Example 10.25.122D0F50D560BC

Example 10.25.12.2D0F50D560BC Fix $\lambda ,\eta ,\beta >0$ and take $u_\beta =\pi _1^*v_\beta $ with the background of lemma 10.25.9. This is a continuous bounded solution of full mass with minimal ordinary singularities, and its metric on $M$ is incomplete, although

To see this, put $K_\beta =\eta \beta (\beta +1)$ and define the density by the equation on $M$. Near $D$ it is

It is integrable, and the global identity follows as before. Fix the other coordinates and let $z_1=r\downarrow 0$ along a radial segment starting at $r_0$, with $0<r_0<r_1$. The length of this segment is at most

Points along this segment approaching $D$ form a Cauchy sequence with no limit in $M$.

Here the smallest admissible exponents in (10.25.6) are $q=\beta +2$, $a=0$ and $b=0$: the density is asymptotic to a positive constant times $R^{-2}\tau ^{-(\beta +2)}$, the potential is bounded, and the tangential determinant is $1$. Their sum is $2+\beta $. Since $\beta >0$ is arbitrary, the threshold $2$ in the criterion cannot be replaced by any larger universal number, even in a Kähler class with a bounded full-mass solution.