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    The greatest Ricci lower bound, conical Einstein metrics and the Chern number inequality

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    We partially confirm a conjecture of Donaldson relating the greatest Ricci lower bound R(X)R(X) to the existence of conical Kahler-Einstein metrics on a Fano manifold XX. In particular, if D∈∣−KX∣D\in |-K_X| is a smooth simple divisor and the Mabuchi KK-energy is bounded below, then there exists a unique conical Kahler-Einstein metric satisfying Ric(g)=βg+(1−β)[D]Ric(g) = \beta g + (1-\beta) [D] for any β∈(0,1)\beta \in (0,1). We also construct unique smooth conical toric Kahler-Einstein metrics with β=R(X)\beta=R(X) and a unique effective Q-divisor D∈[−KX]D\in [-K_X] for all toric Fano manifolds. Finally we prove a Miyaoka-Yau type inequality for Fano manifolds with R(X)=1R(X)=1
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