Coherent sheaves and categorical sl₂ actions

We introduce the concept of a geometric categorical sl2 action and relate it to that of a strong categorical sl2 action. The latter is a special kind of 2-representation in the sense of Lauda and Rouquier. The main result is that a geometric categorical sl2 action induces a strong categorical sl2 action. This allows one to apply the theory of strong sl2 actions to various geometric situations. Our main example is the construction of a geometric categorical sl2 action on the derived category of coherent sheaves on cotangent bundles of Grassmannians.