Derived categories of toric varieties. II. (Q357525)

This article is a direct continuation of [\textit{Y. Kawamata}, Mich. Math. J. 54, No. 3, 517--535 (2006; Zbl 1159.14026)], where it was shown that the derived category \(\mathcal D^b(\mathcal X)\) of a toric stack \(\mathcal X\) is generated by an exceptional collection. The stack \(\mathcal X\) comes into play as a smooth replacement of a pair \((X,B)\) where \(X\) is a \(\mathbb Q\)-factorial toric variety and \(B = \sum (1-1/r_i) D_i\) an invariant divisor. \(\mathcal X\) is actually an orbifold and can be constructed from \(X\) using \(r_i\)-th roots along the divisors \(D_i\) as in [\textit{B. Fantechi} et al., J. Reine Angew. Math. 648, 201--244 (2010; Zbl 1211.14009)].NEWLINENEWLINECompleting the results of the first article, the missing case of a divisorial extraction is treated. This is a birational morphism \(f : X \to Y\) such that \(f^* K_Y = K_X + eE\) where \(E\) is the exceptional divisor and \(e>0\). Moreover, there is a remark that the exceptional sequence obtained for a pair \((X,B)\) will in general involve complexes and not only sheaves.NEWLINENEWLINEAdditionally, the following statement on Fourier-Mukai partners is shown. Let \(X\) be a \(\mathbb Q\)-factorial toric variety and \(Y\) a projective variety with at most quotient singularities, such that the associated canoncial stacks \(\mathcal X\) and \(\mathcal Y\) are derived equivalent, i.e.\ \(\mathcal D^b(\mathcal X) \cong \mathcal D^b(\mathcal Y)\). Then \(Y\) is also a \(\mathbb Q\)-factorial toric variety and the derived equivalence induces a birational map \(X \dashrightarrow Y\). Since \(X\) is especially a Mori dream space, there can be only finitely many \(Y\) with a birational map \(X \dashrightarrow Y\) of above type.