Local regularity for concave homogeneous complex degenerate elliptic equations comparable to the Monge-Amp\`ere equation

In this paper, we establish a local regularity result for $W^{2,p}_{\mathrm{loc}}$ solutions to complex degenerate nonlinear elliptic equations $F(D^2_{\mathbb{C}} u)=f$ when they are comparable to the Monge-Amp\`ere equation. Notably, we apply our result to the so-called $k$-Monge-Amp\`ere equation.