T-duality and homological mirror symmetry for toric varieties
In the paper under review, the authors prove one direction of the homological mirror symmetry conjecture for toric varieties. Let \(X_{\Sigma}\) be a complex \(n-\)dimensional complete toric variety defined by a finite complete fan \(\Sigma\subseteq N_{\mathbb{R}}\simeq\mathbb{R}^{n}\). Write \(M\) for the dual of \(N\) and \(M_{\mathbb{R}}\) for the dual of \(N_{\mathbb{R}}\). Moreover, let \(T_{\mathbb{R}}\) be the maximal compact subgroup of the algebraic torus \(T=N\times\mathbb{C}^{*}\), and \(T^{\vee}_{\mathbb{R}}\) be its dual torus. As shown by the authors [Invent. Math. 186, No. 1, 79--114 (2011; Zbl 1250.14011)], there is a quasi-equivalence of monoidal dg-categories \(\kappa: \mathrm{Per} f_{T}(X_{\Sigma})\longrightarrow Sh_{cc}(M_{\mathbb{R}},\Lambda_{\Sigma})\), where \(\mathrm{Per} f_{T}(X_{\Sigma})\) is the dg-category of perfect complexes of \(T-\)equivariant coherent sheaves (localized with respect to acyclic complexes); \(\Lambda_{\Sigma}\) is a Lagrangian subvariety of \(T^{*}M_{\mathbb{R}}\) associated to the toric fan; \(\mathrm{Sh}_{cc}(M_{\mathbb{R}},\Lambda_{\Sigma})\) is the dg-category of bounded chain complexes of sheaves of \(\mathbb{C}-\)vector spaces on \(M_{\mathbb{R}}\) (localized with respect to acyclic complexes), whose cohomology sheaves are constructible with respect to the Whitney stratification, have compact support, and the singular support is in \(\Lambda_{\Sigma}\). \textit{D. Nadler} and \textit{E. Zaslow} [J. Am. Math. Soc. 22, No. 1, 233--286 (2009; Zbl 1227.32019)], obtained a quasi-equivalence of \(A_{\infty}-\)categories \(\mu:\mathrm{Sh}_{cc}(M_{\mathbb{R}},\Lambda_{\Sigma})\longrightarrow \mathrm{Fuk}(T^{*}M_{\mathbb{R}},\Lambda_{\Sigma}),\) where the last is the unwrapped Fuakaya category of Lagrangian branes. The authors show that on \(\mathrm{Fuk}(T^{*}M_{\mathbb{R}},\Lambda_{\Sigma})\) there is a product \(\diamond\), and that the functor \(\mu\) intertwins the convolution product and \(\diamond\). Combining all these results, the authors get a quasi-equivalence of \(A_{\infty}-\)categories \(\tau:\mathrm{Per}f_{T}(X_{\Sigma})\longrightarrow \mathrm{Fuk}(T^{*}M_{\mathbb{R}},\Lambda_{\Sigma})\) which intertwins the tensor product and \(\diamond\). As a corollary, if \(X_{\Sigma}\) is smooth this induces an equivalence of tensor tringulated categories between the derived category of \(T-\)equivariant coherent sheaves on \(X_{\Sigma}\) and the derived category of \(\mathrm{Fuk}(T^{*}M_{\mathbb{R}},\Lambda_{\Sigma})\), proving one direction of the equivariant homological mirror symmetry for toric varieties. Similar but weaker results are obtained in the non-equivariant setting. A quasi-embedding \(\overline{\kappa}:\mathrm{Per}f(X_{\Sigma})\longrightarrow \mathrm{Sh}_{c}(T^{\vee}_{\mathbb{R}},\overline{\Lambda}_{\Sigma})\) is obtained by \textit{D. Treumann} [``Remarks on the nonequivariant coherent-constructible correspondence'', \url{arxiv:1006.5756}], and a quasi-equivalence of \(A_{\infty}-\)categories \(\overline{\mu}:\mathrm{Sh}_{c}(T^{\vee}_{\mathbb{R}},\overline{\Lambda}_{\Sigma})\longrightarrow \mathrm{Fuk}(T^{*}T^{\vee}_{\mathbb{R}},\overline{\Lambda}_{\Sigma})\) is obtained by \textit{D. Nadler} [Sel. Math., New Ser. 15, No. 4, 563--619 (2009; Zbl 1197.53116)]. Combining all these results one gets a quasi-embedding of \(A_{\infty}-\)categories \(\overline{\tau}:\mathrm{Per}f(X_{\Sigma})\longrightarrow \mathrm{Fuk}(T^{*}T^{\vee}_{\mathbb{R}},\overline{\Lambda}_{\Sigma})\), giving rise, when \(X_{\Sigma}\) is smooth, to an embedding of tensor categories \(\mathrm{DCoh}(X_{\Sigma})\longrightarrow \mathrm{DFuk}(T^{*}T^{\vee}_{\mathbb{R}},\overline{\Lambda}_{\Sigma})\), which is conjectured to be an equivalence (giving then one direction of the homological mirror symmetry for toric varieties). The final result of the paper is devoted to prove that the quasi-equivalence provided by the previous results is compatible with T-duality. More precisely, to any \(T-\)equivariant line bundle \(L\) and to any \(T_{\mathbb{R}}-\)invariant real analytic hermitian metric \(h\) on \(L\): the authors associate a real analytic Lagrangian submanifold \(\mathbb{L}_{h}\) of \(T^{*}M_{\mathbb{R}}\), and they show that \(\tau(L)=\mathbb{L}_{h}\). In the appendix, the authors provide first a review of geometric categories; moreover, they prove a stronger version of a theorem of Seidel, namely they prove that \(\mathrm{Per}f_{T}(X)\) is generated by \(T-\)invariant ample line bundles if \(X\) is smooth and projective; finally, they present the relation of the paper with the works of \textit{P. Seidel} [in: Symplectic geometry and mirror symmetry. Proceedings of the 4th KIAS annual international conference, Seoul, South Korea, August 14--18, 2000. Singapore: World Scientific. 429--465 (2001; Zbl 1079.14529)], \textit{D. Auroux}, \textit{L. Katzarkov} and \textit{D. Orlov} [Ann. Math. (2) 167, No. 3, 867--943 (2008; Zbl 1175.14030)] and in particular with the work of \textit{M. Abouzaid} [Sel. Math., New Ser. 15, No. 2, 189--270 (2009; Zbl 1204.14019)].
scientific article