Inventiones mathematicae

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

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

Article

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 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).

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Copyright information

© 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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