Recall that the metric on the regular part of the cone has the form $\frac{i}{2}\partial\overline{\partial}|z|^{2}$. So, just as in the case of $\mbox{${\mathbb{C}}$}^{n}$, we have a line bundle $\Lambda_{0}$ with connection $A_{0}$, curvature the Kähler form $\Omega_{0}$ and a holomorphic section $\sigma_{0}$ with $|\sigma_{0}|=\exp(-|z|^{2}/4)$. Then $\sigma=\beta\sigma_{0}$ is holomorphic on $D$. Note that $\|\sigma\|_{L^{2}}^{2}$ will now be slightly less than $\kappa^{1/2}(2\pi)^{n/2}$ where $\kappa\leq 1$ is the volume ratio as in (2.1).

As we explained, there certainly is some constant C giving the elliptic estimate (H3) and we use the Lemma to choose $\epsilon,\delta,R$ so that this set of data has Property (H).

The parameters $\rho,\delta,\epsilon,R$ are now all fixed. We set $U=U(\rho,\delta,\epsilon,R)$.

Consider now a $C^{0}$-small perturbation $g,J$ of the metric and complex structure $g_{0},J_{0}$, and hence a perturbation $\Omega$ of $\Omega_{0}$ We suppose that $-i\Omega$ is the curvature of a unitary connection $A$ on a bundle $\Lambda$. If we can choose a bundle isomorphism between $\Lambda$ and $\Lambda_{0}$ such that, under this isomorphism, the connection $A$ is a small perturbation of $A_{0}$ then we can apply Proposition 2.4 to conclude that the data $J,\Omega,A$ also has Property (H), (for suitably small perturbations). The difficulty is that if $H_{1}(U;{\mathbb{Z}})\neq 0$ a connection on a line bundle is not determined by its curvature. Said in another way, we consider the line bundle $\Lambda\otimes\Lambda_{0}^{*}$ with the connection $a$ induced from $A,A_{0}$. The curvature of $a$ is small but $a$ need not be close to a trivial flat connection. There is no real loss of generality in supposing that $Y_{\epsilon}$ has smooth boundary ( because we can always replace it by a slightly enlarged domain). Write $\underline{\nu}$ for the normal vector field on the boundary. We want to recall some Hodge Theory on this manifold with boundary. Fix $p>2n$. .

These are fairly standard results. The first item follows from the fact that the $L^{2}$ extension of the image of $d$ is closed. The second asserts the unique solubility of the Neumann boundary value problem for the Laplacian on functions on $Y_{\epsilon}$. The existence and uniqueness of $\alpha$ in the third item is similar. The $L^{p}$ estimate in the third item follows from general theory of elliptic boundary value problems, see [24] for a detailed treatment of this case. Note that in our application the subtleties of the boundary value theory could be avoided by working on a slightly larger domain. Then we can reduce to easier interior estimates. Alternatively one can adjust the set-up to reduce to the standard Hodge theory over a compact “double”.)

Write $a|$ for the restriction of the connection $a$ to the restricted bundle over $Y_{\epsilon}$. A consequence of item (1) is that there is some number $C_{9}>0$ such that any closed $2$-form $F$ over $Y_{\epsilon}$ which represents an integral cohomology class and with $\|F\|_{L^{2}}\leq C_{9}$ is exact. In particular we can apply this to the curvature $F_{a|}=i(\Omega-\Omega_{0})$ of the connection $a|$, using the fact that this represents an integral class. (Here we are considering $Y_{\epsilon}$ as embedded in $U$ in the obvious way.) Thus there is a $C_{10}>0$ such that if $\|\Omega-\Omega_{0}\|_{U}\leq C_{10}$ we can apply item (3) of Prop. 3.7 to write $F_{a|}=d\alpha$ over $Y_{\epsilon}$ for a small $\alpha=\alpha(a)$. More precisely, $\alpha$ is small in $L^{p}_{1}$ and so in $C^{0}$ by Sobolev embedding. Then $a|-\alpha$ is a flat connection on the restriction of $\Lambda\otimes\Lambda_{0}^{*}$ to $Y_{\epsilon}$. This flat connection is determined up to isomorphism by its holonomy: a homomorphism from $H_{1}(Y_{\epsilon},{\mathbb{Z}})$ to $S^{1}$.

Fix a direct sum decomposition of $H_{1}(Y_{\epsilon},{\mathbb{Z}})$ into torsion and free subgroups. Then we get

$${\rm Hom}(H_{1}(Y_{\epsilon},{\mathbb{Z}}),S^{1})=G\times T,$$

where $G$ is a finite abelian group and $T=H^{1}(Y_{\epsilon},\mbox{${\mathbb{R}}$})/H^{1}(Y_{\epsilon},{\mathbb{Z}})$ is a torus. (We will write the group structures multiplicatively.) Thus for our connection $a$ with suitably small curvature we get two invariants $g(a)\in G,\tau(a)\in T$. If both vanish then the restriction of the connection to $Y_{\epsilon}$ is close to the trivial flat connection. When $a$ is the connection induced from $A,A_{0}$ as above we write $g(A,A_{0}),\tau(A,A_{0})$.

This is straightforward. The hypotheses imply that, for small $W,\psi$, there is a trivialisation of $\Lambda\otimes\Lambda_{0}^{*}$ over $Y_{\epsilon}$ in which the connection form is small in $L^{p}_{1}$ and hence in $C^{0}$. Then extend this to a trivialisation over $U$ by parallel transport along rays. In this trivialisation the radial derivative of the connection form is given by a component of the curvature, so is controlled by $\psi$. From another point of view this trivialisation is a bundle isomorphism between $\Lambda,\Lambda_{0}$ under which $A$ is a small perturbation of $A_{0}$.

Let $m_{1}$ be the order of $G$. Thus for any $g\in G$ we have $g^{m_{1}}=1$. Fix a slightly smaller neighbourhood $W^{\prime}\subset\subset W$ of the identity in $J$. By Dirichlet’s theorem we can find an $m_{2}$ such that for any $\tau\in T$ there is a power $\tau^{q}$ which lies in $W^{\prime}$ where $1\leq q\leq m_{2}$. Write $m=m_{1}m_{2}$. Now return to our connection $a|$ on the bundle $\Lambda\otimes\Lambda_{0}^{*}$ over $Y_{\epsilon}$. Recall that for integer $t$ we write $a|^{\otimes t}$ for the induced connection on $\Lambda^{t}\otimes\Lambda_{0}^{-t}$ over $Y_{\epsilon}$. Suppose that $\|F(a|)\|_{U}\leq C_{10}/m$. Then for $1\leq t\leq m$ the invariants $g(a|^{\otimes t}),\tau(a|^{\otimes t})$ are defined and we have:

With $m$ fixed as above, write

$$\tilde{U}=U(m^{-1/2}\delta,\epsilon,R).$$

For integers $t$ with $1\leq t\leq m$ let $\mu_{t}:U\rightarrow\tilde{U}$ be the map $\mu_{t}(z)=t^{-1/2}z$ (in obvious notation). Thus $\mu_{t}^{*}(t\Omega_{0})=\Omega_{0}$.

Our model structure $g_{0},J_{0},\Lambda_{0},A_{0}$ is defined over $\tilde{U}$ Now consider deformed structures $J,\Omega,\Lambda,A$ as before but which are also defined over $\tilde{U}$. Suppose that

$$\|g-g_{0}\|_{\tilde{U}}\leq\tilde{\psi},\|J-J_{0}\|_{\tilde{U}}\leq\tilde{\psi},$$

where $\|\ \|_{\tilde{U}}$ here denotes $C^{0}$ norms over $\tilde{U}$. For integers $t$ as above, let $g_{t},J_{t},\Lambda_{t},A_{t}$ be the data over $U$ given by pulling back $tg,J,\Lambda^{t},A^{\otimes t}$ using the map $\mu_{t}$. It is clear that if $\tilde{\psi}$ is sufficiently small then for every $t$ we have

$$\|g_{t}-g_{0}\|_{U}\leq\psi\ ,\ \|J_{t}-J_{0}\|_{{U}}\leq\psi.$$

It is also clear that, if $\tilde{\psi}$ is sufficiently small, then the invariants $g(A_{0},A_{t}),\tau(A_{0},A_{t})$ are defined.

We choose $t$ according to Prop. 3.9, so that $g(a|^{\otimes t})=1$ and $\tau(a|^{\otimes t})\in W^{\prime}$.

Write $\tau(a|^{\otimes t})=\tau$. Thus $\tau$ can be regarded as a small element of $H^{1}(Y_{\epsilon},\mbox{${\mathbb{R}}$})$. It follows from our set-up that there is a trivialisation of the bundle $\Lambda^{t}\otimes\Lambda_{0}^{-t}$ over $Y_{0}$ in which the connection $a|^{\otimes t}$ is represented by a $C^{0}$-small connection form. Extend this trivialisation to $\tilde{U}$ using parallel transport along rays. As above, in the proof of Proposition 3.8, the radial derivative of the connection form in this trivialisation is given by the curvature $F_{a^{\otimes t}}$ and it follows easily that if $\tilde{\psi}$ is sufficiently small then in the induced trivialisation the pull-back $\mu_{t}^{*}(a^{\otimes t})$, restricted to $Y_{\epsilon}$ has a $C^{0}$-small connection form. In particular, given that $W^{\prime}\subset\subset W$ we can, by fixing $\tilde{\psi}$ sufficiently small, ensure that the “$\tau$ invariant” of this connection lies in $W$ and the “g-invariant” is $1$. Now the fact that $\mu_{t}^{*}(A_{0}^{\otimes t})$ is isomorphic to $A_{0}$ yields the result stated.

We sum up in the following way.