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Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves

Abstract

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 X k obtained by blowing up ℂℙ2 at k points is equivalent to the derived category of vanishing cycles of a certain elliptic fibration W k :M k →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 X k , and give an explicit correspondence between the deformation parameters for X k and the cohomology class [B+iω]∈H 2(M k ,ℂ).

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Correspondence to Denis Auroux, Ludmil Katzarkov or Dmitri Orlov.

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Auroux, D., Katzarkov, L. & Orlov, D. Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves . Invent. math. 166, 537–582 (2006). https://doi.org/10.1007/s00222-006-0003-4

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  • DOI: https://doi.org/10.1007/s00222-006-0003-4

Keywords

  • Line Bundle
  • Symplectic Form
  • Theta Function
  • Cohomology Class
  • Pezzo Surface