An Aubin continuity path for shrinking gradient Kähler–Ricci solitons

Let 𝐷 be a toric Kähler–Einstein Fano manifold. We show that any toric shrinking gradient Kähler–Ricci soliton on certain toric blowups of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="double-struck">C</m:mi> <m:mo lspace="0.222em" rspace="0.222em">×</m:mo> <m:mi>D</m:mi> </m:mrow> </m:math> C× D satisfies a complex Monge–Ampère equation. We then set up an Aubin continuity path to solve this equation and show that it has a solution at the initial value of the path parameter. This we do by implementing another continuity method.