Deformations of calibrations, {Calabi-Yau, HyperK\"ahler, $G_2$ and   Spin(7) structures}
  We shall develop a new deformation theory of geometric structures in terms of closed differential forms. This theory is a generalization of Kodaira -Spencer theory and further we obtain a criterion of unobstructed deformations. We apply this theory to certain geometric structures: Calabi-Yau, HyperK\"ahler, $\G$ and $\Spin$ structures and show that these deformation spaces are smooth in a systematic way. 
Ryushi Goto