We will discuss enumerative aspects of moduli spaces of sheaves/complexes on Calabi-Yau 4-folds. I will review virtual cycle on moduli spaces of sheaves on Calabi-Yau 4-folds constructed by Borisov-Joyce/Oh-Thomas. This BJ/OT class often vanishes when the second Chern character of the sheaves is nontrivial primitive part. We will fix this issue by studying cosection arising from Hodge theory and construct the reduced virtual cycles. The reduced virtual cycle has the deformation invariance property along the Hodge locus. This property has an implication to variational Hodge conjecture. This is a joint work with Martijn Kool and Hyeonjun Park.
