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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 CP^2 at k points is equivalent to the derived category of vanishing cycles of a certain elliptic fibration W_k:M_k\to\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+i\omega]\in H^2(M_k,C).

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Denis Auroux, Ludmil Katzarkov, Dmitri Orlov. 2005-06-09. Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves. https://doi.org/10.1007/s00222-006-0003-4

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