Abstract: Consider a smooth projective Calabi-Yau 4-fold X(including hyperkahler and abelian 4-folds) over C. By the work of Borisov-Joyce/Oh Thomas, the moduli space of stable sheaves on X with fixed Chern character has a virtual fundamental class. This cycle vanishes when the (2,2)-class of the Chern character is Hodge theoretically `nontrivial'. I will fix this issue by using Kiem-Li/Kiem-Park cosection technique. Also, I will try to explain why the surface counting theory has its own unique feature compared to curve counting theories. This talk is based on joint projects with Martijn Kool and Hyeonjun Park.