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Summary: MIRROR SYMMETRY FOR WEIGHTED PROJECTIVE PLANES AND THEIR
NONCOMMUTATIVE DEFORMATIONS
DENIS AUROUX, LUDMIL KATZARKOV, AND DMITRI ORLOV
Contents
1. Introduction 2
2. Weighted projective spaces 5
2.1. Weighted projective spaces as stacks 5
2.2. Coherent sheaves on weighted projective spaces 6
2.3. Cohomological properties of coherent sheaves on P(a) 8
2.4. Exceptional collection on P(a) 10
2.5. A description of the derived categories of coherent sheaves on P(a) 13
2.6. DG algebras and Koszul duality. 15
2.7. Hirzebruch surfaces Fn 19
3. Categories of Lagrangian vanishing cycles 20
3.1. The category of vanishing cycles of an affine Lefschetz fibration 20
3.2. Structure of the proof of Theorem 1.2 22
3.3. Mirrors of weighted projective lines 24
4. Mirrors of weighted projective planes 26
4.1. The mirror Landau-Ginzburg model and its fiber 0 26
4.2. The vanishing cycles 29
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