The ordinary-double-point gap conjecture for normalized volumes of klt singularities

Theorem 1.1: for $n\ge2$ and a singular closed point of a normal complex algebraic $n$-fold with $K_X$ $\mathbb Q$-Cartier, klt with zero boundary, $\widehat{\mathrm{vol}}(x,X)\le2(n-1)^n$, with equality iff the germ is analytically an ordinary double point; the point need not be isolated. The companion proves dimension four (bound 162), the base case. Corollary 1.2 gives a strict anticanonical volume bound below $2n^n$ for singular K-semistable $\mathbb Q$-Fano $n$-folds, so $(-K_V)^n=2n^n$ off projective space only for the smooth quadric or $\mathbb P^1\times\mathbb P^{n-1}$. Not shown: nonzero boundary, the next gaps, or sharpness of the singular global bound.

Theorem 1.1: for $n\ge2$ and a singular closed point of a normal complex algebraic $n$-fold with $K_X$ $\mathbb Q$-Cartier, klt with zero boundary, $\widehat{\mathrm{vol}}(x,X)\le2(n-1)^n$, with equality iff the germ is analytically an ordinary double point; the point need not be isolated. The companion proves dimension four (bound 162), the base case. Corollary 1.2 gives a strict anticanonical volume bound below $2n^n$ for singular K-semistable $\mathbb Q$-Fano $n$-folds, so $(-K_V)^n=2n^n$ off projective space only for the smooth quadric or $\mathbb P^1\times\mathbb P^{n-1}$. Not shown: nonzero boundary, the next gaps, or sharpness of the singular global bound.