$\mathbb{P}$-objects and autoequivalences of derived categories

We describe new autoequivalences of derived categories of coherent sheaves arising from what we call ${\mathbb{P}}^n$-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 ${\mathbb{P}}^n$ 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.