Interior $C^2$ estimate for Hessian quotient equation in general dimension

In this paper, we study the interior $C^2$ regularity problem for the Hessian quotient equation $\left(\frac{{\sigma }_n}{{\sigma }_k}\right)(D^{2}u)=f$. We give a complete answer to this longstanding problem: for $k=n-1,n-2$, we establish an interior $C^2$ estimate; for $k\le n-3$, we show that interior $C^2$ estimate fails by finding a singular solution.