Decouplings for $d$-dimensional surfaces in $\mathbb{R}^{2d}$

Bourgain and Demeter obtained the sharp $l^p$ decoupling for two-dimensional nondegenerate surfaces in $\mathbb{R}^4$. As a generalization of their results, we study the $l^p$ decoupling for $d$-dimensional surfaces in $\mathbb{R}^{2d}$. Especially, we obtain the sharp $l^p$ decoupling for 3-dimensional nondegenerate quadratic surfaces in $\mathbb{R}^6$.