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We prove a generalized vanishing theorem for certain quasi-coherent sheavesalong the derived blow-ups of quasi-smooth derived Artin stacks. We give fourapplications of the generalized vanishing theorem: we prove a $K$-theoreticversion of the generalized vanishing theorem which verified a conjecture of theauthor and give a new proof of the $K$-theoretic virtual localization theoremfor quasi-smooth derived schemes through the intrinsic blow-up theory ofKiem-Li-Savvas; we prove a desingularization theorem for quasi-smooth derivedschemes and give an approximation formula for the virtual fundamental classes;we give a resolution of the diagonal along the projection map of blow-ups ofsmooth varieties, which strengthens the semi-orthogonal decomposition theoremof Orlov; we illustrate the relation between the generalized vanishing theoremand weak categorifications of quantum loop and toroidal algebra actions on thederived category of Nakajima quiver varieties. We also propose severalconjectures related to birational geometry and the $L_{\infty}$-algebroid ofCalaque-C\u{a}ld\u{a}raru-Tu.

A Generalized vanishing theorem for Blow-ups of Quasi-smooth Stacks