Ambitoric geometry. II: Extremal toric surfaces and Einstein 4-orbifolds (Q2788602)

The existence of extremal Kähler metrics (and its relation to \(K\)-polystability) is an extremely difficult and very actively studied subject. In this paper the authors consider the existence of extremal Kähler metrics for toric symplectic \(4\)-orbifolds \((M, \omega, T)\) whose rational Delzant polytope is quadrilateral (which holds for example if \(b_2(M)=2\)). They prove that \(M\) admits a \(T\)-invariant extremal Kähler metric if and only if \((M, \omega, T)\) is analytically relatively \(K\)-polystable with respect to toric degenerations. They also prove that in this case the extremal metric is ambitoric, i.e., compatible with a conformally equivalent, oppositely oriented toric Kähler metric. Combining this result with their earlier work [``Ambitoric geometry I: Einstein metrics and extremal ambikähler structures'', J. Reine Angew. Math. (to appear)] the authors produce infinite families of compact toric Bach-flat Kähler orbifolds, including examples which are globally conformally Einstein, and examples which are conformal to complete smooth Einstein metrics on an open subset.