19.8 Holomorphic sections and Hermitian mass725C2F97B643

On a projective manifold, holomorphic sections of line bundles connect singularities with geometry: a singular weight selects the sections whose vanishing compensates for its poles, and the growth of the selected section spaces is governed by Monge–Ampère masses. On a compact Hermitian manifold the curvature of a Hermitian line bundle is still closed, whereas the background form of the Monge–Ampère operator need not be. This part of the book asks what section spaces, and the geometric models attached to sections and singularities, can still detect. Gauduchon degrees often take the place of intersection numbers: they bound the vanishing orders of sections, the valuations along a flag and the divisorial parts of positive currents on modifications.

Two kinds of geometric models appear. A complete flag turns vanishing orders into lattice points and section growth into the volume of a convex body, and the elliptic fibration of a Hopf manifold over projective space reduces its sections and its descended potentials to the base. Compact modifications carry the divisorial part of a singularity: on each model, the divisorial Lelong numbers or trace numbers sum to a closed positive current, and subtracting it from the pulled-back background gives residual classes that form b-divisor classes. Finally, positive Hermitian perturbations of a background organize multiplier-ideal envelopes into compatible systems; the limit of such a system in the original background need not recover it.

In this chapter, $X$ is a connected compact complex manifold of dimension $n$, $L$ is a holomorphic line bundle on $X$ with a smooth Hermitian metric $h$, $u\in \mathrm{QPSH}(X)$ is a fixed weight, and $\mathrm{d}V$ is a smooth positive volume form. In a local holomorphic frame $e$ of $L$, write $|e|_ h^2=\mathrm{e}^{-\chi }$. The form $\beta =\mathrm{dd}^{\mathrm{c}}\chi $ does not depend on the frame; it is the smooth closed curvature form of $h$.

The weight selects the global sections of $L^ m$ that are square integrable against $h^ m\mathrm{e}^{-mu}$, and the Bergman kernels of these finite-dimensional Hilbert spaces produce closed positive currents in the curvature class of $L$. Pairing the Poincaré–Lelong formula with a Gauduchon metric bounds the vanishing that a nonzero section can carry; on a submanifold, this gives a necessary condition for realizing the trace data of section 20.3 by restricted sections. When $X$ is a curve, $\beta +\mathrm{dd}^{\mathrm{c}}u\geq 0$ and $u$ has finitely many logarithmic poles, Riemann–Roch identifies the leading growth of the section spaces with the extended Monge–Ampère mass of $u$. In higher dimensions this identification depends on the background. Homogeneous sections on a Hopf manifold grow only like the sections on its projective base, and a potential of infinite Hermitian mass with trivial multiplier ideals leaves every section space unchanged. Section counts therefore cannot recover a mass computed in a Hermitian background unrelated to the curvature of $L$.

19.8.1 Weighted sections and Bergman currents5F8226DC1A97

The multiplier ideals of $mu$ determine which sections of $L^ m$ have finite weighted $L^2$ norm. Evaluation at points of the resulting Hilbert space defines a Bergman kernel, and its extremal characterization turns comparisons of norms into comparisons of the associated potentials.

For every positive integer $m$, set

The integrand is defined off the pole set $\{ u=-\infty \} $, which has volume zero. A section $s\in H^0(X,L^ m)$ belongs to $\mathcal H_ m(u)$ if and only if $\| s\| _{m,u}<\infty $. Indeed, if the local coefficients of $s$ lie in $\mathcal I(mu)$, the integrand is locally integrable, and a finite covering of the compact manifold $X$ gives a finite integral; conversely, a finite integral gives local integrability on every coordinate ball. Thus $\mathcal H_ m(u)$, a subspace of the finite-dimensional space $H^0(X,L^ m)$, is a Hilbert space for this norm. Write $N_ m=\dim \mathcal H_ m(u)$.

Let $v\in \mathrm{QPSH}(X)$. If $u\preceq _{\mathcal I}v$, that is, $\mathcal I(ku)\subseteq \mathcal I(kv)$ for every real $k>0$ (section 5.33), then $\mathcal H_ m(u)\subseteq \mathcal H_ m(v)$ for every $m$. In particular, weights of the same ordinary singularity type (definition 16.7.1) have the same section spaces, by proposition 5.33.2. When the two weights differ by a bounded function, their Hilbert norms are also comparable. Below, $\| s\| _{m,v,\mathrm{d}V'}$ denotes the norm defined with $v$ and $\mathrm{d}V'$ in place of $u$ and $\mathrm{d}V$.

Fix $m\geq 1$ and choose an orthonormal basis $s_1,\ldots ,s_{N_ m}$ of $\mathcal H_ m(u)$. Define $K_{m,u}$ on $X$ and $B_{m,u}$ off the pole set by

and set $B_{m,u}=0$ on the pole set. When $N_ m=0$, both functions are zero. Since two orthonormal bases differ by a unitary matrix, $K_{m,u}$ does not depend on the choice of basis. We call $\Phi _ m=m^{-1}\log K_{m,u}$ the Bergman potential; when $N_ m>0$, $\beta +\mathrm{dd}^{\mathrm{c}}\Phi _ m$ is its Bergman current. For $N_ m=0$, we have $\Phi _ m\equiv -\infty $.

Write $\Phi _ m(u,\mathrm{d}V)$ when the dependence on the weight and the volume form matters. Under the hypotheses of proposition 19.8.2, and provided that the common section space is nonzero, the extremal formula gives

outside the common base locus. Both potentials equal $-\infty $ on that locus. In particular, the two potentials have the same ordinary singularity type, and for a fixed weight a change of the smooth volume form changes the Bergman potential by $O(m^{-1})$.

A simultaneous change of metric and weight leaves the weighted norm unchanged. Let $a\in C^\infty (X,\mathbb R)$, replace the metric by $h'=h\mathrm{e}^{-a}$, so that $|\cdot |_{h'}^2=|\cdot |_ h^2\mathrm{e}^{-a}$, and replace the weight by $u'=u-a$. Then $h'^ m\mathrm{e}^{-mu'}=h^ m\mathrm{e}^{-mu}$, so the Hilbert space, its norm and the density $B_{m,u}\, \mathrm{d}V$ are unchanged, while

Consequently the Bergman current $\beta +\mathrm{dd}^{\mathrm{c}}\Phi _ m$ does not change.

19.8.4 Gauduchon degree and vanishingE5AAFD4B6553

The zero divisor of a holomorphic section is effective, and the Poincaré–Lelong formula expresses its integration current as the curvature form of the line bundle plus $\mathrm{dd}^{\mathrm{c}}$ of a global function. Pairing with a Gauduchon metric removes the $\mathrm{dd}^{\mathrm{c}}$ term, so the size of the divisor is bounded by a degree of the line bundle. Recall that a Hermitian form $\gamma $ is Gauduchon if $\mathrm{dd}^{\mathrm{c}}(\gamma ^{n-1})=0$ (definition 11.4.1); every Hermitian form on a curve is Gauduchon. Write $h^0(X,L^ m)=\dim H^0(X,L^ m)$.

For a Gauduchon metric $\gamma $ on $X$, define the degree

A change of the smooth metric on $L$ changes $\beta $ by $\mathrm{dd}^{\mathrm{c}}a$ for a smooth real function $a$, and the integral of $\mathrm{dd}^{\mathrm{c}}a$ against $\gamma ^{n-1}$ vanishes by Stokes' theorem. Hence the degree does not depend on the smooth metric on $L$.

The degree constrains positive curvature currents in the same way. If $T=\beta +\mathrm{dd}^{\mathrm{c}}u\geq 0$, the trace identity of proposition 11.5.1 gives

If the degree is zero, the positivity of $T$ and the strict positivity of $\gamma $ imply $T=0$.

19.8.6 Restricted sections and trace dataB7AD9A465882

Let $Y\subset X$ be a connected compact complex submanifold of dimension $r\geq 1$, and suppose that the generic Lelong number $\nu (u,Y)=\inf _{y\in Y}\nu (u,y)$ vanishes. Recall from subsection 20.3.1 the image ideals $\mathfrak a_ m=\mathfrak a_ m(u;Y)=\mathcal I(mu)\mathcal O_ Y$ and the divisorial trace data $\tau _{Y,u}(E)$ of a prime divisor $E$ on a smooth compact modification of $Y$, defined in (20.3.3) as the limit of $\operatorname {ord}_ E\mathfrak a_ m/m$. Put

The local coefficients of a restricted section lie in $\mathfrak a_ m$, so $V_ m\subseteq H^0(Y,L^ m|_ Y\otimes \mathfrak a_ m)$; the right-hand space can contain sections that do not extend to $X$.

On a modification of $Y$, the pullback of a nonzero restricted section vanishes along each prime divisor at least to the order imposed by the image ideal. Subtracting finitely many of these orders still leaves an effective divisor, which the Poincaré–Lelong formula turns into a positive current. If such sections exist in arbitrarily large degrees, normalized compactness produces a positive current in the class obtained by subtracting the trace data from the curvature class, and pairing it with a Gauduchon metric gives a degree inequality. Below, $c_1$ denotes the first Chern class in real Bott–Chern cohomology (subsection 20.4.9); it is represented by the curvature form of any smooth Hermitian metric.

Equivalently, if one of these inequalities fails for some choice of $Z$, $E_1,\ldots ,E_ q$ and $\gamma $, then $V_ m=0$ for all sufficiently large $m$.

The converse of the theorem fails in general. Let $E=\mathbb C/(\mathbb Z+\mathrm{i}\mathbb Z)$ be the square elliptic curve, choose an irrational real number $a$, and let $L$ be the flat line bundle on $E$ with multipliers $\mathrm{e}^{2\pi \mathrm{i} a}$ and $1$ along the two generators: its sections lift to holomorphic functions $f$ on $\mathbb C$ with $f(z+1)=\mathrm{e}^{2\pi \mathrm{i} a}f(z)$ and $f(z+\mathrm{i})=f(z)$. The constant metric on $L$ has curvature zero. Take $u=0$ and $Y=X=E$. Then every trace number is zero, and the class in the theorem is zero, hence pseudoeffective. Nevertheless no positive tensor power of $L$ has a nonzero holomorphic section. Indeed, a nonzero section of $L^ m$ would have a zero divisor of degree zero and would therefore be nowhere zero. The logarithm of its norm would be harmonic on $E$, hence constant, so its lift $f$ would be a holomorphic function of constant modulus on $\mathbb C$, and thus constant. Since $ma\notin \mathbb Z$, the relation $f(z+1)=\mathrm{e}^{2\pi \mathrm{i} ma}f(z)$ then forces $f=0$, a contradiction.

19.8.8 Bounded weighted sectionsC9CFDB0867F5

The spaces $\mathcal H_ m(u)$ need not be closed under multiplication: the product of two sections with integrable weighted squared norms need not have an integrable weighted squared norm, as the following logarithmic example shows. Requiring the weighted pointwise norm to be bounded rather than integrable gives spaces that do multiply. They form the graded algebra used in section 19.9.

We return to the general data $(X,L,h,u)$. Define $\mathcal W_0=\mathbb C$ and, for $m\geq 1$,

These are vector spaces, since $|s+t|^2\leq 2|s|^2+2|t|^2$; they are finite-dimensional and do not depend on the smooth metric, because the ratio of two smooth metrics is bounded above and below on $X$. Since $X$ has finite volume, they are contained in $\mathcal H_ m(u)$. For positive integers $m,k$, $s\in \mathcal W_ m(u)$ and $t\in \mathcal W_ k(u)$, the identity

proves $\mathcal W_ m(u)\mathcal W_ k(u)\subseteq \mathcal W_{m+k}(u)$. Thus $\bigoplus _ m\mathcal W_ m(u)$ is a graded subalgebra of the section ring $\bigoplus _{m\geq 0}H^0(X,L^ m)$ of $L$.

On a smooth projective variety, for big line bundles with metrics of positive curvature current and positive multiplier-ideal volume, the almost-semigroup construction of 19.4 Algebraic partial Okounkov bodies works with the integrable section spaces themselves.

19.8.10 Section growth on a curve25780F9C24E0

On a curve, a weight with logarithmic poles imposes vanishing orders at the poles. Integrability rounds the required order down and boundedness rounds it up, so the two resulting divisors differ by a bounded amount, and Riemann–Roch gives the same leading growth for both section spaces.

Throughout this section, $X$ is a compact Riemann surface and $\beta +\mathrm{dd}^{\mathrm{c}}u\geq 0$. Let $p_1,\ldots ,p_ r$ be distinct points such that $u$ is locally bounded on $X\setminus \{ p_1,\ldots ,p_ r\} $ and, in coordinates $z_ i$ centered at $p_ i$,

Set $\ell =\deg L=\int _ X\beta $ and $d=\sum _ i c_ i$. Recall that $\mathrm{MA}^{\mathrm{ext}}_\beta (u)$ denotes the extended non-pluripolar Monge–Ampère measure of theorem 13.14.3.

19.8.12 Homogeneous sections on a Hopf manifold82379C382500

On a scalar Hopf manifold, a holomorphic section of a line bundle lifts to a function on $\mathbb C^ n\setminus \{ 0\} $ with an exact homogeneity property under the contraction. Hartogs extension reduces the section spaces to spaces of homogeneous polynomials, and the invariance of the metric makes the Bergman kernel constant. The resulting growth is of order $m^{n-1}$, reflecting the dimension of the projective base rather than that of the manifold.

Let $n\geq 2$, let $z=(z_1,\ldots ,z_ n)$ be the coordinates on $\mathbb C^ n$, and set

On the covering space put

The contraction changes $s$ by the constant $-2$, so these forms descend to $X_ n$. The calculation proving (23.1.3), written in section 23.1 for $n=3$ and with $\beta $ in place of $\zeta $, applies in every dimension and gives

Thus $\omega $ is a Gauduchon form. Projectivization defines a holomorphic map $\pi :X_ n\to \mathbb P^{n-1}$ with elliptic fibers, and $\alpha $ is the pullback of the Fubini–Study form normalized to have total mass one. Since $\zeta $ restricts to twice the normalized area form on each fiber, as in the proof of proposition 23.1.4, and $\omega ^ n=n\alpha ^{n-1}\wedge \zeta $, we obtain

Set $\mathrm{d}V=\omega ^ n/(2n)$, a probability measure.

For an integer $k\geq 0$, let $L_ k$ be the line bundle on $X_ n$ obtained from $(\mathbb C^ n\setminus \{ 0\} )\times \mathbb C$ through the identification

Give it the metric $|t|_{h_ k}^2=|t|^2|z|^{-2k}$. Under the identification the two factors change by $\mathrm{e}^{-2k}$ and $\mathrm{e}^{2k}$, so this metric descends. In the frame $t=1$ the squared norm is $|z|^{-2k}$, and the curvature form is $k\alpha $. We apply the constructions of subsection 19.8.1 with $X=X_ n$, $L=L_ k$, $h=h_ k$ and this probability volume form.

For $n=3$, $X_3$ is the Gauduchon threefold of theorem 23.1.1, on which Hermitian Monge–Ampère masses are unbounded; for $k>0$, the line bundles $L_ k$ nevertheless have positive degree and quadratic section growth. In the notation of example 11.21.4, $L_ k$ is the line bundle $E_ a$ on $X_ q$ with $q=\mathrm e^{-1}$ and $a=\mathrm e^{-k}$. Hence, for every $n\geq 2$ and $k>0$, the complement $\mathbb P(\mathcal O\oplus L_ k)\setminus \mathbb P(\mathcal O)$ of 11.21 has least smooth convexity index two, that is, it is $2$-convex but not $1$-convex in the sense of 11.17. When $k=0$, the line bundle is trivial and the least smooth convexity index of its complement is $n+1$, by the same example.

19.8.14 Section counts and infinite Hermitian massB32BAEE663D4

We return to a connected compact complex manifold $X$ of dimension $n$ and fix a Hermitian form $\omega $ on $X$. A weight with trivial multiplier ideals leaves every integrable section space unchanged. When $X$ fails the bounded mass property of definition 13.14.14, theorem 13.14.18 provides such a weight $U\in \mathrm{PSH}(X,\omega )$ with infinite extended mass. For the trivial line bundle with constant metric, $\mathcal H_ m(U)$ consists of constants and every Bergman current vanishes. Thus section growth in a closed curvature class does not determine the mass in an unrelated Hermitian background.