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5.34 Analytic types and restriction0D77FC17B23B

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The simplest singularities of quasi-plurisubharmonic functions are logarithmic poles along the common zeros of finitely many holomorphic functions. The logarithmic growth of a general singularity at a point is measured by its Lelong number. Let $u$ be a quasi-plurisubharmonic function on a complex manifold and let $x$ be a point. Choose holomorphic coordinates centered at $x$ and a smooth function $q$ such that $p=u+q$ is plurisubharmonic near $x$, and write $B(x,r)$ for the Euclidean coordinate ball of radius $r$. The Lelong number of $u$ at $x$ is

\[ \nu (u,x)=\lim _{r\downarrow 0} \frac{\sup _{B(x,r)}p}{\log r^2}. \]

By the local growth theorem recalled in Proposition 6.7.1, the limit exists and does not depend on the coordinates or on the correction $q$. With this normalization, $\log |z|^2$ has Lelong number one at the origin. For an irreducible analytic subset $Z$, the generic Lelong number of $u$ along $Z$ is

\[ \nu (u,Z)=\inf _{x\in Z}\nu (u,x). \]

By Siu's theorem, recalled in 5.34.2, $\nu (u,x)=\nu (u,Z)$ for all $x\in Z$ outside a countable union of proper analytic subsets of $Z$.

For analytic singularities we use the convention of Definition 6.14.5: we say that a quasi-plurisubharmonic function $u$ has analytic singularities if near every point there are holomorphic functions $f_1,\ldots ,f_ N$, not all identically zero, and a rational number $c>0$ such that

\[ u=c\log \bigl (|f_1|^2+\cdots +|f_ N|^2\bigr )+O(1), \qquad c\in \mathbb Q_{>0}. \]

The remainder is only required to be bounded; if it can be chosen smooth, we say that $u$ has analytic singularities with smooth remainder. Given a coherent ideal sheaf $\mathfrak a$ and a common coefficient $c$, we say that $u$ has analytic type $(c,\mathfrak a)$ if this holds with local generators $f_1,\ldots ,f_ N$ of $\mathfrak a$. The choice of generators does not matter: two finite systems of generators of the same ideal change the logarithm only by a bounded amount on smaller relatively compact open sets.

Proposition 5.34.1.AAD8908A0CF9 Analytic singularities are preserved by addition, finite maxima, and multiplication by positive rational numbers. Let $u$ have analytic type $(c,\mathfrak a)$, and let $Y\subset X$ be a connected complex submanifold on which $\mathfrak a$ does not vanish identically. Then the restriction $u|_ Y$ is a quasi-plurisubharmonic function on $Y$, in particular not identically $-\infty $, and it has analytic type

\[ (c,\mathfrak a\mathcal O_ Y). \]

Here $\mathfrak a\mathcal O_ Y$ is the image ideal, generated by the restrictions to $Y$ of local sections of $\mathfrak a$.

Proof. For multiplication by a positive rational number, multiply the coefficient. Consider two presentations with coefficients $c_1,c_2$ and ideals $\mathfrak a_1,\mathfrak a_2$, and choose a positive rational number $d$ with $c_1=k d$ and $c_2=\ell d$, where $k,\ell $ are positive integers. Raising a sum of squared moduli of generators of $\mathfrak a_1$ to the power $k$ gives, up to bounded factors, the corresponding sum for $\mathfrak a_1^ k$; similarly for $\mathfrak a_2^\ell $. Products of generators of $\mathfrak a_1^ k$ and $\mathfrak a_2^\ell $ therefore present the sum with coefficient $d$, since a product of sums of squares is a sum of squares of pairwise products.

For the maximum, apply the elementary inequalities

\[ \max \{ A,B\} \leq A+B\leq 2\max \{ A,B\} ,\qquad A,B\geq 0, \]

to the two sums of squared moduli of generators of $\mathfrak a_1^ k$ and $\mathfrak a_2^\ell $. Taking logarithms shows that the maximum differs from $d$ times the logarithm of their total by a bounded amount. These are local assertions; on the compact manifold, finitely many charts give the corresponding comparisons of ordinary singularity types.

For the restriction, restrict to $Y$ the two-sided bounds that define the remainder. Since $Y$ is connected and is not contained in the zero set of $\mathfrak a$, the common zero set of the restricted generators is a proper analytic subset of $Y$, so $u|_ Y$ is finite on a dense open subset of $Y$. After a smooth local correction, the restriction theorem for plurisubharmonic functions shows that $u|_ Y$ is quasi-plurisubharmonic, and the restricted bounds give the asserted presentation. $\square$

For an arbitrary quasi-plurisubharmonic function, a vanishing generic Lelong number along a submanifold does not prevent the restriction from being identically $-\infty $. For example, on a small polydisk in $\mathbb C^ n$ with coordinates $z=(z_1,\ldots ,z_ n)$ and $|z_1|<\mathrm{e}^{-1}$, the function

\[ u(z)=-\sqrt{-\log |z_1|^2} \]

is plurisubharmonic, because $t\mapsto -\sqrt{-t}$ is increasing and convex for $t<0$. Its Lelong number vanishes at every point of $\{ z_1=0\} $, but its restriction to this hyperplane is identically $-\infty $. For the analytic approximants constructed below, the analytic hypothesis in proposition 5.34.1 excludes this behavior.

5.34.2 Lelong level setsE911F3E2C8E9

Siu's analyticity theorem states that, for every quasi-plurisubharmonic function $u$ and every $c>0$, the upper level set $\{ x:\nu (u,x)\geq c\} $ is an analytic subset. After a smooth local correction, this is the theorem for closed positive currents, which we use in the form recalled in Theorem 6.9.5 (Siu); see section 25.3 for the original source.

For a closed positive current, the Lelong monotonicity formula also bounds the trace mass of small balls. The resulting Hausdorff estimate shows that a closed positive $(j,j)$-current puts no mass on sets of zero Hausdorff $2(n-j)$-measure.

Corollary 5.34.3.C6F4FBF73B9E Let $T$ be a closed positive $(j,j)$-current on a complex manifold of dimension $n$, with $1\leq j<n$. Its coefficient measures vanish on every Borel set of zero Hausdorff $2(n-j)$-measure.

Proof. In a coordinate chart put $\kappa =\mathrm{dd}^{\mathrm{c}}|z|^2$ and $\mu =T\wedge \kappa ^{n-j}$. By the Lelong monotonicity formula in bidimension $(n-j,n-j)$, the function $r\mapsto r^{-2(n-j)}\mu (B(x,r))$ is nondecreasing; see [Chapter III, formulas (5.5)–(5.8), pp. 158–159, DemCADG]. Fix a relatively compact inner chart and a radius $r_0>0$ such that all balls of radius $r_0$ centered in it lie in a fixed compact subset of the chart. Their masses are bounded by the mass of that compact set. Hence

\[ \mu (B(x,r))\leq Cr^{2(n-j)},\qquad 0<r\leq r_0, \]

uniformly for centers in the inner chart. A Borel set in the inner chart with zero Hausdorff $2(n-j)$-measure can be covered by balls $B(x_ i,r_ i)$ with $\sum _ ir_ i^{2(n-j)}$ arbitrarily small, so it has zero $\mu $-mass. Since $T$ is positive, the total variation of each of its coefficient measures is bounded by a constant multiple of $\mu $. A countable cover by charts proves the assertion. $\square$

The non-strict inequality in Siu's theorem cannot be replaced by a strict one, even on a compact Kähler manifold.

Example 5.34.4.697E97CED780 Let $X=\mathbb P^2$, with homogeneous coordinates $[x:y:z]$, and let $\sigma $ be the Fubini–Study form normalized by $\int _ X\sigma ^2=1$. Put $\| Z\| ^2=|x|^2+|y|^2+|z|^2$, $a_ j=2^{-j}$ and $\lambda _ j=2^{-j-1}$ for $j\geq 1$. The global function

\[ u([Z])=\log \frac{|y|^2}{\| Z\| ^2} +\sum _{j\geq 1}\lambda _ j \log \frac{|x-a_ jz|^2}{(1+a_ j^2)\| Z\| ^2} \]

belongs to $\mathrm{PSH}(X,\frac32\sigma )$. For the line $D=\{ y=0\} $ and the points $p_ j=[a_ j:0:1]$, it satisfies

\[ \{ \nu (u,\cdot )\geq 1\} =D, \qquad \{ \nu (u,\cdot )>1\} =\{ p_ j:j\geq 1\} . \]

The second set is not analytic.

Proof. For a linear form $\ell $ with coefficient vector of norm one, the function $v_\ell =\log (|\ell (Z)|^2/\| Z\| ^2)$ is nonpositive and satisfies $\mathrm{dd}^{\mathrm{c}}v_\ell =[\ell =0]-\sigma $. Its $L^1(X,\sigma ^2)$ norm is finite and independent of $\ell $, because the unitary group $U(3)$ acts on homogeneous coordinates, preserves $\sigma $, and acts transitively on such forms. Since $\sum _ j\lambda _ j=1/2$, the partial sums of the series defining $u$ decrease to an $L^1$ function. They have the common curvature lower bound $-\frac32\sigma $, so their limit is quasi-plurisubharmonic and

\[ T:=\tfrac 32\sigma +\mathrm{dd}^{\mathrm{c}}u=[D]+\sum _{j\geq 1}\lambda _ j[L_ j], \qquad L_ j=\{ x-a_ jz=0\} . \]

The series of positive closed currents converges, since every line has $\sigma $-trace mass one and the coefficients are summable.

We compute the Lelong numbers from the local mass formula. In affine coordinates $w=(w_1,w_2)$ centered at a point $q$, put $\kappa =\mathrm{dd}^{\mathrm{c}}|w|^2$. An affine complex line $H$ at distance $d$ from $q$ satisfies

\[ \int _{H\cap B(q,r)}\kappa =(r^2-d^2)_+. \]

Consequently $r^{-2}\int _{H\cap B(q,r)}\kappa $ is at most one and tends to $\mathbf1_ H(q)$ as $r\downarrow 0$; a line that does not meet the chart contributes zero. Tonelli's theorem and dominated convergence for the summable coefficients therefore give

\[ \nu (u,q)=\nu (T,q) =\mathbf1_ D(q)+\sum _{j\geq 1}\lambda _ j\mathbf1_{L_ j}(q). \]

Here we used that a smooth local correction does not change the Lelong number. The computation also covers points at which infinitely many lines accumulate.

The lines $L_ j$ meet $D$ at the distinct points $p_ j$ and meet one another only at $p_\infty =[0:1:0]\notin D$. Thus $\nu (u,p_ j)=1+\lambda _ j$, whereas $\nu (u,q)=1$ for $q\in D\setminus \{ p_ j:j\geq 1\} $. Outside $D$, the Lelong number is at most $\sum _ j\lambda _ j=1/2$. This proves the two level-set identities. Finally, $p_ j\to p_*=[0:0:1]$ and $\nu (u,p_*)=1$, so the strict level set is not closed and hence not analytic. $\square$

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