The greatest Ricci lower bound, conical Einstein metrics and Chern number inequality (Q258880)

Let \(X\) be an \(n\)-dimensional compact Fano manifold and \(D\) a smooth divisor in \(X\) given in a local chart by \(\{ z_n =0 \} .\) The singular Kähler form NEWLINE\[NEWLINE\omega = \frac{i}{2}\sum _{j=1}^{n-1} dz_j \wedge d\bar{z}_j + |z_n |^{2\beta -1} dz_n \wedge d\bar{z}_nNEWLINE\]NEWLINE is called conical with cone angle \(2\pi\beta \) along \(D\). The following conical Kähler-Einstein equation was proposed by Donaldson NEWLINE\[NEWLINE\mathrm{Ric} (\omega ) =\beta\omega +(1-\beta )[D]NEWLINE\]NEWLINE for \(D\) in the anticanonical class of \(X\) and \(\beta \in (0,1)\). Donaldson conjectured that the equation has no solution if \(\beta \in (R(X),1]\), and that there exist conical Kähler-Einstein metrics for \(\beta \in (0, R(X) )\). Here the holomorphic invariant \(R(X)\) has been defined by Tian as NEWLINE\[NEWLINER(X) =\sup \big\{ \beta :\mathrm{Ric} (\omega ) \geq \beta\omega \text{ for some Kähler } \omega \in c_1 (X) \big\}.NEWLINE\]NEWLINE In this paper, the authors partially confirm the conjecture. They consider conical metrics along divisors from \(|-m K_X |\) (\(m\) a positive integer) of the anticanonical class. Thus for \(\beta \in (R(X),1)\) and \(D \in |-m K_X |\) there does not exist a conical Kähler-Einstein metric \(\omega\) such that NEWLINE\[NEWLINE\mathrm{Ric} (\omega ) =\beta\omega +\frac{1-\beta }{m} [D] .NEWLINE\]NEWLINE On the other hand if \(\beta \in (0, R(X) )\) then there exists a smooth \(D \in |-m K_X |\) for some positive \(m\) and a conical Kähler-Einstein metric \(\omega\) satisfying the above equation.NEWLINENEWLINEOne would like to get rid of the dependence of \(m\) on \(\beta \) in this statement. This is indeed possible under the assumption that the Mabuchi K-energy is bounded below.NEWLINENEWLINESimilar results have been obtained independently by \textit{C. Li} and \textit{S. Sun} [Commun. Math. Phys. 331, No. 3, 927--973 (2014; Zbl 1296.32008)].NEWLINENEWLINEThe next statement deals with toric Fano manifolds. It says that there exist an effective toric \(\mathbb Q\) divisor \(D\in |- K_X |\) (unique for \(R(X)<1\)) and a smooth toric conical Kähler metric \(\omega \) (unique up automorphisms) satisfying NEWLINE\[NEWLINE\mathrm{Ric} (\omega ) =R(X)\omega +(1-R(X) )[D].NEWLINE\]NEWLINE The number \(R(X)\) is the largest possible with this property.NEWLINENEWLINEFinally the authors show that for \(X\) Fano with \(R(X)=1\), the Miyaoka-Yau type inequality NEWLINE\[NEWLINEc_2 (X)c_1 (X) ^{n-2} \geq \frac{n}{2(n+1)} c_1 (X) ^nNEWLINE\]NEWLINE holds.