In this paper we prove that for a nonsingular projective variety of dimension at most 4 and with non-negative Kodaira dimension, the Kodaira dimension of coherent subsheaves of ${\Omega }^p$ is bounded from above by the Kodaira dimension of the variety. This implies the finiteness of the fundamental group for such an $X$ provided that $X$ has vanishing Kodaira dimension and non-trivial holomorphic Euler characteristic.

We give a one-parameter family of Bridgeland stability conditions on the derived category of a smooth projective complex surface $S$ and describe “wall-crossing behavior” for objects with the same invariants as ${𝒪}_C\left(H\right)$ when $H$ generates Pic$\left(S\right)$ and $C\in \left|H\right|$. If, in addition, $S$ is a $K3$ or Abelian surface, we use this description to construct a sequence of fine moduli spaces of Bridgeland-stable objects via Mukai flops and generalized elementary modifications of the universal coherent sheaf. We also discover a natural...