Theorem 14.29.1DD001DBBE659

Theorem 14.29.1.DD001DBBE659 On $X=\mathbb P^3$, with block homogeneous coordinates $[A:B]$ and Fubini–Study form $\omega _ X$, put

There exist a smooth nonclosed Hermitian form $\omega $ and distinct smooth $\omega $-plurisubharmonic functions $u,v$, normalized by $\sup _ Xu=\sup _ Xv=0$, with

Both solution forms are positive away from the two connected real analytic hypersurfaces $\{ t=a\} $, $\{ t=b\} $, and have complex rank two on them. The density is real analytic on $X$, and it vanishes exactly to second order in the direction transverse to these hypersurfaces. For every $n\geq 4$ the same assertions hold on $\mathbb {P}^3\times \mathbb {P}^{n-3}$, with the pulled-back density and the product Fubini–Study volume, and with rank $n-1$ on the two product hypersurfaces.