Mirror symmetry for weighted projective planes and their noncommutative deformations

By D. Auroux, L. Katzarkov, Dmitri Orlov. 2004

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We study the derived categories of coherent sheaves of weighted projective spaces and their noncommutative deformations, and the derived categories of Lagrangian vanishing cycles of their mirror Landau-Ginzburg models. In particular, we show that the derived category of coherent sheaves (B-branes) on the weighted projective plane $\CP^2(a,b,c)$ is equivalent to the derived category of vanishing cycles (A-branes) on the affine hypersurface $X=\{x^ay^bz^c=1\}\subset (\C^*)^3$ equipped with an exact symplectic form and the superpotential $W=x+y+z$. Hence, the homological mirror symmetry conjecture holds for weighted projective planes. Moreover, we also show that this mirror correspondence between derived categories can be extended to toric noncommutative deformations of $\CP^2(a,b,c)$ where B-branes are concerned, and their mirror counterparts, non-exact deformations of the symplectic structure of $X$ where A-branes are concerned. We also obtain similar results for other examples such as weighted projective lines or Hirzebruch surfaces.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/19d14b702df3e4e2f08afcc60c86485d763ffa45

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Citations in the corpus

Determinantal Barlow surfaces and phantom categories (2012, methodology): “ectures Recently some progress was made towards proving Kontsevich’s Homological Mirror Symmetry (HMS) conjecture, originally proposed for Calabi{Yau, and a little later also for Fano varieties - see [AKO]. The scope of HMS was then extended to varieties of general type in [KKOY] and [GKR]. As noted”
Autoequivalences of derived categories via geometric invariant theory (2013, unknown): “hts are simply di 1,...,d i n i which are all positive. Therefore the stack Z0/C ∗ is a weighted projective space. Its derived category is understood thanks to the following. Theorem (Theorem 2.12 of [2]). Db(P(di 1,...,d i n i)) has a full exceptional collection of d i line bundles. In particular K”
Conjectures on Bridgeland stability for Fukaya categories of Calabi-Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow (2014, background): “ects in DbF(M) are ‘semistable’, so the notion of stability in (c) above is not interesting for m= 1. Example 4.5. Let M be CP2, a Kahler–Einstein manifold with λ>0, and take F= C. Mirror Symmetry [7] predicts that the mirror of Mis the Landau– Ginzburg model f: U= (C∗)2 →C, f(x,y) = x+ y+ 1/xy. ”
The Coherent-Constructible Correspondence for Toric Deligne-Mumford Stacks (2009, background): “This version has been proved for weighted projective planes by Auroux-Katzarkov-Orlov [5], for toric orbifolds of toric del Pezzo surfaces by Ueda-Yamazaki [53], and more recently, for toric orbifolds of projective spaces (of any dimension) by Futaki-Ueda [24].”
Variation of geometric invariant theory quotients and derived categories (2012, methodology): “= hDb(cohP(1 : 1 : 2))i, d) If r>2, we have an SOD, Db(cohP(1 : 1 : r)) = hDb(cohF r),Db(cohSpeck),...,Db(cohSpeck)((r−3)D1)i. Remark 5.1.3. The example above was obtained using tilting objects in [AKO08], where it is used to prove homological mirror symmetry for Hirzebruch surfaces. As an applic”
Fukaya A∞-structures Associated to Lefschetz Fibrations. II (2014, background): “ways be negative. Even in the Lefschetz bration context, it is easy to cook up simple examples for which the answer is negative. On the other hand, there are interesting examples from mirror symmetry [5] in which the deformation is trivial, with the B-eld set to zero (in fact, triviality of the defo”
T-Duality and Homological Mirror Symmetry of Toric Varieties (2008, methodology): “The homological mirror symmetry proofs of Seidel [S1] and Auroux-Katzarkov-Orlov [AKO1, AKO2], formulated in the Fukaya-Seidel version of the mirror, make use of the fact that the mirror categories are generated by a finite collection of objects (Lagrangian thimbles).”
Morse homology, tropical geometry, and homological mirror symmetry for toric varieties (2006, result): “This distinguishes our approach from that taken by Auroux, Katzarkov, and Orlov [2,3] in their proof of the mirror conjecture for Fano surfaces.”
Hypergeometric equations and weighted projective spaces (2006, methodology): “k statement of mirror symmetry between wP and the pencil f: Y → C×. See [Gol01], where MP and MHred are called, respectively, the RiemannRoch and the anticanonical Riemann-Roch D-modules; the article [AKO] contains a much deeper discussion of the case of weighted projective planes. 1.6 Calabi-Yau we”
Gauged Floer Theory Of Toric Moment Fibers (2010, background): “We remark that homological mirror symmetry for the B-model on, in particular, P(1, 3, 5) is proved in Auroux et al [7].”
Homological mirror symmetry for Calabi–Yau hypersurfaces in projective space (2011, unknown): “Complete or partial results in the Fano case are known for toric varieties [13, 14, 15], del Pezzo surfaces [16], and weighted projective planes [17].”
Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves (2005, background): “When qF = 0 (in particular in the toric case) we have fixxfiyyfizz = 0, as in [ 3 ].”