We obtain a residue formula for an obstruction to the existence of coupled Kähler–Einstein metrics described by Futaki–Zhang. We apply it to an example studied separately by Futaki and Hultgren which is a toric Fano manifold with reductive automorphism, does not admit a Kähler–Einstein metric but still admits coupled Kähler–Einstein metrics.
The work of the second author was partially supported by Tsinghua University Initiative Scientific Research Program.
"Residue formula for an obstruction to coupled Kähler–Einstein metrics." J. Math. Soc. Japan 73 (2) 389 - 401, April, 2021. https://doi.org/10.2969/jmsj/83558355