The greatest Ricci lower bound, conical Einstein metrics and Chern number inequality
type: Item
instance of: scholarly article
title: The greatest Ricci lower bound, conical Einstein metrics and Chern number inequality
zbMATH Open document ID: 1342.32018
published in: Geometry \& Topology
publication date: 2016-03-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/1207.4839
Mathematics Subject Classification ID: 32Q20
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 14J45
Mathematics Subject Classification ID: 14M25
zbMATH DE Number: 6553428
zbMATH Keywords: conical metrics
zbMATH Keywords: Fano manifolds
zbMATH Keywords: toric manifolds
zbMATH Keywords: Kähler-Einstein equation
reviewed by: Slawomir Kołodziej
MaRDI profile type: Publication
OpenAlex ID: W3103482135
arXiv ID: 1207.4839
Wikidata QID: Q125995710
DOI: 10.2140/GT.2016.20.49
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review text: 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 …
description: scientific article; zbMATH DE number 6553428