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

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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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Authors and Affiliations

  1. Department of Mathematics, M.I.T., Cambridge, 02139, MA, USA

    Denis Auroux

  2. Department of Mathematics, University of Miami, Coral Gables, 33124, FL, USA

    Ludmil Katzarkov

  3. Algebra Section, Steklov Mathematical Institute,, Russian Academy of Sciences, 8 Gubkin str., Moscow, 119991, Russia

    Dmitri Orlov

Authors
  1. Denis Auroux
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  2. Ludmil Katzarkov
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  3. Dmitri Orlov
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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

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