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K-theoretic generalized Donaldson-Thomas invariants Kiem, Young-Hoon


For the moduli of derived category objects or the partial desingularizations of the moduli stack of semistable sheaves on Calabi-Yau 3-folds, there are no perfect obstruction theories but only semi-perfect obstruction theories. While a semi-perfect obstruction theory is sufficient for the construction of virtual cycles in Chow groups, it seems insufficient for virtual structure sheaves. In this talk, I will introduce the notion of an almost perfect obstruction theory, which lies in between a semi-perfect obstruction theory and an honest perfect obstruction theory. I will show that an almost perfect obstruction theory enables us to construct the virtual structure sheaf and hence K-theoretic virtual invariants. Examples of DM stacks with almost perfect obstruction theories include the Inaba-Lieblich moduli spaces of simple gluable perfect complexes and the partial desingularizations of moduli stacks of semistable sheaves on Calabi-Yau 3-folds. We thus obtain K-theoretic Donaldson-Thomas invariants of derived category objects and K-theoretic generalized Donaldson-Thomas invariants. Based on a joint work with Michail Savvas.

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