Homological Mirror Symmetry for Toric del Pezzo Surfaces

By Kazushi Ueda. 2004

47 citations recorded by the source corpus.

We prove the homological mirror conjecture for toric del Pezzo surfaces. In this case, the mirror object is a regular function on an algebraic torus We show that the derived Fukaya category of this mirror coincides with the derived category of coherent sheaves on the original manifold.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ce9bae8f30d8fffa536acc63565155382971ce96

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves (2005, background): “Ginzburg model include weighted projective planes, Hirzebruch surfaces [3], and toric blow-ups of CP2 [ 19 ].”
Homological mirror symmetry for punctured spheres (2011, background): “One direction of the mirror symmetry conjecture, in which the B-model consists of coherent sheaves on a Fano variety, has been established for toric Fano varieties in [1,8,11,15,46], as well as for del Pezzo surfaces”
Mirror symmetry for toric Fano manifolds via SYZ transformations (2008, unknown): “A version of the Homological Mirror Symmetry Conjecture has also been formulated by Kontsevich [28], which again has been checked in many cases [39], [44], [7], [8], [1], [2].”
T-Duality and Homological Mirror Symmetry of Toric Varieties (2008, background): “Ueda proves it for toric del Pezzo surfaces [34]; Ueda and Yamazaki prove it for toric orbifolds of toric del Pezzo surfaces [35].”
Suspending Lefschetz Fibrations, with an Application to Local Mirror Symmetry (2009, background): “Homological mirror symmetry for such varieties, first considered by Kontsevich [11], is now proved (see [16] for Y = CP (2), [2] for several more cases, and [1, 20] for all remaining ones).”
Lagrangian Floer theory on compact toric manifolds II: bulk deformations (2008, background): “ement (and Theorems 1.3, 1.9) can be regarded as a version of mirror symmetry between the toric A-model and the Landau-Ginzburg B-model. In various literature on mirror symmetry, such as [Ab], [AKO], [Ue], the Bmodel is dealt for Fano or toric manifolds in which the derived category of coherent shea”
Fukaya A∞-structures Associated to Lefschetz Fibrations. II (2014, unknown): “ivial deformation of T. Because of our previous observation concerning F q(ˇ), that deformation must then be trivial. The same argument applies to the mirrors of the other smooth toric Fano surfaces [5, 4, 58]. In higher dimensions, it is still true that smooth toric Fano varieties are rigid [7], he”
Brane tilings and their applications (2008, methodology): “The Fano version (6.2) was proved by Seidel [221] for P2 and P1 × P1, Auroux, Katzarkov and Orlov [222] for Hirzebruch surfaces and Ueda [223] for P2 blown-up at two or three points.”