A generalized Bondal-Orlov full faithfulness criterion for Deligne-Mumford stacks

Abstract

Let $X$, $Y$ be smooth projective varieties over $\mathbf{C}$. Let $K$ be a bounded complex of coherent sheaves on $X\times Y$ and let ${\Phi }_K:{\mathsf{D}}_{\mathsf{Coh}}^b(X)\to {\mathsf{D}}_{\mathsf{Coh}}^b(Y)$ be the resulting Fourier-Mukai functor. There is a well-known criterion due to Bondal-Orlov for ${\Phi }_K$ to be fully faithful. This criterion was recently extended to smooth Deligne-Mumford stacks with projective coarse moduli schemes by Lim-Polischuk. We extend this to all smooth, proper Deligne-Mumford stacks over arbitrary fields of characteristic $0$. Along the way, we establish a number of foundational results for bounded derived categories of proper and tame morphisms of noetherian algebraic stacks (e.g., coherent duality).