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The phenomenon of Mirror Symmetry, in its “classical” version, was first observed for Calabi-Yau manifolds, and mathematicians were introduced to it through a series of remarkable papers [20], [13], [38], [40], [15], [30]. Some very strong conjectures have been made about its topological interpretation – e.g. the Strominger-Yau-Zaslow conjecture. In a different direction, the framework of mirror symmetry was extended by Batyrev, Givental, Hori, Vafa, etc. to the case of Fano manifolds. | Annals of Mathematics Mirror symmetry for weighted projective planes and their noncommutative deformations By Denis Auroux Ludmil Katzarkov and Dmitri Orlov Annals of Mathematics 167 2008 867 943 Mirror symmetry for weighted projective planes and their noncommutative deformations By Denis ÀUROUX Ludmil Katzarkov and Dmitri Orlov Contents 1. Introduction 2. Weighted projective spaces 2.1. Weighted projective spaces as stacks 2.2. Coherent sheaves on weighted projective spaces 2.3. Cohomological properties of coherent sheaves on Pfl a 2.4. Exceptional collection on Pfl a 2.5. A description of the derived categories of coherent sheaves on Pfl a 2.6. DG algebras and Koszul duality. 2.7. Hirzebruch surfaces Fra 3. Categories of Lagrangian vanishing cycles 3.1. The category of vanishing cycles of an affine Lefschetz fibration 3.2. Structure of the proof of Theorem 1.2 3.3. Mirrors of weighted projective lines 4. Mirrors of weighted projective planes 4.1. The mirror Landau-Ginzburg model and its fiber s0 4.2. The vanishing cycles 4.3. The Floer complexes 4.4. The product structures 4.5. Maslov index and grading 4.6. The exterior algebra structure 4.7. Nonexact symplectic forms and noncommutative deformations 4.8. B-fields and complexified deformations 5. Hirzebruch surfaces 5.1. The case of F0 and Fl 5.2. Other Hirzebruch surfaces 6. Further remarks 6.1. Higher-dimensional weighted projective spaces 6.2. Noncommutative deformations of CP2 6.3. HMS for products References 868 DENIS AUROUX LUDMIL KATZARKOV AND DMITRI ORLOV 1. Introduction The phenomenon of Mirror Symmetry in its classical version was first observed for Calabi-Yau manifolds and mathematicians were introduced to it through a series of remarkable papers 20 13 38 40 15 30 . Some very strong conjectures have been made about its topological interpretation - e.g. the Strominger-Yau-Zaslow conjecture. In a different direction the framework of mirror symmetry was extended by Batyrev Givental Hori Vafa etc. to the .
