PomeD95BCCFE2A18The exponent $2$ in corollary 10.25.5 is sharp. On a product torus, two explicit logarithmic profiles give a complete metric with $q+a+b=2$ and incomplete metrics with $q+a+b=2+\beta $ for every $\beta >0$. Both profiles have full mass, and the second one is bounded.
Let $E=\mathbb C/(\mathbb Z+\mathrm{i}\mathbb Z)$, let $\omega _ E=(\mathrm{i}/2\pi )\, \mathrm{d}z\wedge \mathrm{d}\bar z$, and put $X=E^ n$, $D=\{ 0\} \times E^{n-1}$ and $M=X\setminus D$. Write $\pi _ j:X\to E$ for the coordinate projections and $\omega _0=\sum _{j=1}^ n\pi _ j^*\omega _ E$. Choose radii $0<r_1<r_2<\mathrm{e}^{-1}$ and a smooth radial cutoff $\chi $ on the coordinate disk $\{ |z|<r_2\} \subset E$, equal to $1$ on $\{ |z|\leq r_1\} $ and to $0$ near its outer boundary, with $0\leq \chi \leq 1$. For fixed numbers $A,\eta ,\beta >0$, set $L=-\log |z|^2$ and define on $E$
E3D518CD1255
10.25.8
Both functions are extended by zero outside the disk, and at its center we set $v_{\mathrm c}(0)=-\infty $ and $v_\beta (0)=0$. The bounded profile also occurs in example 15.21.4. We write $\mathrm{MA}_\theta (u)$ for the extended non-pluripolar Monge–Ampère measure $\mathrm{MA}^{\mathrm{ext}}_\theta (u)$ of theorem 13.14.3.
Lemma 10.25.9.BCA9F26FE4F4 For $C>0$ sufficiently large, put $\theta =C\omega _0$. For either profile $v$ in (10.25.8), the function $u=\pi _1^*v$ belongs to $\mathrm{PSH}(X,\theta )$, the form $T=\theta +\mathrm{dd}^{\mathrm{c}}u$ is smooth and positive on $X\setminus D$, and
The same $C$ may be used for the two fixed profiles. The bounded profile has minimal ordinary singularities; the profile $v_{\mathrm c}$ does not.
Proof. For a twice differentiable function $G$ of $L=-\log |z|^2$, differentiation away from zero gives
On the inner disk this yields
F60812366089
10.25.10
Both densities are integrable at zero. Both profiles are locally integrable, and their distributional Hessians have no additional atom at zero. To see this, integrate twice by parts outside $\{ |z|=\varepsilon \} $: the boundary terms against a smooth test function are bounded by constant multiples of $\varepsilon |G(-\log \varepsilon ^2)|$ and $|G'(-\log \varepsilon ^2)|$, and both tend to zero for the two profiles. Thus (10.25.10) holds in the sense of distributions across the center. On the cutoff annulus all derivatives are bounded. Hence, for a sufficiently large common $C$,
is a positive measure, smooth and strictly positive away from zero, with no atom at zero. Pairing $\mathrm{dd}^{\mathrm{c}}v$ with the constant test function gives $\int _ E\alpha _ v=C\int _ E\omega _ E$.
Let $Y=E^{n-1}$, let $\pi _ Y:X\to Y$ be the projection onto the remaining factors, and let $\theta _ Y$ be $C$ times the sum of the $n-1$ flat forms of $Y$. On $X\setminus D$ the smooth wedge product is
for $n=1$ this means just $\alpha _ v$, with $Y$ a point. The right side defines a finite product measure on $X$ that gives $D$ zero mass. By theorem 13.14.3, the non-pluripolar measure agrees with this product off $D$ and gives the pluripolar set $D$ zero mass, so the two global measures are equal. Integrating the product proves the full-mass identity. Since $\theta >0$, its envelope is $V_\theta =0$ (section 5.40). The bounded profile therefore has minimal ordinary singularities, whereas $v_{\mathrm c}$ is not bounded below by $0$ minus a constant. $\square$
We apply the preceding results with $\omega =\theta $. Choose a smooth norm on $\mathcal O_ E(0)$, the line bundle of the point $0\in E$, for which its canonical section has squared norm $|z|^2$ near zero; such a norm is obtained from any smooth norm by multiplying by a smooth positive function prescribed on a smaller disk. Pull it back to $X$ and rescale it by a constant. For some fixed $K>0$ we may then arrange
near $D$, and $\tau \geq 1$ on $M$. For both profiles, $E_\tau $ is exactly the tangent space of the remaining torus factors near $D$, so $\det _\theta B_\tau =1$.
Example 10.25.11.36258B2EA152 Fix $\lambda >0$ and $0<A<1/\lambda $, and take $u_{\mathrm c}=\pi _1^*v_{\mathrm c}$ with the background of lemma 10.25.9. This gives a complete metric on $M$ solving
where the right side is extended by zero on $D$. The three exponents in (10.25.6) can be taken to be
so their sum equals the critical exponent $2$.
Indeed, define $f_{\mathrm c}=\mathrm{e}^{-\lambda u_{\mathrm c}}\det _\theta (\theta +\mathrm{dd}^{\mathrm{c}}u_{\mathrm c})$ on $M$. On the inner disk the product structure gives
The substitution $L=-\log r^2$ in the radial integral shows that the second term is integrable precisely when $\lambda A<1$. The first term is integrable as well, and the density is smooth elsewhere. The identity of measures across $D$ follows from the preceding lemma. Since $\tau =L+K$, all three bounds hold with the stated exponents, and completeness follows from corollary 10.25.5. Along a radial normal segment the divergence of the length at the critical exponent is also visible directly:
For arbitrary paths, however, completeness requires the corollary, not just this family of radial paths.
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.
In these examples one may write $f=F/R^2$ with $F=R^2f$ smooth and positive on $M$, but $F$ does not extend smoothly and positively across $D$: the logarithmic factors make it tend to zero there. Thus the examples concern integrable divisorial poles weakened by logarithmic factors. They do not decide completeness for a nonintegrable pole with smooth positive coefficient across the divisor, nor the question of compatibility with the flow discussed in section 25.19.
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