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MIRROR SYMMETRY FOR DEL PEZZO SURFACES: VANISHING CYCLES AND COHERENT SHEAVES
 

Summary: MIRROR SYMMETRY FOR DEL PEZZO SURFACES:
VANISHING CYCLES AND COHERENT SHEAVES
DENIS AUROUX, LUDMIL KATZARKOV, AND DMITRI ORLOV
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 Xk obtained by blowing up CP2
at k points is equivalent to the derived category of
vanishing cycles of a certain elliptic fibration Wk : Mk 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 Xk, and give an explicit correspon-
dence between the deformation parameters for Xk and the cohomology class [B +i] H2
(Mk, C).
1. Introduction
The phenomenon of mirror symmetry has been studied extensively in the case of Calabi-Yau
manifolds (where it corresponds to a duality between N = 2 superconformal sigma models), but
also manifests itself in more general situations. For example, a sigma model whose target space is
a Fano variety is expected to admit a mirror, not necessarily among sigma models, but in the more
general context of Landau-Ginzburg models.
For us, a Landau-Ginzburg model is simply a pair (M, W), where M is a non-compact manifold
(carrying a symplectic structure and/or a complex structure), and W is a complex-valued function

  

Source: Auroux, Denis - Department of Mathematics, Massachusetts Institute of Technology (MIT)

 

Collections: Mathematics