The $(n-2)$-linear restriction estimate in ${\mathbb{R}}^n$

We present a proof for the $(n-2)$-linear extension estimate for curved surfaces in ${\mathbb{R}}^n$, $n\ge 5$, at the local product exponent $2(n-1)\mathrm{/}(n-2{)}^2$, with an arbitrarily small power loss. The hypotheses are transversality of the normals and nondegeneracy of the compressed shape operators on their common two-dimensional orthogonal complement. The argument combines polynomial partitioning, a descent through algebraic varieties, localized multilinear restriction, and positive packet-energy estimates.