Log crepant birational maps and derived categories.

*(English)*Zbl 1095.14014In his paper [J. Differ. Geom. 61, No.1, 147–171 (2002; Zbl 1056.14021)], the author has studied aspects of the conjecture that birationally equivalent smooth projective varieties have equivalent derived categories iff they have equivalent canonical divisors. The article under review is dealing with singular instead of only smooth varieties and moreover with pairs of varieties and \(\mathbb Q\)-divisors, where at most log-terminal singularities are allowed:

Derived equivalence conjecture. Let \((X,B)\) and \((Y,C)\) be such pairs (and suppose some technical conditions). Denote \(\mathcal X\) and \(\mathcal Y\) the associated stacks, and assume there are proper birational morphisms \(\mu :W\to X\) and \(\nu :W\to Y\) such that \(\mu ^* (K_X+B)=\nu ^* (K_y+C)\). Then there exists an equivalence \(D^b({\mathcal X}) \to D^b({\mathcal Y})\) of triangulated categories .

After formulating the conjecture, first of all a converse statement is given. Main result of the article is a proof of the conjecture for toroidal varieties. The theorem generalizes an earlier result of the author [in: Algebraic geometry. A volume in memory of Paolo Francia. Berlin: de Gruyter. 197–215 (2002; Zbl 1092.14023)]. It implies the McKay correspondence for abelian quotient singularities as a special case.

Derived equivalence conjecture. Let \((X,B)\) and \((Y,C)\) be such pairs (and suppose some technical conditions). Denote \(\mathcal X\) and \(\mathcal Y\) the associated stacks, and assume there are proper birational morphisms \(\mu :W\to X\) and \(\nu :W\to Y\) such that \(\mu ^* (K_X+B)=\nu ^* (K_y+C)\). Then there exists an equivalence \(D^b({\mathcal X}) \to D^b({\mathcal Y})\) of triangulated categories .

After formulating the conjecture, first of all a converse statement is given. Main result of the article is a proof of the conjecture for toroidal varieties. The theorem generalizes an earlier result of the author [in: Algebraic geometry. A volume in memory of Paolo Francia. Berlin: de Gruyter. 197–215 (2002; Zbl 1092.14023)]. It implies the McKay correspondence for abelian quotient singularities as a special case.

Reviewer: Marko Roczen (Berlin)

##### MSC:

14E30 | Minimal model program (Mori theory, extremal rays) |

18E30 | Derived categories, triangulated categories (MSC2010) |