Notices of the International Consortium of Chinese Mathematicians

Volume 6 (2018)

Number 2

Volume minimization and obstructions to solving some problems in Kähler geometry

Pages: 51 – 60



Akito Futaki (Graduate School of Mathematical Sciences, University of Tokyo, Japan)

Hajime Ono (Department of Mathematics, Saitama University, Saitama, Japan)


There is an obstruction to the existence of Kähler–Einstein metrics which is used to define the GIT weight for $K$-stability, and it has been extended to various geometric problems. This survey paper considers such extended obstructions to the existence problem of Kähler–Ricci solitons, Sasaki–Einstein metrics and (conformally) Einstein–Maxwell Kähler metrics. These three cases have a common feature that the obstructions are parametrized by a space of vector fields. We see, in these three cases, the obstructions are obtained as the derivative of suitable volume functionals. This tells us for which vector fields we should try to solve the existence problems.

The first author would like to thank Yau Mathematical Sciences Center at Tsinghua University for its hospitality where this survey paper was completed.

Published 16 May 2019