Brunn--Minkowski Inequality for the First Complex ${\sigma }_2$-Hessian Eigenvalue

There are relatively few results on the convexity of solutions to complex equations. In this paper, We prove a strict real log-concavity theorem for the first eigenfunction of the complex ${\sigma }_2$-Hessian operator on smooth, bounded, real uniformly strictly convex domains in ${\mathbb{C}}^n$. As an application, we obtain a Brunn–Minkowski inequality for the first complex ${\sigma }_2$-Hessian eigenvalue. The proof combines a Bian–Guan constant-rank argument, a new inverse-convexity lemma for the compressed real Hessian, and Salani’s viscosity admissible-test-function method.