Abstract
We provide an explicit resolution of the existence problem for extremal Kähler metrics on toric 4orbifolds M with second Betti number b_2(M)=2. More precisely we show that M admits such a metric if and only if its rational Delzant polytope (which is a labelled quadrilateral) is Kpolystable in the relative, toric sense (as studied by S. Donaldson, E. Legendre, G. Székelyhidi et al.). Furthermore, in this case, the extremal Kähler metric is ambitoric, i.e., compatible with a conformally equivalent, oppositely oriented toric Kähler metric, which turns out to be extremal as well. These results provide a computational test for the Kstability of labelled quadrilaterals. Extremal ambitoric structures were classified locally in Part I of this work, but herein we only use the straightforward fact that explicit Kähler metrics obtained there are extremal, and the identification of Bachflat (conformally Einstein) examples among them. Using our global results, the latter yield countably infinite families of compact toric Bachflat Kähler orbifolds, including examples which are globally conformally Einstein, and examples which are conformal to complete smooth Einstein metrics on an open subset, thus extending the work of many authors.
Original language  English 

Pages (fromto)  1075–1112 
Number of pages  38 
Journal  Annales Scientifiques de l'École Normale Supérieure 
Volume  48 
Issue number  5 
DOIs  
Publication status  Published  31 Dec 2015 
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David Calderbank
 Department of Mathematical Sciences  Professor
 EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
Person: Research & Teaching