Note on black hole no hair theorems for massive forms and spin-$1/2$ fields

We give a proof of the nonperturbative no hair theorems for a massive 2-form field with 3-form field strength for general stationary axisymmetric and static (anti)–de Sitter or asymptotically flat black hole spacetimes with some suitable geometrical properties. The generalization of this result for higher form fields is discussed. Next, we discuss the perturbative no hair theorems for massive spin-$1/2$ fields for general static backgrounds with electric or magnetic charge. Some generalization of this result for stationary axisymmetric spacetimes are also discussed. All calculations are done in arbitrary spacetime dimensions.