The coherent-constructible correspondence and Fourier-Mukai transforms
The purpose of the paper under review is to investigate some of Kawamata's results concerning derived equivalences between toric orbifolds using the so-called coherent constructible correspondence (CCC). Given a toric \(n\)-fold \(X\), the coherent-constructible correspondence (CCC) is an equivalence between a category of coherent sheaves on \(X\) and a category of constructible sheaves on a compact \(n\)-torus. There also exists an equivariant version. It is furthermore possible to extend this correspondence to toric Deligne-Mumford stacks, in particular to toric orbifolds, that is, toric DM-stacks with generically trivial stabilizers. Let \(\mathcal{X}_{\pmb\Sigma}\) be a toric orbifold defined by a so-called stacky fan \({\pmb\Sigma}=(N,\Sigma,\beta)\). This contains the torus \(T\cong (\mathbb{C}^*)^n\) as a dense open subset. The CCC for \(\mathcal{X}_{\pmb\Sigma}\) is a quasi-equivalence of the following triangulated dg-categories: On the one hand, one has the category of \(T\)-equivariant perfect complexes on the orbifold and on the other hand a category of constructible sheaves on \(M_\mathbb{R}\) (the dual space of \(N_\mathbb{R}\)) characterized by a conical Lagrangian \(\Lambda_{\pmb{\Sigma}} \in T^*M_\mathbb{R}\) determined by the above stacky fan. The toric orbifold \(\mathcal{X}_{\pmb\Sigma}\) is the DM-stack associated with a log pair \((X_\Sigma,B)\) in the sense of \textit{Y. Kawamata} [J.\ Math.\ Sci., Tokyo 12, No.\ 2, 211--231 (2005; Zbl 1095.14014)]. Given two toric orbifolds \(\mathcal{X}_1\) and \(\mathcal{X}_2\) associated with projective toric log pairs, one says that the orbifolds are \(K\)-equivalent if there exists an orbifold \(\mathcal{W}\) and proper birational morphisms \(\mu_i: \mathcal{W}\rightarrow \mathcal{X}_i\) of orbifolds such that \(\mu_1^*K_{\mathcal{X}_1}= \mu_2^*K_{\mathcal{X}_2}\). Kawamata proved that the Fourier--Mukai functor \((\mu_1)_*\circ \mu_2^*\) is an equivalence of triangulated categories. To be more precise, he proved that if \(\mu_1^*K_{\mathcal{X}_1}\geq \mu_2^*K_{\mathcal{X}_2}\) and the birational map \(f: X_1 \rightarrow X_2\) is (1) the identity, (2) a divisorial contraction, (3) the inverse of a divisorial contraction or (4) a flip, then the above Fourier--Mukai functor if fully faithful; it is an equivalence when \(\mu_1^*K_{\mathcal{X}_1}= \mu_2^*K_{\mathcal{X}_2}\). The paper under review proves the above result in the equivariant setting (in cases (1), (2) and (3)) using the CCC. The paper is organized as follows. Section 2 presents an introduction to toric orbifolds, Section 3 gives an exposition of the CCC of toric orbifolds and in Section 4 the authors elaborate Kawamata's theorem in the equivariant setting from the perspective of constructible sheaves.
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