We begin by recalling that, under our hypotheses, there is a uniform Sobolev inequality

(2.2)

$$\|f\|_{L^{2n/(n-1)}}\leq C_{1}\|\nabla f\|_{L^{2}}+C_{2}\|f\|_{L^{2}},$$

for functions $f$ on a manifold $X$ in the class ${\mathcal{K}}(n,c,V)$, where $C_{1},C_{2}$ depend only on $n,c,V$ [8]. Here of course we are referring to norms defined by the metric $g$. When working with the line bundle $L^{k}$ it will be convenient to use the norms defined by the rescaled metrics $kg$ (for integers $k\geq 1$). Thus lengths are scaled by $\sqrt{k}$ and volumes by $k^{n}$. We will use the notation $L^{2,\sharp}$ etc. to denote norms defined by these rescaled metrics. Then the scaling weight gives

(2.3)

$$\|f\|_{L^{2n/(n-1),\sharp}}\leq C_{1}\|\nabla f\|_{L^{2,\sharp}}+C_{2}k^{-1/2}\|f\|_{L^{2,\sharp}}.$$

So the scaling only helps in the Sobolev inequality. Of course the Ricci tensor ${\rm Ric}^{\sharp}$ of the rescaled metric is bounded between $-1/(2k)$ and $1/k$.

In the second item $\Delta_{\overline{\partial}}=\overline{\partial}^{*}\overline{\partial}+\overline{\partial}\overline{\partial}^{*}$, with adjoints defined using the rescaled metric, and the statement is that for all $\phi$

(2.4)

$$\langle\Delta_{\overline{\partial}}^{-1}\phi,\phi\rangle_{\sharp}\leq 2\|\phi\|^{2}_{L^{2,\sharp}}$$

This Proposition summarises results which are well-known to workers in the field and which all hinge on various formulae of Bochner-Weitzenbock type. We use the rescaled metrics throughout the discussion. First on $C^{\infty}$ sections $s$ of $L^{k}$ we have

$$\nabla^{*}\nabla s=2\overline{\partial}^{*}\overline{\partial}s+s,$$

so when $s$ is holomorphic $\nabla^{*}\nabla s=s$ which implies that

(2.5)

$$\Delta|s|\leq|s|,$$

where the lack of differentiability of $|s|$ at the zero set is handled in a standard way. (Note that we use the “geometers convention” for the sign of the Laplacian in this paper.) Now the bound on the $L^{\infty}$ norm follows from the Moser iteration argument applied to this differential inequality, using the uniform Sobolev inequality (see [22]).

The first derivative bound is obtained in a similar way. Changing notation slightly, for a holomorphic section $s$ with $\overline{\partial}s=0$ we write $\nabla s=\partial s$ where

$$\partial:\Omega^{p,q}(L^{k})\rightarrow\Omega^{p+1,q}(L^{k})$$

is defined using the connection. Since $\partial^{2}=0$ we have

$$\Delta_{\partial}\partial s=\partial\Delta_{\partial}s,$$

where $\Delta_{\partial}=\partial^{*}\partial+\partial\partial^{*}$. Then for a holomorphic section $s$ , $\Delta_{\partial}s=\nabla^{*}\nabla s=s$ and

$$\Delta_{\partial}(\partial s)=\partial s.$$

Now the Bochner-Weitzenbock formula comparing $\Delta_{\partial}$ and $\nabla^{*}\nabla$ on $\Omega^{1,0}(L^{k})$ has the form

$$\Delta_{\partial}=\nabla^{*}\nabla-1+{\rm Ric}^{\sharp},$$

so

$$(\nabla^{*}\nabla(\partial s),\partial s)\leq\frac{5}{2}|\partial s|^{2}.$$

It follows that

$$\Delta|\partial s|\leq\frac{5}{2}|\partial s|,$$

and the Moser argument applies as before. Notice that, with some labour, the constants $K_{0},K_{1}$ could be computed explicitly in terms of $n,c,V$.

For the second item in the Proposition we need a Bochner-Weizenbock formula on $\Omega^{0,1}(L^{k})$ i.e. sections of the bundle $\overline{T}^{*}\otimes L^{k}$. We decompose the covariant derivative on this bundle into (0,1) and (1,0) parts: $\nabla=\nabla^{\prime}+\nabla^{\prime\prime}$. Then the formula we want is

(2.6)

$$\Delta_{\overline{\partial}}=(\nabla^{\prime\prime})^{*}\nabla^{\prime\prime}+{\rm Ric}^{\sharp}+1$$

Given this we have, in the operator sense, $\Delta_{\overline{\partial}}\geq 1/2$ since ${\rm Ric}^{\sharp}\geq-1/2$ from which the invertibility and bound on the inverse follow immediately. An efficient way to derive (2.6) is to make the identification

$$\Omega^{0,1}(L^{k})=\Omega^{n,1}(K^{*}\otimes L^{k}),$$

under which $\nabla^{\prime\prime}$ becomes identified with

$$\partial^{*}:\Omega^{n,1}(K_{X}^{*}\otimes L^{k})\rightarrow\Omega^{n-1,1}(K_{X}^{*}\otimes L^{k}).$$

The formula (2.6) then becomes a special case of the Kodaira-Nakano formula ([13] p.154), using the fact that the Ricci form is the curvature of $K_{X}^{*}$.

With this background in place we move on to recall a version of the “Hörmander” construction of holomorphic sections. Suppose we have the following data

• •

  A (non-compact) manifold $U$, a base point $u_{*}\in U$ and an open neighbourhood $D\subset\subset U$ of $u_{0}$.

• •

  A $C^{\infty}$ Hermitian line bundle $\Lambda\rightarrow U$.

• •

  A complex structure $J$ and Kähler metric $g$ on $U$ with Kähler form $\Omega$.

• •

  A connection $A$ on $\Lambda$ having curvature $-i\Omega$.

We use this connection to define a $\overline{\partial}$-operator on sections of $\Lambda$, and hence a holomorphic structure.

We define a “Property (H)” which this data might have. Fix any $p>2n$.

Property (H):

There is a number $C>0$ and a compactly supported section $\sigma$ of $\Lambda\rightarrow U$ such that the following hold.

H1: $\|\sigma\|_{L^{2}}<(2\pi)^{n/2};$

H2: $|\sigma(u_{*})|>3/4;$

H3: For any smooth section $\tau$ of $\Lambda$ over a neighbourhood of $\overline{D}$ we have

$$|\tau(u_{*})|\leq C\left(\|\overline{\partial}\tau\|_{L^{p}(D)}+\|\tau\|_{L^{2}(D)}\right);$$

H4: $\|\overline{\partial}\sigma\|_{L^{2}}<\min(1/(8\sqrt{2}C),(2\pi)^{n/2}/10\sqrt{2});$

H5: $\|\overline{\partial}\sigma\|_{L^{p}(D)}<1/(8C).$

Many of the specific numbers here are arbitrary but it is convenient to fix some definite numbers.

We have

Notice first that for any choice of data there is some constant $C$ for which the bound in (H3) holds. This follows from the elliptic estimate

(2.7)

$$\|\tau\|_{L^{p}_{1}(D_{0})}\leq C_{3}\left(\|\overline{\partial}\tau\|_{L^{p}(D)}+\|\tau\|_{L^{2}(D_{0})}\right),$$

and the Sobolev inequality

$$|\tau(u_{*})|\leq C_{4}\|\tau\|_{L^{p}_{1}(D_{0})}.$$

Here $D_{0}\subset D$ is some interior domain containing $u_{*}$. We can write the $\overline{\partial}$-operator on functions for a perturbed complex structure as $\overline{\partial}+\mu\partial$ where $\mu$ is a “Beltrami differential”. Similarly if the variation of the connection is given by a $1$-form $a$ then the perturbed $\overline{\partial}$-operator on sections can be written as

$$(\overline{\partial}+\mu\partial)+(a^{\prime\prime}+\mu a^{\prime}),$$

where $a=a^{\prime}+a^{\prime\prime}$ is the decomposition into type. It follows that if $\mu$ and $a$ are small in $C^{0}$ then the perturbation of the $\overline{\partial}$ operator is small in the $L^{p}_{1}\rightarrow L^{p}$ operator norm and it is then clear that the inequality in the third item holds for the perturbed operator, with a slightly larger constant $C$. For the perturbed structure we use the same section $\sigma$, so the first and second item is automatic. Then it is also clear that, for sufficiently small perturbations, the bounds in the fourth and fifth item (with a slightly larger constant $C$) are also preserved, since we impose strict inequality.

For a connection $A$ on a line bundle $L$ write $A^{\otimes k}$ for the induced connection on $L^{k}$. The following proposition—basically well-known—will provide the core of our proof of Theorem 1.

To prove this we transport the section $\sigma$ using the maps $\chi,\hat{\chi}$ and regard it as a smooth section of $L^{k}$ over $X$, extending by zero. The norms we considered over $U$ match up with the $\sharp$-norms over $X$. We write $s=\sigma-\tau$ where $\tau=\overline{\partial}^{*}\Delta_{\overline{\partial}}^{-1}\overline{\partial}\sigma$. By simple Hodge Theory we have $\overline{\partial}s=0$. Now

$$\|\tau\|_{L^{2,\sharp}}=\langle\Delta_{\overline{\partial}}^{-1}\overline{\partial}\sigma,\overline{\partial}\ \overline{\partial}^{*}\Delta_{\overline{\partial}}^{-1}\overline{\partial}\sigma\rangle=\langle\Delta_{\overline{\partial}}^{-1}\overline{\partial}\sigma,\overline{\partial}\sigma\rangle,$$

since $\overline{\partial}\ \overline{\partial}\sigma=0$. Thus

(2.8)

$$\|\tau\|_{L^{2,\sharp}}\leq\sqrt{2}\|\overline{\partial}\sigma\|_{L^{2,\sharp}}\leq\min((8C)^{-1},\frac{1}{10}(2\pi)^{n/2}).$$

Hence in particular

$$\|s\|_{L^{2,\sharp}}\leq\|\sigma\|_{L^{2,\sharp}}+\|\tau\|_{L^{2,\sharp}}\leq\frac{11}{10}2\pi^{n/2}.$$

Now work over the image $\chi(D)$. Applying item (H3) to the section $\tau$ and using (H4), (H5) we get $|\tau(\chi(u_{*}))|\leq 1/4$, so $|s(\chi(u_{*}))|\geq 1/2$. By the derivative bound, $|s|$ exceeds $1/4$ at points a distance less than $(4K_{1})^{-1}$ from $\chi(u_{*})$.

To sum up we have the following.

This is just a direct combination of Lemma 2.2 and Proposition 2.3. (Here we use the notation $\|\ \|_{U}$ to indicate the $C^{0}$-norm over $U$.)

To illustrate this, take the case when $U$ is the ball of radius $R>2$ in $\mbox{${\mathbb{C}}$}^{n}$ with the standard flat metric and standard Kähler form $\Omega_{0}$. Let $\Lambda$ be the trivial holomorphic line bundle with metric $\exp(-|z|^{2}/2)$ so the trivialising section, $\sigma_{0}$ say, has norm $\exp(-|z|^{2}/4)$ and the induced connection $A_{0}$ has curvature $-i\Omega_{0}$ as required. Let $u_{*}$ be the origin and $D$ be the unit ball. Let $\beta_{R}$ be a standard cut-off function of $|z|$, equal to $1$ when $|z|\leq R/2$ and vanishing when $|z|\geq 9R/10$. Define $\sigma=\beta_{R}\sigma_{0}$. Then we have $\overline{\partial}\sigma=(\overline{\partial}\beta_{0})\sigma_{0}$. The $L^{2}$ norm of $\sigma$ is slightly less than $(2\pi)^{n}$ and $|\sigma(0)|=1$. The section $\sigma$ is holomorphic over $D$, so we get (H4) and there certainly is some constant $C$ as in item (H3) of Property (H), independent of $R$. It is clear that, because of the exponential decay, we can fix $R$ so that item (H5) is satisfied. So we have a set of data satisfying Property (H). Now let $x$ be a point in some X in ${\mathcal{K}}(n,c,V)$. Since the ball is simply connected $U(1)$ connections over it are determined up to isomorphism by their curvature tensors. It is then clear that, when $k$ is sufficiently large, we can find a map $\chi$ with $\chi(0)=x$ and such that the pull back of $kg_{X},J_{X},A_{X}^{\otimes k}$ differs by an arbitrarily small amount from the model $g_{0},J_{0},A_{0}$. Then we construct a holomorphic section of $L^{k}\rightarrow X$, of controlled $L^{2}$ norm and of a definite positive size on a definite neighbourhood of $x$.