Moduli spaces of polarized irreducible symplectic manifolds are not   necessarily connected
  We show that the moduli space of polarized irreducible symplectic manifolds of $K3^{[n]}$-type, of fixed polarization type, is not always connected. This can be derived as a consequence of Eyal Markman's characterization of polarized parallel-transport operators of $K3^{[n]}$-type. 
Apostol Apostolov
math.AG