Abstract
In this paper we will describe an approach to mirror symmetry for appropriate one-dimensional DM stacks of arithmetic genus g ≤ 1, called tcnc curves, which was developed by the author with Treumann and Zaslow in Sibilla et al. (Ribbon Graphs and Mirror Symmetry I, arXiv:1103.2462). This involves introducing a conjectural sheaf-theoretic model for the Fukaya category of punctured Riemann surfaces. As an application, we will investigate derived equivalences of tcnc curves, and generalize classic results of Mukai on dual abelian varieties (Mukai, Nagoya Math. J. 81, 153–175, 1981).
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Notes
- 1.
- 2.
- 3.
From now on, we will refer to objects in Sh(X) simply as ‘constructible sheaves.’ See [11] for a comprehensive introduction to the subject.
- 4.
CPM stands for ‘constructible plumbing model,’ as this framework can be more generally applied to investigate the Fukaya category of a plumbing of cotangent bundles, for which see also [1].
- 5.
Lagrangian branes in Fuk(Σ) are further required to be, in a suitable sense, ‘adapted’ to the skeleton, and thus in particular compact when Γ Σ is. Also, when referring to the ‘Fukaya category,’ we shall mean the split closure of the category of twisted complexes over the Fukaya category, see [24].
- 6.
- 7.
Note that, if π: T ∗ X → X is the natural projection, then \(\pi _{{\ast}}\mathit{MSh}(-)\mathop{\cong}\mathit{Sh}_{\varLambda }(-)\).
- 8.
In fact, any ribbon graph is deformation equivalent, in the above sense, to a ribbon graph admitting chordal structure. This gives us a concrete, although not ‘functorial,’ way of computing the global sections of CPM(−) on a general ribbon graph up to quasi-equivalence.
- 9.
Note that our conventions differ from the ones commonly found in the literature. Weighted projective lines, which are denoted \(\mathbb{P}^{1}(a_{1},a_{2})\), are usually defined as quotients of \(\mathbb{C}^{2} -\{ 0\}\) by \(\mathbb{C}^{{\ast}}\) acting with weights a 1, a 2. According to the latter definition, if gcd(a 1, a 2) ≠ 1, \(\mathbb{P}^{1}(a_{1},a_{2})\) has non-trivial generic isotropy group. However, the two definitions agree if \(\mathit{gcd}(a_{1},a_{2}) = 1\).
- 10.
The scheme-theoretic notions employed in the definition, such as ‘normalization,’ can be easily adapted to DM stacks. We leave it to the reader to fill in the details.
- 11.
Note that Z is a disjoint union of classifying stacks of the form \([Spec(\mathbb{C})/\mu _{a_{i}}]\).
- 12.
Note that \(\mathit{Hom}_{\mathit{HoE}}^{1}(\psi (\mathcal{O}_{C}),\psi (\mathcal{F}))\) vanishes, since it is isomorphic to the quotient of \(\mathit{Hom}_{Z}^{0}(\sigma ^{{\ast}}\mathcal{O}_{\tilde{C}},\tau ^{{\ast}}\pi ^{{\ast}}\mathcal{F})\mathop{\cong}\mathbb{C}\) by the image of the differential, which is easily seen to be surjective.
- 13.
Both the set of characters and the set of up-/down-ward spokes come with natural cyclic orders (the spokes inherit it from the ribbon structure on \(\varLambda _{a_{1},a_{2}}\)). The labelling cannot therefore be entirely arbitrary, as it must preserve this cyclic order, see [32].
- 14.
It is important to point out that, as shown in Fig. 2, in a dualizable ribbon graph the strands joining together the components of the chordal basis cannot be (non-trivially) ‘braided.’ This can be translated in appropriate conditions of coherency on the maps \(R_{a_{i}} \rightarrow \varLambda _{i}\). We refer the reader to [32] for further details.
- 15.
The genus of a ribbon graph D can be described geometrically as the genus of any surface in which D can be embedded, in a way compatible with the ribbon structure, as a deformation retract. Thus \(D_{A}^{0}\) in Example 4(1) has genus 0, while \(D_{A}^{1}\) in Example 4(2), has genus 1. For a formal, combinatorial definition of the genus of a ribbon graph, see [32].
- 16.
- 17.
Note that this is true, without further specifications, only if we are considering the Fukaya category of compact Lagrangians in Σ.
- 18.
Note that any such equivalence would extend to an equivalence of the full derived categories, see Theorem 1.2 in [2].
- 19.
In [30], extending results of [6], we defined an action of the mapping class group of a torus with n punctures on D b(Coh(X n )). The argument we shall describe below can be interpreted, roughly, as defining an action of an appropriate version of the mapping class groupoid. For a definition of spherical functor, see [28].
- 20.
- 21.
- 22.
It might be surprising that the strands of A n are ‘non-trivially braided.’ The existence of the equivalence Φ n depends, in fact, in a crucial way on the choice of this particular geometry. Note however that the chordal basis of a dualizable ribbon graph is a union of loops. Considered as edges of the larger graph, the strands in a subgraph of type A can therefore be un-braided, cf. also Footnote 14.
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Sibilla, N. (2014). Mirror Symmetry in Dimension 1 and Fourier–Mukai Transforms. In: Castano-Bernard, R., Catanese, F., Kontsevich, M., Pantev, T., Soibelman, Y., Zharkov, I. (eds) Homological Mirror Symmetry and Tropical Geometry. Lecture Notes of the Unione Matematica Italiana, vol 15. Springer, Cham. https://doi.org/10.1007/978-3-319-06514-4_10
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