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

Denis Auroux, Ludmil Katzarkov, Dmitri Orlov

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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).

Original languageEnglish (US)
Pages (from-to)537-582
Number of pages46
JournalInventiones Mathematicae
Volume166
Issue number3
DOIs
StatePublished - Dec 1 2006

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ASJC Scopus subject areas

  • Mathematics(all)

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