Homological mirror symmetry for the genus two curve

By P. Seidel. 2008

202 citations recorded by the source corpus.

Katzarkov has proposed a generalization of Kontsevich's mirror symmetry conjecture, covering some varieties of general type. We prove a version of this conjecture in the simplest example, relating the Fukaya category of a genus two curve to the category of Landau-Ginzburg branes on a certain singular rational surface.

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Citations in the corpus

Mirror symmetry between orbifold curves and cusp singularities with group action (2011, methodology): “deﬁning a cusp singularity. The group GT leaves F invariant and we can consider a crepant resolution Y → U/GT given by the GT-Hilbert scheme and the proper transform Xb⊂ Y of X= F−1(0)/GT ⊂ U/GT (cf. [Se]). Let HMFGe S (f) be the stable homotopy category of Ge-graded matrix factorizations of f. Let ”
A Beginner’s Introduction to Fukaya Categories (2013, background): “[44, 46] for an illustration of this approach.”
Conjectures on Bridgeland stability for Fukaya categories of Calabi-Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow (2014, methodology): “ed). (b) We would like to deﬁne a Fukaya category DbF(M) of balanced Lagrangians in a Kahler–Einstein manifold (M,J,g). When λ<0 (negative scalar curvature), a model for this is provided by Seidel [74], who deﬁnes DbF(M) when Mis the genus two curve. When λ>0 (positive scalar curvature, monoto”
Generating the Fukaya categories of Hamiltonian 𝐺-manifolds (2015, background): “li split-generates then K is not displaceable from any Lagrangian submanifold L with HF(L,L) 6= 0. Finding a split-generator is also the ﬁrst step in many modern proofs of homological mirror symmetry [6, 51, 52, 54, 55, 56]. The aim of this paperis todeterminewhen a free Lagrangian orbit of a Hamilt”
Homological mirror symmetry for Brieskorn–Pham singularities (2009, background): “special case when n= 2 in Theorem 1.3 is proved in [20]. Homological mirror symmetry is proposed by Kontsevich [7] for Calabi-Yau manifolds, and later generalized to more general classes of manifolds [8, 5, 14]. Homological mirror symmetry for singularities are studied by Takahashi and his collabora”
A category of kernels for equivariant factorizations and its implications for Hodge theory (2011, background): “tevich, [Kon95]. Today, HMS plays a central role in geometry, and extensive progress has been made applying matrix factorizations to HMS, including a proof of HMS for higher genus curves by P. Seidel [Sei08b] and A. Eﬁmov [Eﬁ09] and for punctured spheres by M. Abouzaid, D. Auroux, Eﬁmov, the third a”
Quiver algebras as Fukaya categories (2013, background): “We should point out that analogous finite determinacy arguments have been used to pin down A∞-structures in symplectic topology elsewhere, going back at least to Seidel’s [48].”
Dynamical systems and categories (2013, background): “Let DπFuk(M) denote the triangulated hull, which is locally proper (in the Z/2-graded sense), and has a generator given by a collection of loops [47, 24].”
Khovanov homology from Floer cohomology (2015, methodology): “The case in which some of the Lagrangians are equal is handled by counting configurations of holomorphic discs and (perturbed) gradient flow lines, as in [10, 46, 52].”
Gauged Floer Theory Of Toric Moment Fibers (2010, background): “[53] P. Seidel.”
Fukaya A∞-structures Associated to Lefschetz Fibrations. II (2014, background): “over a category which is Morita equivalent to that associated to the cubic pencil on CP2: more precisely, (5.39) A~ qo(Z=3)2 ˘=A q End(C3): This is the same trick mentioned in Remark 5.7 (we refer to [52, 45, 54], where parallel arguments appear, for further explanation). The triviality of the defor”
Towards Mirror Symmetry for Varieties of General Type (2012, background): “ial, so by [LP11],Thm.4.1, the subcategory of compact objects is a Calabi-Yau category and it is expected that this generates the entire category. There is a suggestion of how to reﬁne the grading in [Sei08],§8 for a genus two curve. It is currently unknown whether there is a general way to reﬁne th”
On the Fukaya category of a Fano hypersurface in projective space (2013, methodology): “, let A be a noncurved A 1algebra of type An a , where n 1 a2 and n4. We will compute two versions of the Hochschild cohomology HH (A). Our strategy is suggested by [38, Section 3.6] (compare also [35]). It relies on a slightly modied version of [9, Proposition 1]. Let us start by stating this modie”
Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces (2012, background): “These results provide a geometric foundation for mirror constructions that have appeared in the recent literature [11, 28, 27, 40, 42, 1, 20].”
On the homological mirror symmetry conjecture for pairs of pants (2010, background): “See [48] for a formula for the grading that holds in the 1-dimensional case.”