Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves
By D. Auroux, L. Katzarkov, Dmitri Orlov. 2005
216 citations recorded by the source corpus.
We study homological mirror symmetry for Del Pezzo surfaces and their mirror Landau-Ginzburg models. In particular, we show that the derived category of coherent sheaves on a Del Pezzo surface Xk obtained by blowing up ℂℙ2 at k points is equivalent to the derived category of vanishing cycles of a certain elliptic fibration Wk:Mk→ℂ with k+3 singular fibers, equipped with a suitable symplectic form. Moreover, we also show that this mirror correspondence between derived categories can be extended to noncommutative deformations of Xk, and give an explicit correspondence between the deformation parameters for Xk and the cohomology class [B+iω]∈H2(Mk,ℂ).
Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/97ebc991faccddae1db8847652a8d84cb023cc6e
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Citations in the corpus
- 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.”
- 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).”
- Birational models of moduli spaces of coherent sheaves on the projective plane (2016, methodology): “We first recall the following definitions from [3; 20; 31].”
- 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.”
- 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].”
- 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).”
- Mirror symmetry and T-duality in the complement of an anticanonical divisor (2007, methodology): “The best understood case so far is that of Del Pezzo surfaces [4]; however, in that example the construction of the mirror is motivated by entirely ad hoc considerations.”
- An update on semisimple quantum cohomology and F-manifolds (2008, background): “7 produces a weak version of the Landau–Ginzburg model for such manifolds and thus gives a partial solution of the mirror problem for them: compare with a much more detailed pictures in [1, 18].”
- Fukaya A∞-structures Associated to Lefschetz Fibrations. II (2014, background): “than our previous observation). Instead of addressing this by direct computation, we will give a mirror symmetry explanation, which is not quite rigorous (it is a degenerate version of arguments from [4]). For a complex number q, 0 <jqj<1, consider the following elliptic curve Y q together wit”
- SMOOTH AND PROPER NONCOMMUTATIVE SCHEMES AND GLUING OF DG CATEGORIES (2014, background): “projective plane have been described in [ATV]. The category Perf–P2 has a full exceptional collection pO,Op1q,Op2qq. Note also that mirror symmetry relations for noncommutative planes is described in [AKO]. Any deformation of the category Perf–P2 is a category with three ordered objects F0,F1,F2 and”