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- Title
Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves.
- Authors
Auroux, Denis; Katzarkov, Ludmil; Orlov, Dmitri
- 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).
- Subjects
ALGEBRAIC surfaces; MIRROR symmetry; ALGEBRAIC cycles; HOMOLOGY theory; ALGEBRAIC topology; COHERENT analytic sheaves; NONCOMMUTATIVE algebras; GEOMETRIC modeling
- Publication
Inventiones Mathematicae, 2006, Vol 166, Issue 3, p537
- ISSN
0020-9910
- Publication type
Article
- DOI
10.1007/s00222-006-0003-4