Recently, Y. Xu has shown Holder regularity for weak solutions to complex $m$-Hessian equations on compact Hermitian manifolds. We present a new, effective proof of this result, using a two-point maximum principle argument. We then demonstrate how to go further, and obtain optimal Holder regularity on compact Kahler manifolds, when the right-hand side is bounded and $m\geq 3$; this is new even for the complex Monge-Ampere equation. For $m=2$ or right-hand sides in $L^p$, the method is nearly optimal, giving all Holder exponents below the best possible one.

Bio: Nicholas McCleerey earned his doctorate from Northwestern University in 2020, under the supervision of Valentino Tosatti, and was then a postdoctoral professor at the University of Michigan, under Mattias Jonsson. Since 2023, he is an assistant professor at Purdue University. His research is in complex geometry and PDEs, with a focus on pluripotential theory for complex Hessian and Monge-Ampere equations.