FORMALITY FOR ALGEBROIDS II: FORMALITY THEOREM FOR GERBES PAUL BRESSLER, ALEXANDER GOROKHOVSKY, RYSZARD NEST, AND BORIS TSYGAN Abstract. We extend the formality theorem of M. Kontsevich from deformations of the structure sheaf on a manifold to deformations of gerbes. Contents 1. Introduction 2 2. Formality 3 2.1. Hochschild cochains 4 2.2. Outline of the proof of formality for cochains 4 2.3. Some (super-)symmetries 5 2.4. Extensions and generalizations 5 2.5. Deformations of O and Kontsevich formality 6 3. Calculus in the presence of distribution 6 3.1. Complex distributions 6 3.2. The Hodge filtration 6 3.3. ∂ -operators 7 3.4. Calculus 7 3.5. Jets 8 4. Formality for the algebroid Hochschild complex 10 4.1. Hochschild cochains in formal geometry 10 4.2. Formality for jets 10 4.3. Formality for jets with a twist 11 5. L ∞ -structures on multivectors 12 5.1. L ∞ -deformation complex 12 5.2. L ∞ -structures on multivectors 12 5.3. L ∞ -structures on multivectors via formal geometry 13 5.4. Dolbeault complexes 14 5.5. Formal geometry vs. Dolbeault 15 6. Deformations of algebroid stacks 16 6.1. Algebroid stacks 16 6.2. Twisted forms of O 17 6.3. Deformations of linear stacks 17 6.4. Deformations of twisted forms of O 18 References 19 A. Gorokhovsky was partially supported by NSF grant DMS-0900968, R. Nest was partially supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92), B. Tsygan was partially supported by NSF grant DMS-0906391. 1