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The goal of the present paper is to show the transformation formula of Donaldson–Thomas invariants on smooth projective Calabi–Yau 3-folds under birational transformations via categorical method. We also generalize the non-commutative Donaldson–Thomas invariants, introduced by B. Szendrői in a local (−1, −1)-curve example, to an arbitrary flopping contraction from a smooth projective Calabi–Yau 3-fold. The transformation formula between such invariants and the usual Donaldson–Thomas invariants are also established. These formulas will be deduced from the wall-crossing formula in the space of weak stability conditions on the derived category.

Curve counting theories via stable objects Ⅱ : DT/ncDT flop formula