We show the rigidity of Sp(m+1)-equivariant harmonic maps from the quaternionic projective space HPm into a complex Grassmann manifold Grp(Cp+q) when q is less than 2m. Our method depends on the differential geometry of vector bundles and connections; a generalization of the do Carmo–Wallach theory based on a generalization of Takahashi’s theorem implies the result. We also obtain the rigidity of those maps into the real or quaternionic Grassmannians.
Given a finitely generated module M over a Noetherian local ring R, we give a characterization for the first syzygy of the associated graded module Gm(M) to be equigenerated. As an application of this, we identify a complex of free Gm(R)-modules arising from given free resolution of M over R, which is a resolution of Gm(M) if and only if Gm(M) is a pure Gm(R)-module. We also give several applications of the purity of Gm(M). Our results demonstrate that while not all algebraic properties of a module carry over to its associated graded module, the purity of the minimal free resolution of Gm(M) ensures that several important invariants are inherited. In addition, we provide sufficient conditions for Cohen–Macaulayness and purity of Gm(M), and we provide a local version of the Herzog–Kühl equations.
Inspired by the recent work of Wu, we extend the classical compactness theorem for complete Riemannian manifolds via Ricci curvature due to Cheeger, Gromov, and Taylor by replacing their asymptotic condition on the Ricci curvature by some upper bounds on integral radial m-Bakry–Émery Ricci curvatures when m is a positive constant, a negative constant, or infinity. Our results generalize the Cheeger–Gromov–Taylor-type compactness theorems for complete Riemannian manifolds via m-Bakry–Émery Ricci curvatures obtained by Soylu and the author to the case of integral settings.
