To begin we fix some sequence $k_{\nu}\rightarrow\infty$ so that the scalings of the based space $(X_{\infty},p)$ by $\sqrt{k_{\nu}}$ converge to a tangent cone $C(Y)$. For a while we focus attention on this cone. Write $|z|$ for the distance from the vertex. Let $\Sigma_{Y}\subset Y$ be the singular set and $Y^{{\rm reg}}=Y\setminus\Sigma$.

The only information about the singular set which we need is contained in the following proposition. This is very likely a standard fact but the proof is quite short so we include it.

Recall that $Y$ has dimension $2n-1$. For clarity in this proof we write $N=2n-1$. A simple argument using the noncollapsing condition (1.2) and the Bishop inequality in the original manifolds shows that there are fixed numbers $\overline{c},\underline{c}>0$ such that for $r\leq 1$ and any metric ball $B_{r}$ in $Y$ we have

(3.1)

$$\underline{c}r^{N}\leq\text{Vol}(B_{r})\leq\overline{c}r^{N}.$$

We know that $\Sigma_{Y}$ is a compact set of Hausdorff dimension strictly less than $N-2$. By the definition of Hausdorff dimension we can find a number $\lambda\in(0,N-2)$ with the following property. For any $\epsilon>0$ there is a cover of $\Sigma_{Y}$ by a finite number of balls $B_{r_{i}/2}(p_{i})$ such that

(3.2)

$$\sum r_{i}^{N-2-\lambda}<\epsilon.$$

We write $B_{i}=B_{r_{i}}(p_{i})$ so, in an obvious notation, the cover is by the balls $\frac{1}{2}B_{i}$. By the Vitali argument we can suppose that the balls $\frac{1}{10}B_{i}$ are disjoint. We take $\epsilon<1$ so for each $i$ we obviously have $r_{i}\leq\epsilon^{1/(N-2-\lambda)}<1$.

Let $\phi(t)$ be a standard cut-off function, vanishing for $t\geq 2$, equal to $1$ when $t\leq 1$ and with derivative bounded by $2$. Define

$$f_{i}(y)=\phi(r_{i}^{-1}d(y,p_{i})).$$

Thus $f_{i}$ is supported in $B_{i}$ and equal to $1$ in $\frac{1}{2}B_{i}$. This function need not be smooth but it is Lipschitz and differentiable almost everywhere, with $|\nabla f_{i}|\leq 2r_{i}^{-1}$. Set $f=\sum f_{i}$. Let $\Psi(t)$ be a cut-off function, equal to $1$ when $t\geq 9/10$, with $\Psi(0)=0$ and with derivative bounded by $2$. Put $g_{0}=\Psi\circ f$. Then $g_{0}$ is equal to $1$ on a neighbourhood of $\Sigma_{Y}$ and is supported in the $2\epsilon^{1/N-2-\lambda}$-neighbourhood of $\Sigma_{Y}$. Also we have

$$\|\nabla g_{0}\|_{L^{2}}\leq 2\|\nabla f\|_{L^{2}}.$$

We claim that $\|\nabla f\|^{2}_{L^{2}}\leq C_{5}\epsilon$ for some fixed $C_{5}$, depending only on $\underline{c},\overline{c}$. Given this claim we can make $\|\nabla g_{0}\|_{L^{2}}$ as small as we please and then finally approximate $g_{0}$ by a smooth function $g$ to achieve our result. (Note that this approximation only involves working over a compact subset of $Y^{{\rm reg}}$. )

To establish the claim, divide the index set into subsets

$$I_{\alpha}=\{i:2^{-\alpha-1}\leq r_{i}<2^{-\alpha}\},$$

for $\alpha\geq 0$. A simple packing argument, using the fact that the balls $\frac{1}{10}B_{i}$ are disjoint, shows that there is a fixed number $C_{6}$ with the following property. If $j\in I_{\alpha}$ then for each fixed $\beta\leq\alpha$ there are at most $C_{6}$ balls $B_{i}$ with $i\in I_{\beta}$ which intersect $B_{j}$. Now we have

$$\|\nabla f\|^{2}_{L^{2}}\leq\sum_{i,j}\int|\nabla f_{i}|\ \ |\nabla f_{j}|.$$

Thus

$$\|\nabla f\|^{2}_{L^{2}}\leq 2\sum_{i,j:r_{j}\leq r_{i}}\int|\nabla f_{i}|\ |\nabla f_{j}|.$$

For fixed $j$ there are at most $C_{6}(1+\log_{2}(r_{j}^{-1}))$ terms which contribute to this last sum. For each term $|\nabla f_{i}|\leq 2r_{i}^{-1}\leq 2r_{j}^{-1}$ and the integrand is supported on the ball $B_{j}$ of radius $2r_{j}$. So for fixed $j$ the contribution to the sum is bounded by

$$8C_{6}(1+\log_{2}(r_{j}^{-1}))r_{j}^{-2}\overline{c}(2r_{j})^{N}.$$

Hence, summing over $j$,

$$\|\nabla f\|^{2}_{L^{2}}\leq 2^{N+3}C_{6}\overline{c}\sum r_{j}^{N-2}(\log_{2}(r_{j}^{-1})+1).$$

We can find a number $C_{7}$ such that for $t\geq 1$ we have $1+\log t\leq C_{7}t^{\lambda}$. Thus

$$\|\nabla f\|^{2}_{L^{2}}\leq 2^{N+3}C_{6}C_{7}\overline{c}\sum r_{j}^{N-2-\lambda}\leq 2^{N+3}C_{6}C_{7}\overline{c}\epsilon.$$

We pick some base point $y_{0}$ in $Y^{{\rm reg}}$. We will need 4 parameters $\rho,\epsilon,\delta,R$ in our basic construction, where $\rho,\epsilon,\delta$ will be “small” and $R$ “large”. In particular $\delta<<\rho<<1<<R$.

First we fix $\rho$ so that $\exp(-\rho^{2}/4)\geq 3/4$ and $\rho\leq(16K_{1})^{-1}$ where $K_{1}$ is the constant in our first derivative estimate. We take $u_{*}=\rho y_{0}\in C(Y)$, with the obvious notation. Fix any neighbourhood $D$ of $u_{*}$ whose closure does not meet the singular set in $C(Y)$. For any $\epsilon$ let $Y_{\epsilon}$ be the set of points of distance greater than $\epsilon$ from $\Sigma$. Let $U_{\epsilon,\delta,R}$ be the set of points $z$ in $C(Y_{\epsilon})$ such that $\delta<|z|<R$. We choose the parameters so that $U_{\epsilon,\delta,R}$ contains the closure of $D$. We consider a smooth compactly supported cut-off function $\beta$ on $U_{\epsilon,\delta,R}$. For such a function we set

$$E_{\beta}=\int e^{-|z|^{2}/2}|\nabla\beta|^{2}.$$

To see this we take $\beta=\beta_{\delta}\beta_{R}\beta_{\epsilon}$ where

• •

  $\beta_{\delta}$ is a standard cut-off function of $|z|$ equal to $1$ for $|z|>2\delta$.

• •

  $\beta_{R}$ is likewise a standard cut-off function of $|z|$, equal to $1$ for $|z|<R/2$.

• •

  $\beta_{\epsilon}=1-(g\circ\varpi)$ where $g$ is a function on $C(Y)$ of the kind constructed in Proposition (3.5) and $\varpi$ is the radial projection from the cone minus the vertex to $Y$.

Then the lemma follows from elementary calculations.

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.

With this lengthy discussion involving the tangent cone in place, we return to the limit space $X_{\infty}$. Recall that we have a sequence of scalings $\sqrt{k_{\nu}}$. We consider embeddings $\chi_{\nu}:\tilde{U}\rightarrow X^{{\rm reg}}_{\infty}$. Given such a $\chi_{\nu}$ we write $J^{\nu}$ for the pull-back of the complex structure on $X_{\infty}^{{\rm reg}}$ and $g^{\nu}$ for the pull-back of $k_{\nu}$ times the metric.

This follows easily from the general assertions in Section 2.1 about convergence. We now fix this $k_{\nu}$ and define $k(p)=mk_{\nu}$ and $r(p)=\rho k(p)^{-1/2}$. We write $\chi_{k_{\nu}}=\chi$.

Let $X_{i}\in{\mathcal{K}}(n,C,V)$ be a sequence converging to $X_{\infty}$. We fix distance functions on $X_{\infty}\sqcup X_{i}$. We consider embeddings $\chi^{i}:\tilde{U}\rightarrow X_{i}$. Given such maps we write $g_{i},J_{i}$ for the pull backs of the metric and complex structure, $\Lambda_{i}$ for the pull-back of $L^{k_{\nu}}$ and $A_{i}$ for the pulled back connection.

Again this follows from our general discussion of convergence.

Fix $i$ large enough, as in Proposition 3.13. We apply Proposition 3.11 to find a $t$ such that the pull-back by $\mu_{t}$ of the data $tg_{i},J_{i},\Lambda_{i}^{t},A^{\otimes t}_{i}$ has Property (H) over $U$. Now write $k=tk_{\nu}$ so $k\leq k(p)$. We apply Proposition 2.4 to construct a holomorphic section $s$ of $L^{k}\rightarrow X_{i}$, with a fixed bound on the $L^{2,\sharp}$ norm and with $|s(x)|\geq 1/4$ at points $x$ with $d^{\sharp}(x,\chi_{i}(u_{*}))<(4K_{1})^{-1}$. Here we are writing $d^{\sharp}$ for the scaled metric, so in terms of the original metric the condition is $d(x,\chi_{i}(u_{*}))<k^{-1/2}(4K_{1})^{-1}$.

To finish, suppose $q\in X_{i}$ has $d(q,p)\leq r(p)$. By construction $r(p)\leq\rho k^{-1/2}$. Note also that if we set $p^{\prime}=\chi^{i}(t^{-1/2}z_{\rho})$. then

$$d(q,p^{\prime})\leq d(q,p)+d(p,\chi(t^{-1/2}z_{\rho}))+d(\chi(t^{-1/2}z_{\rho}),\chi^{(i)}(t^{-1/2}z_{\rho}))\leq 4\rho\sqrt{k}.$$

This means that $d^{\sharp}(q,p^{\prime})\leq 4\rho$ which is less than $(4K_{1})^{-1}$ by our choice of $\rho$.