_2 \ne 0$ for the symplectic structure and many more;

$ b_{2k+1}$ are even and $b_{k-2} \leq b_k$ for $k \leq n$ for the Hodge structure, which date back to Lefschetz;

Various integral results on Chern numbers by Hirzebruch-Riemann-Roch on Kaehler manifolds;

and etc.

To make the question precise, I would like to ask:

Has there been any "higher" or essentially new invariants discovered so far? Particularly, one may observe that the above invariants are all torsion-free(as Yau did in his problem list), so the torsion invariants would be rather interesting, suppose they do exist.

Any comments are welcomed and thanks a lot!