The McKay Correspondence, Tilting, and Rationality

Abstract

We consider the problem of comparing $t$-structures under the derived McKay correspondence and for tilting equivalences. In low-dimensional cases, we relate the $t$-structures via torsion theories arising from additive functions on the triangulated category. As an application, we give a criterion for rationality for surfaces with a tilting bundle. We also show that every smooth projective surface that admits a full, strong, and exceptional collection of line bundles is rational.