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The generalized homological mirror symmetry conjecture states that for mirror dual models $(X_E, w)$ and $(X_E', w')$ , if $L$, a lattice polytope which is a Newton polytope of a nonsingular projective toric variety, is in $M_{R}$ , then isomorphisms exist between: $D^b(X_E,w)$ and $DFS(X_E',w')$, $D^b(X_E',w')$ and $DFS(X_E,w)$, where $DFS$ is the derived Fukaya category and $D^b$ is derived category of coherent sheaves. What cases of this conjecture are open? Any references are greatly appreciated.

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First - note that the way you state HMS in the setting you ask about (toric) is not exactly the way it actually is (because - in the "A to B direction" - you actually need to consider the singular derived category - see below).

Also you state your question as a general question about mirror symmetry - while actually (if understood correctly) eventually ask specifically about mirror symmetry for toric manifolds.

Pay attention that (compact) toric manifolds are not Calabi-Yau - Hence it is not exactly the setting HMS was originally defined by Kontsevich in 1994 - that's no problem - because Kontsevich latter extended HMS to a more general setting (Lectures at ENS Paris, 1998) relating to ideas on Landau-Ginzburg theory of sigma models due to Hori-Vafa.

In relation to your question toric manifolds really turn to be a fantastic class of manifolds to study HMS (98). First - you need to make clear what is $X'$ - i.e the mirror in the toric setting. This is especially straight forward to describe for the subclass of toric Fano manifolds due to a beautiful geometric construction due to Batyrev involving the notion of reflexive polytopes. (From now on let's assume we are talking about toric Fano manifolds). In the HMS setting (a) the mirror is actually taken to be the open algebraic torus $(\mathbb{C}^{\ast})^n$ in Batyrev's $X'$ (note that this is a non-compact manifold and that in general it's compactification $X'$ is a singular manifold) (b) it is also associated with a Laurent polynomial (or a generalization of such) $ W: (\mathbb{C}^{\ast}) \rightarrow \mathbb{C}$ called the Landau-Ginzburg mirror potential which is the same as taking a hyperplane section of $X'$.

Now you have to introduce the categories you are working on. Let's start with the B-side on X goes to A-side on mirror $((\mathbb{C}^{\ast})^n, W)$ case. Here the B-side is clear - you should practically look at $\mathcal{D}^b(X)$ (or some natural extension of it - to meet the $A_{\infty}$ structure on the other side). While on the A-side on the mirror you need to take into account what you are looking at as the Fukaya category of $((\mathbb{C}^{\ast})^n,W)$. A famous and remarkable construction is the Fukays-Seidel category which strongly uses Picard-Lefschetz theory. One of the features of the Fukaya-Seidel category is that it comes together with natural exceptional collections - determined by a collection of vanishing thimbles. One of the ways to prove HMS (98) (an approach which works for a much wider class of manifolds than toric) is to find the appropriate analog of the A-exceptional collection of thimbles in the B-side - that is, the derived category $D^b(X)$ - as once you have such an equivalence - you actually get a HMS-functor. One of the first applications of this approach (aside from original applications due to Seidel for $X=\mathbb{P}^2$, for instance) is the work of Auroux-Katzarakov-Orlov on toric Del-Pezzo surfaces which established such a functor in this setting.

Specifically in the toric case, one also has a remarkable approach due to Abouzaid involving tropical geometry - which works for any toric (Fano) manifold. (Abouzaid's approach to the Fukaya category on $((\mathbb{C}^{\ast})^n,W)$ is slightly different, at least in the level of the definition, from the Fukaya-Seidel setting). Abouzaid notes that in the toric case $D^b(X)$ is generated (as a triangulated category) by $Pic(X)$ and identifies the mirror analogs of $Pix(X)$ in $((\mathbb{C}^{\ast})^n,W)$ (W here is a slight modification of a Laurent polynomial) using an ingenious construction involving tropical geometry. This in particular, gives a HMS-functor, which is practically a proof of HMS in this setting.

The other case of A-side on $X$ goes to B-side on $((\mathbb{C}^{\ast})^n, W)$ - this is, traditionally, a bit slower to follow in terms of development than the B to A side. One of the features (mentioned in the beginning) is that as $X'$ is singular \ W is an algebraic fibration with singular fibers - you actually need to consider the singular derived category $D^b_{sing}(X')$ and not just $D^b(X')$ this adds some interesting features (relation to matrix factorizations) - and there are various open questions.

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The conjecture has been solved for elliptic curves, abelian varieties, non-singular torus bundles over affine manifolds, and quartic surfaces. It remains to find a unification from algebraic geometry. A few references for this subject are the following:

Kontsevich, Maxim (1994), Homological algebra of mirror symmetry, arXiv:alg-geom/9411018. Kontsevich, Maxim; Soibelman, Yan (2000), Homological Mirror Symmetry and torus fibrations, arXiv:math.SG/0011041. Seidel, Paul (2003), Homological mirror symmetry for the quartic surface, arXiv:math.SG/0310414. Hausel, Tamas; Thaddeus, Michael (2002), Mirror symmetry, Langlands duality, and the Hitchin system, arXiv:math.DG/0205236

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    $\begingroup$ Why not put this information in the original question? $\endgroup$
    – user5117
    Jul 13 '12 at 19:25

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