 
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 CalabiYau
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 LandauGinzburg models.
For us, a LandauGinzburg model is simply a pair (M, W), where M is a noncompact manifold
(carrying a symplectic structure and/or a complex structure), and W is a complexvalued function
