Twisted Kähler-Einstein metrics on flag varieties (Q6648895)

Consider a generalized flag manifold \(X=G/P\), that is, a projective, Fano even, variety which is homogeneous under a semisimple complex Lie group \(G\). It is well-known that the Kähler cone of \(X\) may be described Lie-theoretically in terms of a cone generated by of certain dominant weights of \(G\) associated with the (parabolic) subgroup \(P\). Let now \(K\) be a maximal compact subgroup of \(G\), then it is well-known that any \((1,1)\)-class \(\beta\) on \(X\) contains a unique \(K\)-invariant \((1,1)\)-form \(\omega_{\beta}\), which is Kähler if and only if \(\beta\) is. Any \(K\)-invariant Kähler form \(\omega\) on \(X\) thus has Ricci form equal to \(\omega_{c_1(X)}\). These observations were used in 1972 [\textit{Y. Matsushima}, Nagoya Math. J. 46, 161--173 (1972; Zbl 0249.53050)] to show that \(X\) admits a \(K\)-invariant Kähler-Einstein metric, unique up to scaling.\N\NIn the paper under review, the authors derive from the above that a \(K\)-invariant Kähler form \(\omega_{\alpha}\) in a Kähler class \(\alpha\) on \(X\) satisfy a form of twisted Kähler-Einstein equation\N\[\N\omega_{c_1(X)}=\operatorname{Ric}(\omega_{\alpha})= \omega_{\alpha}+\omega_{c_1(X)-\alpha}\N\]\NFrom this they derive the equality, in this setting, of the nef threshold, that is, the sup of \(s\) such that \(c_1(X)-s\alpha\) is Kähler, with the greatest Ricci lower bound \(R(\alpha)\) of \(\alpha\), and they compute its expression Lie-theoretically. Using this expression, and the well-known Lie-theoretical formula for the degree of a Kähler class on \(X\) the authors prove:\N\[\NR(\alpha)^n \alpha^n \leq (c_1(X))^n \leq (n+1)^n\N\]