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Mirror symmetry for weighted projective planes and their noncommutative deformations. (English) Zbl 1175.14030

The paper deals with homological mirror symmetry for Fano manifolds. In particular, the authors prove HMS for weighted projective lines and planes, and Hirzebruch surfaces, extending a result by [P. Seidel, More about vanishing cycles and mutation. 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)] on the complex projective plane. Moreover, they give the first explicit extension of HMS to noncommutative deformations of Fano varieties.
Mirror symmetry relates Fano varieties with certain Landau-Ginzburg models. The B-branes on the Fano variety are described by the derived category of coherent sheaves, and under mirror symmetry they correspond to A-branes in the mirror Landau-Ginzburg model \(W:X \to \mathbb{C}\). P. Seidel [in: Vanishing cycles and mutation, Casacuberta, Carles (ed.) et al., 3rd European congress of mathematics (ECM), Barcelona, Spain, 2000. Volume II. Basel: Birkhäuser. Prog. Math. 202, 65–85 (2001; Zbl 1042.53060)] rigourously defined, in the case of nondegenerate critical points, the category of Lagrangian vanishing cycles \(D(\mathrm{Lag}_{vc}(W))\), whose objects represent A-branes on \(W:X \to \mathbb{C}\). Starting from a complete intersection \(Y\) in a toric variety, the author use a procedure they call toric mirror ansatz, which was described originally by V. Batyrev [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, vol. II, 239–248 (1998; Zbl 0933.14020)], [A. Givental, in: Topological Field Theory, Primitive Forms and related Topics(Kyoto 1996), Progr. Math. 160, Birkäuser, Boston, 141–175 (1998, Zbl 0936.14031)] and K. Hori, and C. Vafa [Mirror Symmetry, arXiv:hep-th/0002222], to produce an open symplectic manifold \((X,\omega)\) as an affine subset of its mirror LG model and a symplectic fibration \(W: X \to \mathbb{C}\). Then the HMS conjecture states that the category \(D(\mathrm{Lag}_{vc}(W))\) is equivalent to the category \(D(\mathrm{Coh}(Y))\). This is the conjecture proved by the authors in various examples.
The main case discussed in this paper is a complex weighted projective plane \({\mathbb C\mathbb P}(a,b,c)\). Its mirror is the affine hypersurface \(X = \{ x^a y^b z^c = 0 \}\) in \(({\mathbb C}^*)^3\) equipped with the superpotential \(W = x + y +z\). In the proof of the conjecture, it turns out to be very important to treat singular toric variety as smooth stacks. In this case, such categories are indeed generated by an exceptional collection of line bundles. This is proved in Section 2, where the authors first describe weighted projective spaces and the stacky point of view, then describe coherent sheaves and their derived category and finally show the existence of the required collection. After constructing explicitely the category \(D(\mathrm{Lag}_{vc}(W))\) of Lagrangian vanishing cycles on the mirror Landau-Ginzburg model, they describe carefully the equivalence predicted by the HMS conjecture. This is done by studying the vanishing cycles and their intersection properties, which allow to describe the category \(\mathrm{Lag}_{vc}(W)\). The description of Floer products is determined by the study of moduli spaces of pseudo-holomorphic discs. Finally, the authors establish an explicit correspondence between noncommutative deformations of the weighted projective plane and complexified Kähler class on the mirror (after a discussion on the Maslov index and grading) to complete the proof of the HMS mirror conjecture for weighted projective spaces and their noncommutative deformations. All this is performed in Section 4.
Another example in which the HMS conjecture is proved is the case of Hirzebruch surfaces \({\mathbb F}_n\). Indeed, the category of Lagrangian venishing cycles on the mirror of \({\mathbb F}_n\) is related to the category of Lagrangian vanishing cycles on the mirror of \({\mathbb C\mathbb P}(n,1,1)\). The HMS conjecture is proved directly for the Fano cases, i. e. for \(n= 0,1,2\). In the non-Fano case, a certain degenerate limit of the Landau-Ginzburg model describes a subcategory of \(\mathrm{Lag}_{vc}(W)\) whose derived category is equivalent to the derived category of coherent sheaves on \({\mathbb F}_n\).

MSC:

14J33 Mirror symmetry (algebro-geometric aspects)
14A22 Noncommutative algebraic geometry
53D40 Symplectic aspects of Floer homology and cohomology
18E30 Derived categories, triangulated categories (MSC2010)
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