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On Landau-Ginzburg systems, co-tropical geometry, and \(\mathcal{D}^b(X)\) of various toric Fano manifolds. (English) Zbl 1475.14034

Let \(X\) be a Fano manifold and denote by \(f: Y\to \mathbb{C}\) its Landau-Ginzburg mirror. If \((Y,f)\) is a Lefschetz fibration, i.e. \(f\) has isolated non-degenerate critical points, then the homological mirror symmetry conjecture states that the bounded derived category \(\mathcal{D}^b(X)\) of coherent sheaves on \(X\) is equivalent to the Fukaya-Seidel category \(\mathcal{FS}(Y,f)\) of Lefschetz thimbles of \((Y,f)\). Note that the latter depends on choices of paths in \(\mathbb{C}\), relating a reference point in \(\mathbb{C}\) to the images of critical points of \(f\). The author of the present paper relates – in some cases – the critical points of \(f\) directly to full exceptional collections of line bundles of \(\mathcal{D}^b(X)\).
If \((Y,f)\) is a Lefschetz fibration, then \(\mathcal{FS}(Y,f)\) is generated by an exceptional collection of vanishing Lagrangian spheres. On the \(B\)-side, the analogous question is whether \(\mathcal{D}^b(X)\) admits a full exceptional collection of objects, or even a full exceptional collection of line bundles. It was proven by Kawamata that the bounded derived category of a smooth projective toric Deligne-Mumford stack has a full exceptional collection, albeit not necessarily in line bundles. Which ones do admit a full exceptional collection in line bundles is open.
If \(X\) is toric, then there is additional structure as \(X\) admits a moment map to its Fano polytope \(\Delta\). Moreover, \(f=f_u\) is a member of the space of Laurent polynomials whose Newton polytope is \(\Delta^\circ\), the polar dual of \(\Delta\), and with parameter \(u\). It is a general theme in mirror symmetry that the critical locus \(\mathrm{Crit}(f_u)\) recognizes a lot – if not all – of the complex geometry of \((Y,f_u)\).
The author of the present paper connects \(\mathrm{Crit}(f_u)\) with the \(B\)-side through a natural morphism \[ D_u : \mathrm{Crit}(f_u) \to \mathrm{Pic}(X), \] which the author introduced in previous work. The main question then is under what circumstances and under what conditions on \(u\) the image of \(D_u\) consists of a full exceptional collection generating \(\mathcal{D}^b(X)\). The author proves existence for certain families of simple toric Fano manifolds. In some of theses cases the author additionally proves that \(D_u\) interacts well with the monodromy associated to loops around the critical points.
The techniques of the article pass through co-tropical geometry. Writing the critical locus \(\mathrm{Crit}(f_u)\) as an intersection of hyperplanes \(W_i\), the image of the \(W_i\) under the argument map describes some co-amoebas. A suitable limit \(u\to\infty\) of this leading to co-tropical hyperplanes is used in order to find suitable exceptional maps \(D_u\).
This article is providing another piece of the puzzle of how the full homological mirror symmetry conjecture is reduced to 0-dimensional pieces in some relatively simple examples, providing further insight into mirror symmetry.

MSC:

14F08 Derived categories of sheaves, dg categories, and related constructions in algebraic geometry
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
14T20 Geometric aspects of tropical varieties
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