We focus on generalizations of the Witten genus on so-called spin$ ^k $ manifolds (that is, oriented manifolds embeddable into spin manifolds with codimension $ k $), and applications of these generalized genera to vanishing theorems of the Witten genus on related spin manifolds.

We utilize two Dirac operators constructible on a spin$ ^k $ manifold $ M $, one previously constructed by Mayer and one not before considered, to construct two generalized Witten genera on such $ M $. We show that these genera are rigid with respect to particular circle actions on $ M $. We then show that these two Witten genera are equal to the standard Witten genus on certain related spin manifolds $ N $ and $ \widetilde{M} $ constructed geometrically from $ M $: $ N $ a codimension $ k $ submanifold of $ M $ and $ \widetilde{M} $ a branched cover of $ M $. We finally show that the rigidity of our generalized Witten genera on $ M $ implies vanishing theorems for said genera in certain cases where $ M $ admits a group action, which thus gives vanishing theorems for the standard Witten genera on $ N $ and $ \widetilde{M} $.

These vanishing theorems for the standard Witten genus are qualitiatively unlike any of those already known in the literature, in the sense that they don't require the spin manifolds to either be equipped themselves with a Lie group action or to be complete intersections. We show this also provides new evidence for the Stolz conjecture, particularly in the case of certain complete intersections in weighted projective space.