Notices of the International Congress of Chinese Mathematicians

This is an introduction to a particular class of auxiliary complex Monge–Ampère equations which have been instrumental in $L^\infty$ estimates for fully non-linear equations and various questions in complex geometry. The essential comparison inequalities are reviewed and shown to apply in many contexts. Adapted to symplectic geometry, with the auxiliary equation given now by a real Monge–Ampère equation, the method gives an improvement of an earlier theorem of Tosatti–Weinkove–Yau, reducing Donaldson’s conjecture on the Calabi–Yau equation with a taming symplectic form from an exponential bound to an $L^1$ bound.