ℙ-objects and autoequivalences of derived categories

We describe new autoequivalences of derived categories of coherent sheaves arising from what we call ℙn-objects of the category. Standard examples arise from holomorphic symplectic manifolds. Under mirror symmetry these autoequivalences should be mirror to Seidel's Dehn twists about lagrangian ℙn submanifolds. We give various connections to spherical objects and spherical twists, and include a simple description of Atiyah and Kodaira-Spencer classes in an appendix.

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