On self-maps of complex flag manifolds

It was conjectured in Glover (Trans Am Math Soc 267:423–434, 1981) that for a complex flag manifold F every endomorphism φ:H∗(F;Z)→H∗(F;Z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi :H^*(F;\mathbb Z)\rightarrow H^*(F;\mathbb Z)$$\end{document} is either a grading endomorphism or a projective endomorphism. In this paper, we verify this conjecture for a new class of complex flag manifolds that captures all cases for which the conjecture was previously known to be true. This allows us to calculate the noncoincidence index (invariant that naturally generalizes the fixed-point property) for these manifolds.