Inventiones mathematicae

, Volume 166, Issue 3, pp 537–582 | Cite as

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

  • Denis Auroux
  • Ludmil Katzarkov
  • Dmitri Orlov


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).


Line Bundle Symplectic Form Theta Function Cohomology Class Pezzo Surface 
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© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of MathematicsM.I.T.CambridgeUSA
  2. 2.Department of MathematicsUniversity of MiamiCoral GablesUSA
  3. 3.Algebra Section, Steklov Mathematical Institute,Russian Academy of SciencesMoscowRussia

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