Pome4F870C2646784F870C264678
To prove completeness near a divisor, one needs a lower bound for the metric in the normal direction. A lower bound for the determinant alone does not give it, because the tangential eigenvalues may absorb the volume. Dividing the determinant by the determinant of the tangential block gives the normal Schur complement. A lower bound for it by the inverse square of a defining function of the divisor, weakened by at most the square of the logarithm of that function, forces completeness. For solutions of exponential equations, this bound follows from bounds on the density, the potential and the tangential determinant, and an example on a torus shows that the logarithmic exponent $2$ is sharp.
In this section $X$ is again a connected compact complex manifold of dimension $n\geq 1$ with a Hermitian form $\omega $. Let $D\subset X$ be a nonempty smooth effective divisor, and put $M=X\setminus D$. Let $s_ D$ be the canonical section of $\mathcal O_ X(D)$, equipped with a smooth Hermitian norm, and set
Rescale the norm so that $\tau \geq 1$ on $M$. Write $J$ for the complex structure. For a smooth positive real $(1,1)$-form $T$ on $M$, we use the Riemannian metric $g_ T(v,v)=\pi T(v,Jv)$ on real tangent vectors $v$. Thus, if $T=(\mathrm{i}/2\pi )\sum H_{j\bar k}\, \mathrm{d}z_ j\wedge \mathrm{d}\bar z_ k$, then $g_ T(v,v)=\sum H_{j\bar k}\, \mathrm{d}z_ j(v)\overline{\mathrm{d}z_ k(v)}$, and Hermitian norms below use this normalization of coefficients.
Near $D$, the form $\partial \tau $ does not vanish (see (10.25.3) below), and we define the tangential hyperplane field, the tangential block and the determinant
Similarly, $\det _\omega B_\tau $ denotes the determinant of $B_\tau $ relative to the restriction of $\omega $ to the complex hyperplane $E_\tau $; when $n=1$, it is $1$.
Proposition 10.25.1.D05A6736BCFF Suppose that there are constants $c>0$, $\tau _0\geq 1$ and $p\leq 2$ such that, on $\{ \tau >\tau _0\} $,
C974FBD89E14
10.25.2
Then $(M,g_ T)$ is complete.
Proof. In a coordinate neighborhood meeting $D$, write $s_ D=z_1 e$, where $e$ is a local holomorphic frame with $|e|^2=\mathrm{e}^{-\psi }$ and $\psi $ is smooth. Then
Covering the compact divisor by finitely many such neighborhoods and comparing with the smooth background, we obtain, after increasing $\tau _0$,
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10.25.3
for positive constants $c_1,C_1$. In particular, $E_\tau $ is a hyperplane there.
At a point of this set, choose an $\omega $-unitary frame whose last $n-1$ vectors span $E_\tau $. Let $H$ be the Hermitian matrix that represents $g_ T$ as the quadratic form $w^*Hw$ in the complex components $w$, and write
where $a$ is a column vector and $a^*$ its conjugate transpose. Completing the square gives, for a vector with components $(\xi ,\eta )$,
For $n=1$ the same identity holds without the second term. For the complex components of a real vector $v$ we have $|\xi |=|\partial \tau (v)|/|\partial \tau |_\omega $, so (10.25.2) and (10.25.3) imply
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10.25.4
where $c_2>0$ is uniform. The last inequality uses $\mathrm{d}\tau (v)=2\operatorname {Re}\partial \tau (v)$.
Define a continuous function $\Phi _ p$ on $[1,\infty )$ by
Integrating (10.25.4) along piecewise smooth curves shows that $\Phi _ p\circ \tau $ is Lipschitz for the length distance of $g_ T$; this also holds for curves crossing $\{ \tau =\tau _0\} $, since $\Phi _ p$ is locally Lipschitz and constant below that level. Since $p\leq 2$, $\Phi _ p(t)$ tends to infinity with $t$, so $\Phi _ p\circ \tau $ is a proper function on $M$.
A Cauchy sequence for the length distance has bounded values of this Lipschitz function and hence lies in a compact subset of $M$. It therefore has a subsequence converging in the manifold topology. On a coordinate ball about the limit, the smooth positive form $T$ is comparable to the Euclidean metric, so this subsequence also converges for the length distance. The Cauchy property then gives convergence of the whole sequence, which proves completeness. $\square$
Corollary 10.25.5.A12B874B6590 In the preceding setting, let $\theta $ be a smooth real $(1,1)$-form on $X$, and let $u\in \mathrm{PSH}(X,\theta )$ be smooth on $M$, with $T=\theta +\mathrm{dd}^{\mathrm{c}}u>0$ there. Suppose that, for some $\lambda >0$ and some positive $f\in C^\infty (M)\cap L^1(X,\omega ^ n)$,
Assume that, near $D$, there are constants $c,C>0$, $q>1$, $a\geq 0$ and $b\in \mathbb R$ such that
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10.25.6
If $q+a+b\leq 2$, then $(M,g_ T)$ is complete.
Proof. The equation and the first two inequalities give $\det _\omega T\geq c\mathrm{e}^{-\lambda C}R^{-2}\tau ^{-(q+a)}$. Dividing by the third bound gives (10.25.2) with $p=q+a+b$, and proposition 10.25.1 applies. $\square$
Since $u$ is bounded above on the compact manifold $X$, integrability of $f$ also makes the measure $\mathrm{e}^{\lambda u}f\omega ^ n$ finite. The relevant tangential determinant is that of the intrinsic hyperplane field $E_\tau $, not that of an arbitrary block of coordinate directions.
D95BCCFE2A18The 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 $E$
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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
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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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