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

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

2005-06-09

arxiv.org/abs/math/0506166v1

Invent. Math. 166 (2006), no. 3, 537--582

Mathematics

Algebraic Geometry

40 pages, 9 figures

Scientific paper

10.1007/s00222-006-0003-4

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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