PomeLemma 10.6.1.DC3C9650BBBE Let $L$ be a holomorphic line bundle on a complex manifold $M$, and let $B$ be a coordinate ball whose closure is contained in the domain of its chart. Then $L$ has a holomorphic frame over $B$, that is, a nowhere vanishing holomorphic section of $L$ on $B$.
10.6 Curvature and metrics defined by sections01A60CA78CF8
01A60CA78CF8
This section collects the notions for line bundles on $X$ used in the rest of the chapter, from the curvature class to the Fubini–Study metrics defined by holomorphic sections, and shows that spaces of global holomorphic sections are finite dimensional.
Let $L\to X$ be a holomorphic line bundle, given as in definition 6.14.9 by frames $e_ i$ over an open cover $(V_ i)$ of $X$, and let $h_0$ be a smooth quadratic Hermitian metric on $L$. Its curvature form, which we denote by $c_1(L,h_0)$, equals $-\mathrm{dd}^{\mathrm{c}}\log |e_ i|_{h_0}^2$ on $V_ i$. It also equals $-\mathrm{dd}^{\mathrm{c}}\log |e|_{h_0}^2$ for every local holomorphic frame $e$ of $L$: on $V_ i$ we have $e=ge_ i$ with $g$ holomorphic and nowhere zero, so $\log |e|_{h_0}^2=\log |g|^2+\log |e_ i|_{h_0}^2$, and $\log |g|^2$ is pluriharmonic (section 5.4). Any other smooth metric is $h_1=h_0\mathrm e^{-f}$ with $f$ smooth and real, and then $c_1(L,h_1)=c_1(L,h_0)+\mathrm{dd}^{\mathrm{c}}f$ (lemma 6.14.10(3)). Hence the class $c_1(L)=\{ c_1(L,h_0)\} \in H^{1,1}(X,\mathbb R)$ does not depend on the metric. A holomorphic section of $L$ is given by holomorphic functions $f_ i$ on $V_ i$ with $f_ ie_ i=f_ je_ j$ on $V_ i\cap V_ j$.
If $L$ and $L'$ are given by frames $e_ i$ and $e'_ i$ over a common cover, with transition functions $g_{ij}$ and $g'_{ij}$, their tensor product $L\otimes L'$ is given by the frames $e_ i\otimes e'_ i$ with the transition functions $g_{ij}g'_{ij}$, and the dual $L^{-1}$ of $L$ by the dual frames $e_ i^*$ with the transition functions $g_{ij}^{-1}$. Metrics on $L$ and $L'$ induce metrics with $|e_ i\otimes e'_ i|^2=|e_ i|^2|e'_ i|^2$ and $|e_ i^*|^2=|e_ i|^{-2}$. The curvature of a tensor product of metrics is the sum of the curvatures, and the curvature of a pullback metric is the pullback of the curvature, because the logarithms of the squared norms of local frames add, respectively compose.
Line bundles are trivial over coordinate balls; we use this to work in a single frame on balls of a fixed size.
Proof. We make the metric flat on $B$ and obtain the frame by parallel transport along the segments issuing from the center of $B$. Choose a smooth metric $h_0$ on $L$ (lemma 6.14.10(1)). By lemma 6.14.2 there is a smooth real function $p$ on $B$ with $\mathrm{dd}^{\mathrm{c}}p=c_1(L,h_0)$, and the metric $h_1=h_0\mathrm e^{p}$ on $L|_ B$ has curvature $c_1(L,h_0)-\mathrm{dd}^{\mathrm{c}}p=0$, by lemma 6.14.10(3). We call a holomorphic frame of $h_1$-length one over an open subset of $B$ a unit frame. Unit frames exist near every point of $B$. Indeed, every point of $B$ has a ball neighborhood $V\subset B$ over which $L$ has a holomorphic frame $e$, and $\log |e|_{h_1}^2$ is smooth and pluriharmonic, since $\mathrm{dd}^{\mathrm{c}}\log |e|_{h_1}^2=-c_1(L,h_1)=0$, so by lemma 6.14.2 it equals $\operatorname{Re}F$ for a holomorphic function $F$ on $V$, and $\mathrm e^{-F/2}e$ is a unit frame. Two unit frames on a connected open set differ by a holomorphic factor $g$ with $|g|=1$; then $\bar g=1/g$ is both holomorphic and antiholomorphic, so $\mathrm d\bar g=0$ and $g$ is constant.
Let $c$ be the center of $B$ and fix a unit vector $e_ c\in L_ c$. For $z\in B$, a flat lift along the segment from $c$ to $z$ is a continuous map $t\mapsto \varepsilon (t)\in L_{c+t(z-c)}$, $t\in [0,1]$, with $\varepsilon (0)=e_ c$, which near every $t$ is a constant multiple of a unit frame. There is at most one flat lift: if $\varepsilon '$ is another one, then $\varepsilon '=g\varepsilon $ for a continuous function $g$ on $[0,1]$ which is locally constant, by the preceding paragraph, and equals $1$ at $t=0$, so $g=1$.
It remains to show that flat lifts exist and that their values at $t=1$ depend holomorphically on $z$. Let $z_0\in B$, and choose $\delta _0>0$ with $\overline{B(z_0,\delta _0)}\subset B$. The set $\Gamma _0=\{ c+t(z-c):t\in [0,1],\ |z-z_0|\leq \delta _0\} $ is a compact subset of the convex set $B$. Cover it by finitely many balls $V_ i\subset B$ carrying unit frames $e_ i$, let $\lambda >0$ be a Lebesgue number of this cover, choose $\delta \leq \delta _0$ with $2\delta <\lambda /2$, and choose $0=t_0<t_1<\cdots <t_ N=1$ with $(t_ k-t_{k-1})(|z_0-c|+\delta _0)<\lambda /2$. For $W=B(z_0,\delta )$, each set $\Gamma _ k=\{ c+t(z-c):t\in [t_{k-1},t_ k],\ z\in W\} $ has diameter less than $\lambda $, because $|(c+t(z-c))-(c+t'(z'-c))|\leq |t-t'|\, |z-c|+|z-z'|$; so $\Gamma _ k$ lies in some $V_{i_ k}$. The convex set $V_{i_ k}\cap V_{i_{k+1}}$ is connected and contains $c+t_ k(z_0-c)$, so $e_{i_ k}=\gamma _ ke_{i_{k+1}}$ there for a constant $\gamma _ k$. Let $a$ be the number with $e_ c=a\, e_{i_1}(c)$. For $z\in W$, the map equal to $a\gamma _1\cdots \gamma _{k-1}e_{i_ k}(c+t(z-c))$ for $t\in [t_{k-1},t_ k]$ is well defined at the points $t_ k$ and is a flat lift along the segment from $c$ to $z$. Hence, on $W$, the value $E(z)$ of the unique flat lift at $t=1$ is $a\gamma _1\cdots \gamma _{N-1}e_{i_ N}(z)$, a constant multiple of the unit frame $e_{i_ N}$. Since $z_0$ was arbitrary, $E$ is a nowhere vanishing holomorphic section of $L$ on $B$. $\square$
Families of holomorphic sections that are bounded in $L^2$ are relatively compact for uniform convergence.
94EC30DAF1A3Let $U\subset \mathbb C^ n$ be open, $n\geq 1$, and let $(f_ j)$ be a sequence of holomorphic functions on $U$ with $\sup _ j\int _ U|f_ j|^2\, \mathrm dV<\infty $. Then a subsequence converges locally uniformly on $U$ to a holomorphic function.
Let $E$ be a holomorphic line bundle on $X$ with a smooth quadratic metric $h_ E$, and let $(s_ j)$ be a sequence of holomorphic sections of $E$ over $X$ with $\sup _ j\int _ X|s_ j|_{h_ E}^2\, \mathrm dV<\infty $. Then a subsequence converges uniformly on $X$ to a holomorphic section $s$ of $E$, that is, $\sup _ X|s_ j-s|_{h_ E}\to 0$ along the subsequence.
For every holomorphic line bundle $E$ on $X$, the space $H^0(X,E)$ of holomorphic sections of $E$ over $X$ is finite dimensional.
Assertions (2) and (3) hold more generally on every compact complex manifold $X'$ of positive dimension in place of $X$, with $\mathrm dV$ in (2) replaced by any smooth positive volume form on $X'$.
Proof. (1) Let $M$ be the supremum in the statement. If $\overline{B(z,\delta )}\subset U$, then $|f_ j(z)|^2\leq |B(z,\delta )|^{-1}M$ by (6.8.7). Hence the $f_ j$ are uniformly bounded on every compact subset of $U$: if $K\subset U$ is compact and $\delta >0$ is smaller than the distance from $K$ to $\mathbb C^ n\setminus U$ (any $\delta >0$ if $U=\mathbb C^ n$), then $|f_ j|^2\leq |B(0,\delta )|^{-1}M$ on $K$.
We next show that the sequence is equicontinuous on compact sets. Let $K\subset U$ be compact, and choose $r>0$ such that the compact set $K'$ of points at distance at most $2\sqrt n\, r$ from $K$ lies in $U$; let $M'$ be a bound for all $|f_ j|$ on $K'$. For $z$ at distance at most $\sqrt n\, r$ from $K$, the closed polydisc of radius $r$ centered at $z$ lies in $K'$, and the coefficient bound in (5.2.1) gives $|\partial _{z_ l}f_ j(z)|\leq M'/r$ for $1\leq l\leq n$. Since $\mathrm df_ j=\sum _ l\partial _{z_ l}f_ j\, \mathrm dz_ l$, the $f_ j$ are Lipschitz with a common constant on every ball of radius $\sqrt n\, r$ centered at a point of $K$, so the sequence is equicontinuous on $K$. By proposition 2.7.4, applied on the compact sets $K_ m=\{ z\in U:|z|\leq m,\ \operatorname {dist}(z,\mathbb C^ n\setminus U)\geq 1/m\} $, which exhaust $U$, and a diagonal argument, a subsequence converges locally uniformly on $U$ to a continuous function $f$. Then $\bar\partial f=0$ in the sense of distributions, as a limit of $\bar\partial f_ j=0$, so $f$ agrees almost everywhere with a holomorphic function by lemma 6.4.10(1); both being continuous, $f$ is holomorphic.
(2) Choose finitely many coordinate balls $B_ k$, with closures in the domains of their charts, such that the concentric balls $B'_ k$ of half the radius cover $X$, and let $B''_ k$ be the concentric balls of three quarters of the radius. By lemma 10.6.1, $E$ has a holomorphic frame $e_ k$ over $B_ k$; write $s_ j=f_{j,k}e_ k$ on $B_ k$. On the compact set $\overline{B''_ k}$ the function $|e_ k|_{h_ E}^2$ is bounded below by a positive constant and $\mathrm dV$ is bounded below by a positive multiple of Lebesgue measure in the chart, so the integrals of $|f_{j,k}|^2$ over $B''_ k$ with respect to Lebesgue measure in the chart are bounded independently of $j$. Applying (1) successively for the finitely many $k$, we obtain a subsequence along which $f_{j,k}$ converges locally uniformly on $B''_ k$, hence uniformly on $\overline{B'_ k}$, to a holomorphic function $f_ k$, for every $k$. On $B''_ k\cap B''_ l$ the sections $f_ ke_ k$ and $f_ le_ l$ are both the pointwise limit of $s_ j$, so they define a holomorphic section $s$ of $E$ over $X$, and $\sup _{B'_ k}|s_ j-s|_{h_ E}\leq \sup _{\overline{B'_ k}}|e_ k|_{h_ E}\cdot \sup _{\overline{B'_ k}}|f_{j,k}-f_ k|\to 0$ for every $k$.
(3) Fix a smooth quadratic metric $h_ E$ on $E$ (lemma 6.14.10(1)). The integral $\| s\| ^2=\int _ X|s|_{h_ E}^2\, \mathrm dV$ defines an inner product on $H^0(X,E)$: if $\| s\| =0$, then the continuous function $|s|_{h_ E}$ vanishes almost everywhere, hence everywhere. If $H^0(X,E)$ were infinite dimensional, the Gram–Schmidt process would give an orthonormal sequence $(s_ j)$. By (2), a subsequence converges uniformly on $X$, hence in $L^2$, since $X$ has finite volume; but $\| s_ i-s_ j\| ^2=2$ for $i\neq j$. This contradiction proves (3).
The proofs of (2) and (3) use only that $X$ is a compact complex manifold of positive dimension and that $\mathrm dV$ is a smooth positive volume form; hence they also prove the last assertion of the lemma. $\square$
A line bundle is positive if it admits a smooth metric with positive definite curvature form. It is very ample if its full space of global holomorphic sections defines a holomorphic embedding into a projective space: this space is finite dimensional by lemma 10.6.2(3), and for a basis $s_0,\ldots ,s_ N$ we require that the sections have no common zero and that $x\mapsto [s_0(x):\cdots :s_ N(x)]$ is a holomorphic embedding of $X$ into $\mathbb P^ N$. A compact complex manifold is projective if it admits a holomorphic embedding into a projective space.
Definition 10.6.3.A68F1B40919C A plurisubharmonic metric on $L$ is a quadratic metric of the form $h=h_0\mathrm e^{-\varphi }$ with $\varphi \in \mathrm{PSH}(X,c_1(L,h_0))$. It assigns the value zero to the zero vector, also over the pole set of $\varphi $, and the value $+\infty $ to a nonzero vector over a pole of $\varphi $. Its Chern current is
\[ c_1(L,h)=c_1(L,h_0)+\mathrm{dd}^{\mathrm{c}}\varphi . \]
Proposition 10.6.4.95892535D04F For a nonzero holomorphic section $s$ of $L$ and a smooth metric $h_0$,
A8F577E012B3
10.6.5
\begin{equation} c_1(L,h_0)+\mathrm{dd}^{\mathrm{c}}\log |s|_{h_0}^2=[Z(s)], \end{equation}
where $[Z(s)]$ is the current of integration along the zero divisor of $s$, counted with multiplicities. In particular $[Z(s)]$ represents $c_1(L)$.
Proof. In a local frame write $s=fe$. Since $\log |s|_{h_0}^2=\log |f|^2+\log |e|_{h_0}^2$, the left side equals $\mathrm{dd}^{\mathrm{c}}\log |f|^2$, which is the current of integration along the zero divisor of $f$ by theorem 5.5.6. These local divisors agree on overlaps, because the transition functions have no zeros. $\square$
Example 10.6.6.01B4AD44BAC3 Let $N\geq 0$. The projective space $\mathbb P^ N$ is the quotient of $\mathbb C^{N+1}\setminus \{ 0\} $ by the action of $\mathbb C^*$ by multiplication, with the quotient topology; we write $q$ for the quotient map and $[Z]=[Z_0:\cdots :Z_ N]$ for the class of $Z$. For $0\leq i\leq N$ the sets $U_ i=\{ [Z]:Z_ i\neq 0\} $ are open and cover $\mathbb P^ N$, and the map $\phi _ i\colon U_ i\to \mathbb C^ N$, $[Z]\mapsto (Z_ k/Z_ i)_{k\neq i}$, is a homeomorphism: it is continuous because $\phi _ i\circ q$ is, and its inverse is $q$ composed with the map inserting the coordinate $1$ in the $i$th place. On $\phi _ i(U_ i\cap U_ j)=\{ w_ j\neq 0\} $, where we put $w_ i=1$, the transition map $\phi _ j\circ \phi _ i^{-1}$ is $w\mapsto (w_ k/w_ j)_{k\neq j}$, which is holomorphic, and so is its inverse $\phi _ i\circ \phi _ j^{-1}$. The space $\mathbb P^ N$ is compact and connected, as the image under $q$ of the unit sphere of $\mathbb C^{N+1}$. It is Hausdorff, because the continuous map $[Z]\mapsto Z\overline Z^{\mathsf T}/|Z|^2$ to the space of complex matrices is injective, the matrix being the orthogonal projection onto the line $\mathbb CZ$; and being compact, it is paracompact. Thus $\mathbb P^ N$ is a connected compact complex manifold of dimension $N$ (definition 5.2.3).
The tautological line bundle $\mathcal O_{\mathbb P^ N}(-1)$ has the fiber $\mathbb CZ$ over $[Z]$. In the sense of definition 6.14.9 it is given by the frames $e_ i([Z])=Z/Z_ i$ over $U_ i$, the points of the line with $i$th coordinate one. Since $e_ i=(Z_ j/Z_ i)e_ j$ on $U_ i\cap U_ j$, its transition functions are $g_{ij}=Z_ j/Z_ i$, which are holomorphic and nowhere zero on $U_ i\cap U_ j$ and satisfy $g_{ij}g_{jk}=g_{ik}$. The Euclidean norm of $\mathbb C^{N+1}$ gives the smooth metric $|e_ i|^2=|Z|^2/|Z_ i|^2$, which satisfies $|e_ i|^2=|g_{ij}|^2|e_ j|^2$. The dual line bundle $\mathcal O_{\mathbb P^ N}(1)$ has the dual frames $e_ i^*$, with $e_ i^*(e_ i)=1$ and transition functions $g_{ij}^{-1}$, and the dual metric $|e_ i^*|^2=|e_ i|^{-2}=|Z_ i|^2/|Z|^2$, the Fubini–Study metric $h_{\mathrm{FS}}$. A linear form $\ell $ on $\mathbb C^{N+1}$ defines a holomorphic section $s_\ell $ of $\mathcal O_{\mathbb P^ N}(1)$, the restriction of $\ell $ to the fibers of $\mathcal O_{\mathbb P^ N}(-1)$: on $U_ i$ it is $s_\ell =\ell (e_ i)e_ i^*=(\ell (Z)/Z_ i)\, e_ i^*$, and these expressions agree on overlaps, because $\ell (e_ i)e_ i^*=\ell (g_{ij}e_ j)\, g_{ij}^{-1}e_ j^*$. For the coordinate forms we write $X_ k=s_{Z_ k}$. Thus $X_ i=e_ i^*$ on $U_ i$, and
\[ |X_ k|_{\mathrm{FS}}^2([Z])=\frac{|Z_ k|^2}{|Z|^2}. \]
The tensor power $\mathcal O_{\mathbb P^ N}(k)=\mathcal O_{\mathbb P^ N}(1)^{\otimes k}$, $k\geq 1$, has the frames $(e_ i^*)^{\otimes k}$, the transition functions $g_{ij}^{-k}$ and the metric $h_{\mathrm{FS}}^ k$; products of $k$ sections of $\mathcal O_{\mathbb P^ N}(1)$, such as the monomials $X_0^{a_0}\cdots X_ N^{a_ N}$ with $a_0+\cdots +a_ N=k$, are holomorphic sections of $\mathcal O_{\mathbb P^ N}(k)$.
Example 10.6.7.4CE1D9D7D803 Let $\mathcal O_{\mathbb P^ N}(1)$ carry the Fubini–Study metric $h_{\mathrm{FS}}$ of example 10.6.6, dual to the Euclidean norm on the tautological line. On the chart $U_0=\{ [1:z_1:\cdots :z_ N]\} $ the tautological line is spanned by $(1,z)$, so the frame $X_0=e_0^*$ has squared norm $|X_0(1,z)|^2/|(1,z)|^2=(1+|z|^2)^{-1}$, and the curvature is
\[ \omega _{\mathrm{FS}}=\mathrm{dd}^{\mathrm{c}}\log (1+|z|^2). \]
The matrix of $\partial _ j\partial _{\bar k}\log (1+|z|^2)$ is $\bigl ((1+|z|^2)\delta _{jk}-\bar z_ jz_ k\bigr )/(1+|z|^2)^2$, which is positive definite by the Cauchy–Schwarz inequality $|\sum _ j\bar z_ j\xi _ j|^2\leq |z|^2|\xi |^2$. The same computation in the chart $U_ i$, with the frame $X_ i$, shows that $\omega _{\mathrm{FS}}$ is a Kähler form on $\mathbb P^ N$; thus $\mathbb P^ N$ is a compact Kähler manifold. On $\mathbb P^1$, integration in polar coordinates gives $\int _{\mathbb P^1}\omega _{\mathrm{FS}}=\frac1\pi \int _{\mathbb C}(1+|z|^2)^{-2}\, \mathrm dV=1$. The function $\log |X_1|_{\mathrm{FS}}^2=\log \bigl (|z|^2/(1+|z|^2)\bigr )$, which vanishes at the point at infinity, is the weight of a singular plurisubharmonic metric whose Chern current is the Dirac current $[0]$ at the point $[1:0]$, by (10.6.5).
Definition 10.6.8.9CC891317C85 A Fubini–Study metric on $L$ is a metric $h_0\mathrm e^{-\varphi }$ with
\[ \varphi =\frac1m\max _{1\leq i\leq N} \bigl (\log |s_ i|_{h_0^ m}^2+\lambda _ i\bigr ), \qquad m,N\in \mathbb Z_{\geq 1},\quad \lambda _ i\in \mathbb Q, \]
where $s_1,\ldots ,s_ N$ are holomorphic sections of $L^ m$, not all zero. We denote the set of these metrics by $\mathrm{FS}(L)$. The set of generalized Fubini–Study metrics is the smallest set of plurisubharmonic metrics that contains $\mathrm{FS}(L)$ and is closed under minima of metrics and under convex combinations of weights with rational coefficients. A convex combination of weights corresponds to the weighted geometric mean of the quadratic metrics, with value zero on the zero vector and the value $+\infty $ on a nonzero vector over a pole of a factor with positive coefficient.
Proposition 10.6.9.E5F620306821 Fubini–Study metrics do not depend on the choice of $h_0$, they have analytic singularities, and $\mathrm{FS}(L)$ is closed under minima and rational convex combinations of weights. Consequently the generalized Fubini–Study metrics are exactly the metrics in $\mathrm{FS}(L)$. The set $\mathrm{FS}(L)$ is empty if no positive tensor power of $L$ has a nonzero holomorphic section, even when $L$ admits plurisubharmonic metrics.
Proof. In a local frame $e$ write $s_ i=f_ ie^ m$. The local weight $-\log |e|_ h^2=\varphi -\log |e|_{h_0}^2$ of the metric equals $m^{-1}\max _ i(\log |f_ i|^2+\lambda _ i)$, which does not involve $h_0$; it is plurisubharmonic and not identically $-\infty $, since $X$ is connected and some $s_ i\ne 0$. It differs by a bounded function from $m^{-1}\log \sum _ i\mathrm e^{\lambda _ i}|f_ i|^2$, so it has analytic singularities with the rational coefficient $1/m$; the pole set is the common zero set of the $s_ i$.
Consider two such metrics, given by sections $s_ i$ of $L^ m$ with constants $\lambda _ i$ and sections $r_ j$ of $L^ l$ with constants $\mu _ j$. Replacing $s_ i$ by $s_ i^{M/m}$ and $\lambda _ i$ by $(M/m)\lambda _ i$, and similarly for the $r_ j$, where $M$ is a common multiple of $m$ and $l$, does not change the potentials, so we may assume that both are given in degree $M$. The maximum of the two potentials, which corresponds to the minimum of the metrics, is then given by the union of the two lists. For a rational $t=a/b\in (0,1)$, the combination $t\varphi +(1-t)\psi $ of the two maxima is the maximum over all pairs $(i,j)$ of the corresponding combinations, that is,
\[ t\varphi +(1-t)\psi =\frac1{Mb}\max _{i,j}\Bigl (\log \bigl |s_ i^ ar_ j^{b-a}\bigr |_{h_0^{Mb}}^2+a\lambda _ i+(b-a)\mu _ j\Bigr ), \]
where the sections $s_ i^ ar_ j^{b-a}$ of $L^{Mb}$ are not all zero and the constants are rational. Thus $\mathrm{FS}(L)$ is closed under both operations, and by induction under every finite sequence of them, so the generated set is $\mathrm{FS}(L)$ itself. If no positive tensor power of $L$ has a nonzero section, no presentation as in definition 10.6.8 exists. $\square$
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