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

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 ^sub k^ obtained by blowing up ^sup 2^ at k points is equivalent to the derived category of vanishing cyc...

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Bibliographic Details
Published in:Inventiones mathematicae 2006-12, Vol.166 (3), p.537-582
Main Authors: Auroux, Denis, Katzarkov, Ludmil, Orlov, Dmitri
Format: Article
Language:English
Subjects:
General Mathematics
Mathematics
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Summary: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 ^sub k^ obtained by blowing up ^sup 2^ at k points is equivalent to the derived category of vanishing cycles of a certain elliptic fibration W ^sub k^:M ^sub k^[arrow right] 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 ^sub k^, and give an explicit correspondence between the deformation parameters for X ^sub k^ and the cohomology class [B+iω]H ^sup 2^(M ^sub k^,). [PUBLICATION ABSTRACT]
ISSN:0020-9910
1432-1297
DOI:10.1007/s00222-006-0003-4