14.29.14 Zeros and higher dimensionsE82948C4DB2A

It remains to examine the regularity of the density and the geometry of its zero levels. Taking a product with projective space then preserves the equality of the Monge–Ampère measures and gives examples in every higher dimension.

The function $t$ is real analytic in every affine chart of $X$, being a quotient of real polynomials with positive denominator. Hence $f_*(t)$ is real analytic on $X$. Its two zero levels are regular hypersurfaces because $t=\sigma (r)$, $\sigma '>0$, and $\mathrm{d}r\ne 0$. Each level is connected, since it is the quotient of the product of two three-spheres of fixed nonzero radii by the diagonal action of the circle. Each level is compact and has zero volume. The divided differences in (14.29.5) show near the two levels that

respectively, with smooth strictly positive functions $g_\pm $.

For $n\geq 4$, let $\theta $ be the Fubini–Study form on $Y=\mathbb {P}^{n-3}$, let $\pi _1:X\times Y\to X$ and $\pi _2:X\times Y\to Y$ be the projections, and set, on $X\times Y$,

In the expansion of an $n$-fold product, only the terms with exactly three factors pulled back from $X$ survive. Consequently

and the same holds for $v_ n$. On the two product levels, the solution forms have rank $2+(n-3)=n-1$. Restriction to a slice $X\times \{ y\} $ recovers $\mathrm{d}\omega \ne 0$, so $\omega _ n$ is not closed. Normalization, distinctness and real analyticity are preserved by pullback.

Write $\operatorname {Vol}$ for the volume with respect to $\omega _ X^3$, or $\Omega _ n^ n$ in dimension $n$. The quadratic zeros complete the proof of the sharpness assertion in 14.21.13. In finitely many tubular charts around the zero levels, the density is bounded above and below by positive multiples of the square of a real transverse coordinate, and the volume form has a bounded positive density with respect to coordinate volume. Away from these charts, the density is bounded below by a positive constant. Thus, for $s>0$ and $\alpha >0$,

Thus, for every $0<\alpha <1/2$ and every dimension $n\geq 3$, there are smooth examples of nonuniqueness whose density $f$ satisfies $f^{-\alpha }\in L^1$; this condition is weaker than the hypothesis $f^{-1/2}\in L^1$ of 14.21.13.