Given any sequence of $n$-dimensional complex projective manifolds $\{X_i\}_{i=1}^{\infty}$ such that $X_i$ are diffeomorphic to each other and have ample canonical divisors $K_{X_i}$. Are the volumes $K_{X_i}^n$ bounded from the above?
Remark that this is true for $n\leq 4$. In this case, the volume can be bounded by Betti numbers and Pontryagin numbers of the varieties.
