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

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