A note on singular Hermitian Yang-Mills connections

TL;DR

This paper presents an example of a homogeneous reflexive sheaf over complex three-space that admits a non-conical Hermitian Yang-Mills connection, providing insight into bubbling phenomena in complex geometry.

Contribution

It introduces a specific example of a sheaf with a non-conical Hermitian Yang-Mills connection, advancing understanding of singularity models in complex differential geometry.

Findings

Example of a homogeneous reflexive sheaf with a non-conical Hermitian Yang-Mills connection

Insights into bubbling phenomena along complex codimension 2 submanifolds

Modeling of bubbling behavior in complex geometric settings

Abstract

We give an example of a homogeneous reflexive sheaf over $\mathbb{C}^3$ which admits a non-conical Hermitian Yang-Mills connection. This is expected to model bubbling phenomenon along complex codimension 2 submanifolds when the Fueter section takes zero value.

Equations

$$⟨F_{E}s,s^{\prime }⟩=⟨F_{E_1}s,s^{\prime }⟩-⟨(\beta {\beta }^{†}{)}^{-1}(\nabla \beta )s,(\nabla \beta )s^{\prime }⟩-⟨({\alpha }^{†}\alpha {)}^{-1}(\nabla {\alpha }^{†})s,(\nabla {\alpha }^{†})s^{\prime }⟩,$$

$$\underline{\mathbb{C}}\stackrel{\alpha =(x,y,a_1,a_2{)}^t}{\to }\underline{{\mathbb{C}}^4}\stackrel{\beta =(-y,x,b_1,b_2)}{\to }\underline{\mathbb{C}},$$

$$a_1{b}_1+a_2{b}_2=0,|a_1{|}^2+|a_2{|}^2=|b_1{|}^2+|b_2{|}^2>0.$$

$$(a_1,a_2,b_1,b_2)\mapsto (a_1{e}^{i\theta },a_2{e}^{i\theta },b_1{e}^{-i\theta },b_2{e}^{-i\theta }).$$

$$\{a_1{b}_1+a_2{b}_2=0,|a_1{|}^2+|a_2{|}^2=|b_1{|}^2+|b_2{|}^2\}/U(1)\simeq \{a_1{b}_1+a_2{b}_2=0\}/{\mathbb{C}}^{\ast }\simeq {\mathbb{C}}^2/{\mathbb{Z}}_2.$$

$$\underline{\mathbb{C}}\stackrel{\alpha =(x,y,1,0{)}^t}{\to }\underline{{\mathbb{C}}^4}\stackrel{\beta =(-y,x,0,z)}{\to }\underline{\mathbb{C}},$$

$$0\to {\Omega }_{{\mathbb{P}}^2}\to \mathcal{O}(-1{)}^{\oplus 3}\to \mathcal{O}\to 0,$$

$$\ell \sim \{\begin{array}{ll}\frac{1}{(|x{|}^2+|y{|}^2+|z|)|\vec{x}|},& |\vec{x}|\gtrsim 1,\\ \frac{1}{|x{|}^2+|y{|}^2+|z{|}^2},& |\vec{x}|\lesssim 1.\end{array}$$

$${\alpha }^{†}=(\bar{x}(|\vec{x}{|}^2+1{)}^{-1/2},\bar{y}(|\vec{x}{|}^2+1{)}^{-1/2},1,0),{\beta }^{†}=(-\bar{y}\sqrt{|\vec{x}{|}^2+1},\bar{x}\sqrt{|\vec{x}{|}^2+1},0,\bar{z}{)}^t,$$

$${\alpha }^{†}\alpha =(|x{|}^2+|y{|}^2)(|\vec{x}{|}^2+1{)}^{-1/2}+1,\beta {\beta }^{†}=(|x{|}^2+|y{|}^2)\sqrt{|\vec{x}{|}^2+1}+|z{|}^2.$$

$$\{\begin{array}{l}\nabla {\alpha }^{†}=\bar{\partial }{\alpha }^{†}=(d\bar{x}(|\vec{x}{|}^2+1{)}^{-1/2}-\frac{\bar{x}(xd\bar{x}+yd\bar{y}+zd\bar{z})}{2(|\vec{x}{|}^2+1{)}^{3/2}},d\bar{y}(|\vec{x}{|}^2+1{)}^{-1/2}-\frac{\bar{y}(xd\bar{x}+yd\bar{y}+zd\bar{z})}{2(|\vec{x}{|}^2+1{)}^{3/2}},0,0),\\ \nabla \beta =(\nabla {\beta }^{†}{)}^{†}=(\bar{\partial }{\beta }^{†}{)}^{†}=(-dy-\frac{y(\bar{x}dx+\bar{y}dy+\bar{z}dz)}{2(|\vec{x}{|}^2+1)},dx+\frac{x(\bar{x}dx+\bar{y}dy+\bar{z}dz)}{2(|\vec{x}{|}^2+1)},0,dz).\end{array}$$

$$|s_1|+|s_2|\le \frac{(|s_3|+|s_4|)(|\vec{x}|+1)}{|x|+|y|}.$$

$$|s{|}_h^2=(|s_1{|}^2+|s_2{|}^2)(|\vec{x}{|}^2+1{)}^{-1/2}+(|s_3{|}^2+|s_4{|}^2),$$

$$|s_1|+|s_2|\le C\min \{(|\vec{x}|+1{)}^{1/2},\frac{|\vec{x}|+1}{|x|+|y|}\}|s{|}_h.$$

$$⟨\sqrt{-1}{F}_{E_1}s,s⟩=O(\frac{|s_1{|}^2+|s_2{|}^2}{(|\vec{x}{|}^2+1{)}^{3/2}}).$$

$$\begin{array}{rl}⟨\sqrt{-1}\Lambda F_{E}s,s⟩=& -({\alpha }^{†}\alpha {)}^{-1}(|\vec{x}{|}^2+1{)}^{-1}\sqrt{-1}\Lambda (s_{1}d\bar{x}+s_{2}d\bar{y})\wedge ({\bar{s}}_{1}dx+{\bar{s}}_{2}dy)\\ & -(\beta {\beta }^{†}{)}^{-1}\sqrt{-1}\Lambda (-s_{1}dy+s_{2}dx)\wedge (-{\bar{s}}_{1}d\bar{y}+{\bar{s}}_{2}d\bar{x})+O(\ell |s{|}_h^2).\end{array}$$

$$\begin{array}{r}(|\vec{x}{|}^2+1{)}^{-1}({\alpha }^{†}\alpha {)}^{-1}-(\beta {\beta }^{†}{)}^{-1}=O(\frac{1}{(\beta {\beta }^{†})\sqrt{|\vec{x}{|}^2+1}}),\end{array}$$

$$G(x,y,z)={\int }_{{\mathbb{C}}^3}\frac{\ell ({\vec{x}}^{\prime })}{|(x-x^{\prime },y-y^{\prime },z-z^{\prime }){|}^4}d\text{Vol}({\vec{x}}^{\prime }).$$

$$\{\begin{array}{l}\Delta G=\text{const}\cdot \ell ,\\ |G|\le C|\vec{x}{|}^{-1}\max (\log \frac{|\vec{x}|}{|x|+|y|+|z{|}^{1/2}},1),|\vec{x}|\gtrsim 1.\end{array}$$

$${\int }_{|{\vec{x}}^{\prime }|\sim 2^k}\frac{1}{(|x^{\prime }{|}^2+|y^{\prime }{|}^2)|{\vec{x}}^{\prime }|}d\text{Vol}({\vec{x}}^{\prime })\lesssim 2^{3k},$$

$$|G-{\int }_{|\vec{x}|\sim |{\vec{x}}^{\prime }|}\frac{\ell ({\vec{x}}^{\prime })d\text{Vol}({\vec{x}}^{\prime })}{|(x-x^{\prime },y-y^{\prime },z-z^{\prime }){|}^4}|\lesssim |\vec{x}{|}^{-1}.$$

$${\int }_{|\vec{x}|\sim |{\vec{x}}^{\prime }|}\frac{1}{|(x-x^{\prime },y-y^{\prime },z-z^{\prime }){|}^4}\frac{d\text{Vol}({\vec{x}}^{\prime })}{(|x^{\prime }{|}^2+|y^{\prime }{|}^2)|\vec{x}|}\lesssim |\vec{x}{|}^{-1}\max (1,\log \frac{|\vec{x}|}{|x|+|y|}),$$

$$s_{(1)}=(0,0,1,0{)}^t,s_{(2)}=(z/y,0,0,1{)}^t.$$

$$\{\begin{array}{l}{s}_1^{\prime }=s_{(1)}-\alpha ({\alpha }^{†}\alpha {)}^{-1}{\alpha }^{†}{s}_{(1)}=s_{(1)}-\frac{1}{(|x{|}^2+|y{|}^2)(|\vec{x}{|}^2+1{)}^{-1/2}+1}(x,y,1,0{)}^t,\\ s_2^{\prime }=s_{(2)}-\alpha ({\alpha }^{†}\alpha {)}^{-1}{\alpha }^{†}{s}_{(2)}=s_{(2)}-\frac{\bar{x}z/y}{|x{|}^2+|y{|}^2+(|\vec{x}{|}^2+1{)}^{1/2}}(x,y,1,0{)}^t.\end{array}$$

$$H_0(s_{(i)},s_{(j)})=h_{{\mathbb{C}}^4}(s_i^{\prime },s_j^{\prime })={\delta }_{ij}+O(\frac{|\vec{x}|}{|y{|}^2}).$$

$$|{\partial }^k{H}_0|=O(\frac{|\vec{x}|}{|y{|}^{2+k}}),k\ge 1,$$

$$|{\partial }^k⟨\Lambda F_E{s}_i^{\prime },s_j^{\prime }⟩|\le C(k)|\vec{x}{|}^{-1}|y{|}^{-2-k},$$

$$H_0={\delta }_{ij}+O(\frac{|\vec{x}|}{|x{|}^2}),|{\partial }^k{H}_0|\le C(k)\frac{|\vec{x}|}{|x{|}^{2+k}},k\ge 1,$$

$$(|\vec{x}{|}^2+1{)}^{1/2}{h}_{{\mathbb{C}}^4}=\text{diag}(1,1,(|\vec{x}{|}^2+1{)}^{1/2},(|\vec{x}{|}^2+1{)}^{1/2}),$$

$$\mathbb{C}\stackrel{(x,y,{\zeta }^{1/2},0{)}^t}{\to }\underline{{\mathbb{C}}^4}\stackrel{(-y,x,0,{\zeta }^{1/2})}{\to }\underline{\mathbb{C}},$$

Full text

The guiding wisdom in the study of Hermitian Yang-Mills (HYM) connections is the Hitchin-Kobayashi correspondence principle, namely metric properties of HYM connections translate into algebro-geometric properties of holomorphic vector bundles and vice versa. For instance, the celebrated Donaldson-Uhlenbeck-Yau theorem [13] states that over compact Kähler manifolds, a holomorphic vector bundle admits a unique HYM connection if and only if it is polystable. This has been extended to reflexive sheaves by Bando and Siu [1], and by now there is a developed theory comparing the compactified moduli spaces of HYM connections versus the stable vector bundles [7]. In a more local setting, recent works [9][2] show that for a large class of reflexive sheaf singularities, the local tangent cones of the HYM connections can be read off algebro-geometrically from the Harder-Narasimhan filtration.

The purpose of this paper is to provide a surprising example which shows that in the noncompact setting the holomorphic structre alone does not need to capture everything about HYM connections:

Complex geometrically, the Euler sequence shows that the reflexive sheaf is isomorphic over $\mathbb{C}^{3}\setminus\{0\}$ to the pullback of the cotangent bundle of $\mathbb{CP}^{2}$ via $\mathbb{C}^{3}\setminus\{0\}\to\mathbb{CP}^{2}$. Indeed, by [9][2] the local tangent cone at the origin must be the pullback of the Levi-Civita connection on $\Omega_{\mathbb{P}^{2}}$ up to twisting by a central $U(1)$-connection. The surprise is that the HYM metric is not conical in our example, and the complex geometry does not predict the tangent cone at infinity. Thus the roles of the tangent cone at infinity and the local tangent cone must be fundamentally asymmetrical. This has a similar flavour to the recent discovery of exotic Calabi-Yau metrics on $\mathbb{C}^{n}$ [10][3][14].

Despite the appearance our main result does not violate the Price monotonicity formula, which states that $r^{4-\text{dim}}\int_{B(r)}|F|^{2}$ must be an increasing function in the radius $r$. Geometrically, the curvature has faster than quadratic decay in the generic region near infinity, resulting in a flat tangent cone at infinity, but has slower than quadratic decay close to the $z$-axis, transverse to which the HYM connection is modelled on scaled copies of the standard one-instanton. Thus on large spheres the curvature becomes concentrated in a very small solid angle, and when we take the tangent cone the $L^{2}$ curvature is lost in the limit.

Our strategy is to produce an ansatz from the monad construction, and then use some fairly standard nonlinear existence machinery to find a HYM connection asymptotic to the ansatz near infinity. The monad construction is motivated by studying a family of HYM connections bubbling along a complex curve $S$ inside a Calabi-Yau 3-fold [6]. Near a local patch of $S$, the HYM connections restricted to the normal directions to $S$ are modelled on (framed) ASD instantons, whose variation along $S$ is governed by a holomorphic map (called the Fueter map) from $S$ to the framed instanton moduli space. We are interested in the simplest case of instantons with rank 2 and charge 1, so the framed moduli space is $\mathbb{C}^{2}/\mathbb{Z}_{2}$. Now the most generic kind of zero for a holomorphic map $\mathbb{C}\to\mathbb{C}^{2}/\mathbb{Z}_{2}$ is up to linear change of coordinates given by $z\mapsto(z^{1/2},0)$ to leading order. Via the ADHM construction, this particular Fueter map translates into a monad ansatz for the HYM connection. In our actual construction we take a regularized version of the monad ansatz with better curvature decay properties.

1 An ansatz from monad construction

We begin with a general curvature formula for the cohomology of a monad over a complex manifold.

Lemma 1.1**.**

Consider a monad $E_{0}\xrightarrow{\alpha}E_{1}\xrightarrow{\beta}E_{2}$, namely a complex of Hermitian holomorphic vector bundles with $\alpha$ injective fibrewise and $\beta$ surjective fibrewise. Let $E=\ker\beta/\text{coker}(\alpha)$ be the cohomology bundle. Then the curvature $F_{E}$ of the natural induced connection on $E$ satisfies

[TABLE]

where $F_{E_{1}}$ is the Chern connection on $E_{1}$, and $s,s^{\prime}$ are representing smooth sections of $E$ satisfying $\alpha^{\dagger}s=\alpha^{\dagger}s^{\prime}=\beta s=\beta s^{\prime}=0$, and $\nabla\alpha^{\dagger}$, $\nabla\beta$ are covariant derivatives computed on the Hom bundles.

Example 1.2**.**

(ADHM construction of one-instantons) Start from the monads over Euclidean $\mathbb{C}^{2}_{x,y}$

[TABLE]

where the underlines signify trivial vector bundles, and the parameters $a_{1},a_{2},b_{1},b_{2}$ satisfy the ADHM equation

[TABLE]

The natural connections on cohomology bundles $E_{a,b}$ are ASD instantons on $\mathbb{C}^{2}$ with rank 2, charge 1 and curvature scale $\sim\sqrt{|a_{1}|^{2}+|a_{2}|^{2}}$. The situation with $a_{1}=a_{2}=b_{1}=b_{2}=0$ is viewed as a degenerate case. As Hermitian vector bundles $E_{a,b}$ are identified as $\ker\beta\cap\ker\alpha^{\dagger}\subset\underline{\mathbb{C}^{4}}=\underline{\mathbb{C}^{2}}\oplus\underline{\mathbb{C}^{2}}$, and the monad construction provides a natural projection map into the second $\underline{\mathbb{C}^{2}}$ factor, giving a trivialisation of $E_{a,b}$ near infinity known as framing data. Notice the framed instantons are isomorphic under the $U(1)$-symmetry

[TABLE]

The moduli space of framed instantons centred at the origin is

[TABLE]

We now describe our main ansatz. Take the monad over Euclidean $\mathbb{C}^{3}_{x,y,z}$

[TABLE]

and equip the trivial bundle $\underline{\mathbb{C}^{4}}$ with the nonstandard Hermitian structure given by the diagonal matrix $h_{\mathbb{C}^{4}}=\text{diag}((|\vec{x}|^{2}+1)^{-1/2},(|\vec{x}|^{2}+1)^{-1/2},1,1)$ where $|\vec{x}|^{2}=|x|^{2}+|y|^{2}+|z|^{2}$. The monad has a unique singular point at the origin in $\mathbb{C}^{3}$ where $\beta$ fails to be surjective; in fact simple linear algebra shows the cohomology sheaf $E$ is isomorphic to the homogeneous coherent sheaf $\ker(\underline{\mathbb{C}^{3}}\xrightarrow{(x,y,z)}\underline{\mathbb{C}})$, so is in particular reflexive. Using the Euler sequence over $\mathbb{P}^{2}$

[TABLE]

$E$ is isomorphic as vector bundles over $\mathbb{C}^{3}\setminus\{0\}$ to the pullback of $\Omega_{\mathbb{P}^{2}}$ via $\mathbb{C}^{3}\setminus\{0\}\to\mathbb{P}^{2}$.

The key point is that the mean curvature on $E$ has fast enough decay at infinity.

Lemma 1.3**.**

The curvature $F_{E}$ of the natural connection on $E$ admits the estimate $|\Lambda F_{E}|\leq C\ell,$ where $\ell$ is a fixed function uniformly equivalent to

[TABLE]

Proof.

We shall compute the curvature $F_{E}$. Taking into account the nonstandard Hermitian structure, the adjoint maps are given by

[TABLE]

hence

[TABLE]

Since $\alpha,\beta$ are holomorphic, $\alpha^{\dagger},\beta^{\dagger}$ are antiholomorphic,

[TABLE]

Let $s=(s_{1},s_{2},s_{3},s_{4})^{t}$ be a smooth local section of $E$ represented as a section of $\underline{\mathbb{C}^{4}}$ with $\beta s=\alpha^{\dagger}s=0$, so by expressing $s_{1},s_{2}$ in terms of $s_{3},s_{4}$,

[TABLE]

Under the Hermitian structure

[TABLE]

we have

[TABLE]

Now the Chern curvature on $\underline{\mathbb{C}^{4}}$ is given by $F_{E_{1}}=\bar{\partial}(\partial h_{\mathbb{C}^{4}}h_{\mathbb{C}^{4}}^{-1}),$ so

[TABLE]

Substituting (3)(4)(5) into the curvature formula in Lemma 1.1,

[TABLE]

Combined with a cancellation effect expressed by the inequality

[TABLE]

this implies $\langle\sqrt{-1}\Lambda F_{E}s,s\rangle=O(\ell|s|_{h}^{2}),$ or equivalently $|\Lambda F_{E}|=O(\ell)$ as required. ∎

Remark**.**

From the proof one can readily extract estimates on $|F_{E}|$. In particular near the origin $|F_{E}|=O(|\vec{x}|^{-2})$ is locally in $L^{2}$. For $|x|+|y|\gtrsim|z|^{1/2}+1$, the curvature decays like $|F_{E}|=O(\frac{|\vec{x}|}{(|x|^{2}+|y|^{2})^{2}})$. In particular when $|x|+|y|\gtrsim|\vec{x}|\gg 1$, the curvature has cubic decay, so the tangent cone at infinity is flat. The curvature becomes concentrated near the $z$-axis near spatial infinity.

For later usage, we estimate the potential integral

[TABLE]

Lemma 1.4**.**

The function $G>0$ is well defined on $\mathbb{C}^{3}\setminus\{0\}$, and satisfies

[TABLE]

Proof.

The singularity of the source $\ell$ at the origin is $O(\frac{1}{|\vec{x}|^{2}})$, which is mild enough to guarantee the potential integral is well defined and has the correct Laplacian. We focus on the estimates for $|\vec{x}|\gtrsim 1$. The $L^{1}$ integral of the source $\ell$ at a scale $|\vec{x}^{\prime}|\sim 2^{k}$ is bounded by

[TABLE]

so by summing over all dyadic scales,

[TABLE]

Thus the only important contribution comes from the dyadic scale $|\vec{x}|\sim|\vec{x}^{\prime}|$, which for $|x|^{2}+|y|^{2}\gtrsim|z|$ is controlled by

[TABLE]

and for $|x|^{2}+|y|^{2}\lesssim|z|$ is controlled by $O(|\vec{x}|^{-1}|\log\frac{|\vec{x}|}{|z|^{1/2}}|)$. Combining the above shows the claim. ∎

1.1 Asymptotic geometry near infinity

First we examine the asymptotic geometry for $|x|+|y|\gg|z|^{1/2}+1$, namely the generic region near spatial infinity. Consider the case $|x|\lesssim|y|$, so $|y|\gg|\vec{x}|^{1/2}+1$. A basis of holomorphic sections on $E$ can be represented by sections of $\ker\beta$:

[TABLE]

The projections of $s_{1},s_{2}$ to the orthogonal complement of $\text{Im}(\alpha)$ are respectively

[TABLE]

The Hermitian metric $H_{0}$ on the cohomology bundle is represented by the matrix

[TABLE]

It follows from elementary Taylor expansions that

[TABLE]

where $\partial^{k}$ refers to the $k$-th partial derivatives. The natural connection $\nabla_{E}$ on $E$ is just the Chern connection induced by the Hermitian structure $H_{0}$, so in particular $|F_{E}|=O(\frac{|\vec{x}|}{|y|^{4}})$ compatible with Remark Remark. The mean curvature $\Lambda F_{E}$ has better decay properties. For this, we can apply the more accurate formula in Lemma 1.1 to derive an explict expression for the matrix $(\langle\Lambda F_{E}s_{i}^{\prime},s_{j}^{\prime}\rangle)$ as formula (6). The same cancellation effect as in (7) happens. The higher order version of Lemma 1.3 then follows from Taylor expansion considerations:

[TABLE]

or equivalently $|\nabla^{k}_{E}(\Lambda F_{E})|\leq C(k)|\vec{x}|^{-1}|y|^{-2-k}$.

Similarly in the case $|x|\gtrsim|y|$, we can find another basis of holomorphic sections on $E$, with

[TABLE]

and $|\nabla_{E}^{k}(\Lambda F_{E})|\leq C(k)|\vec{x}|^{-1}|x|^{-2-k}$. To summarize, the Hermitian structure on $E$ in the generic region $|x|+|y|\gg|z|^{1/2}+1$ is approximately flat.

Next we turn to the vicinity of the $z$-axis $1\ll|z|^{1/2}\lesssim|x|+|y|$. Observe that if the ambient Hermitian structure on $\underline{\mathbb{C}^{4}}$ is changed from $h_{\mathbb{C}^{4}}$ to

[TABLE]

then the induced connection on $E$ is twisted by a $U(1)$ connection with curvature $\frac{1}{2}\bar{\partial}\partial\log(|\vec{x}|^{2}+1)$. We shall focus on this twisted situation around a given point $(0,0,\zeta)\in\mathbb{C}^{3}$ with $|\zeta|\gg 1$, and choose a square root $\zeta^{1/2}$. After rescaling the basis vectors on $\underline{\mathbb{C}^{4}}$, the twisted monad can be written as

[TABLE]

where the Hermitian structure on $\underline{\mathbb{C}^{4}}$ is $\tilde{h}_{\mathbb{C}^{4}}=\text{diag}(1,1,\frac{(|\vec{x}|^{2}+1)^{1/2}}{|\zeta|},\frac{|\zeta|(|\vec{x}|^{2}+1)^{1/2}}{|z|^{2}})$. For $|x|+|y|+|z-\zeta|\lesssim|\zeta|^{1/2}$, Taylor expansion shows

[TABLE]

To leading order, the ambient Hermitian metric on $\underline{\mathbb{C}^{4}}$ is Euclidean, and the twisted monad (9) dimensionally reduces to the monad in the ADHM construction with parameters $(a_{1},a_{2},b_{1},b_{2})=(\zeta^{1/2},0,0,\zeta^{1/2})$ (cf. Example 1.2). In particular, the connection on the cohomology bundle of the twisted monad is approximately a framed instanton whose moduli parameter is identified as $(\zeta^{1/2},0)\in\mathbb{C}^{2}/\mathbb{Z}_{2}$.

Remark**.**

From a more global viewpoint, the twisted monad connection in the normal direction to the $z$-axis is described by the Fueter map into the moduli space of framed instantons:

[TABLE]

Notice the Fueter map is independent of the choice of square root $\pm\zeta^{1/2}$. As mentioned in the introduction, our monad construction (1) is best thought as a regularized version of the monad induced by this Fueter map. The extra issue of twisting by a $U(1)$ connection is required for the almost flatness of the connection far away from the $z$-axis.

One can then estimate the difference between $\nabla_{E}$ and the instanton connection $\nabla_{\zeta}$ associated with the ADHM monad, in the region $\{|x|+|y|\lesssim|\zeta|^{1/2},z=\zeta\}$, by taking into account both the deviation of $\tilde{h}_{\mathbb{C}^{4}}$ from being Euclidean, and the effect of $U(1)$-twisting:

[TABLE]

and the regularity scale of $\nabla_{E}$ in this region is comparable to the regularity scale of $\nabla_{\zeta}$ which is $\sim|\zeta|^{1/2}$.

Combining the above, we have a unified higher order estimate for $\Lambda F$:

Corollary 1.5**.**

In the region $|x|\gtrsim 1$,

[TABLE]

2 Perturbation into HYM metric

We seek a nonlinear perturbation of the ansatz to a genuine HYM connection. The strategy is to solve Dirichlet boundary value problems on larger and larger domains exhausting $\mathbb{C}^{3}$, obtain uniform estimates and then extract limits as in [11]. The analysis involved is by now fairly standard.

Theorem 2.1**.**

Let $E$ be a reflexive sheaf over a compact Kähler manifold $(\overline{Z},\omega)$ with nonempty boundary $\partial Z$, which is locally free near the boundary. For any Hermitian metric $f$ on the restriction of $E$ to $\partial Z$ there is a unique Hermitian $H$ on $E$, which is smooth on the locally free locus, has finite $L^{2}$ curvature, and

[TABLE]

Proof.

(Sketch) Donaldson [4] proved the special case when $E$ is a vector bundle using the heat flow method. The key point is that $|\Lambda F|$ is a subsolution to the heat equation with zero boundary data, which forces $|\Lambda F|$ to decay exponentially in time. This together with uniform $C^{k}$ control on the connection in the flow leads to long time existence and convergence to HYM connection at infinite time. Here the issue of stability does not appear.

The singularity problem can be addressed by the continuity method due to Bando and Siu [1]; we follow the exposition in [8], Section 6. The idea is to take repeated blow ups $\tilde{Z}$ in the interior of $\bar{Z}$ so that $E|_{\bar{Z}\setminus\text{Sing}(E)}$ extends as a vector bundle $\tilde{E}$ across the exceptional locus. Equip $\tilde{E}$ with a fixed reference Hermitian metric $\tilde{H}_{1}$ admitting the given boundary data. One can find a sequence of degenerating Kähler metrics $\omega_{\epsilon}\to\omega$ on $\tilde{Z}$ as $\epsilon\to 0$, such that

• •

$\omega_{\epsilon}$ agrees with $\omega$ on the $O(\epsilon)$ neighbourhood of the exceptional locus.

• •

Near the blow up loci $\omega_{\epsilon}$ is locally modelled on a rescaling of the standard metric on $\text{Bl}_{0}\mathbb{C}^{k}\times\mathbb{C}^{n-k}$.

• •

The curvature of $\tilde{H}_{1}$ has uniform $L^{2}$ curvature bound as $\epsilon\to 0$.

• •

The metrics $\omega_{\epsilon}$ have uniform Dirichlet-Sobolev constants.

Applying Donaldson’s result to solve the Dirichlet problem, we obtain HYM connections $\tilde{H}_{\epsilon}$ on $\tilde{E}$ for the background metrics $\omega_{\epsilon}$. Using the almost subharmonicity estimate (in the analyst’s Laplacian convention)

[TABLE]

and the uniform Dirichlet-Sobolev inequality, there is a uniform $L^{2}$ bound on $\text{Tr}\log(\tilde{H}_{\epsilon}\tilde{H}_{1}^{-1})$ and $\text{Tr}\log(\tilde{H}_{1}\tilde{H}_{\epsilon}^{-1})$. On any compact subset of the locally free locus of $E$, the almost subharmonicty implies furthermore $L^{\infty}$ estimates on $\text{Tr}\log(\tilde{H}_{\epsilon}\tilde{H}_{1}^{-1})$ and $\text{Tr}\log(\tilde{H}_{1}\tilde{H}_{\epsilon}^{-1})$, so that $\tilde{H}_{\epsilon}$ is locally uniformly equivalent to $\tilde{H}_{1}$ independent of $\epsilon$. Then the Bando-Siu interior estimate (cf. Appendix C, D in [8]) gives $C^{k}_{loc}$-estimates on $\tilde{H}_{\epsilon}$ over the locally free locus of $E$, uniform in $\epsilon$. Furthermore, there are uniform $L^{2}$ curvature bounds on $\tilde{H}_{\epsilon}$ because of the topological energy formula for HYM connections

[TABLE]

Now taking a subsequential weak limit as $\epsilon\to 0$, we obtain a HYM metric $H$ over the locally free locus of $E$, which must have bounded $L^{2}$ curvature. The uniqueness statement follows from the fact that if $H$, $H^{\prime}$ are two solutions to the Dirichlet problems with finite $L^{2}$-curvature, then $\text{Tr}\log(HH^{\prime-1})$ and $\text{Tr}\log(H^{\prime}H^{-1})$ are both subharmonic. ∎

Theorem 2.2**.**

There is a HYM connection $H$ on $E$ over Euclidean $\mathbb{C}^{3}$ with locally finite $L^{2}$ curvature, which admits the decay estimates on $\{|x|\gtrsim 1\}$:

[TABLE]

In particular, the tangent cone at infinity is flat. The asymptotic estimate (12) and the HYM condition $\Lambda F_{H}=0$ determine $H$ uniquely.

Proof.

Recall $H_{0}$ is the natural Hermitian metric for the cohomology sheaf $E$ of our monad (1), whose curvature is $F_{E}$. We solve the Dirichlet problem on large balls $B(R)\subset\mathbb{C}^{3}$ with boundary data $H_{0}$ on $\partial B(R)$, and denote the solution as $H_{R}$. Crucially we need the almost subharmonicity estimate (cf. Lemma 2.5 in [11], and Lemma 1.3):

[TABLE]

Notice near the origin (13) continues to hold in the distributional sense, since the $L^{2}$ curvature of $H_{0}$ and $H_{R}$ are finite, and the singularity has complex codimension 3 (cf. proof of Proposition 3.1 in [9]). Using the boundary condition $\text{Tr}(H_{R}H_{0}^{-1})=\text{Tr}(H_{0}H_{R}^{-1})=\text{rank}(E)=2$, we apply Lemma 1.4 and the comparison principle to get

[TABLE]

or equivalently there is a pointwise estimate on $B(R)$:

[TABLE]

which is uniform as $R\to\infty$. Using the upper bound for $|G|$ in $\{|x|\gtrsim 1\}$ provided by Lemma 1.4,

[TABLE]

Recall also the higher order control on the mean curvature of the ansatz in Lemma 1.5. Applying Bando and Siu’s interior regularity estimate (cf. Appendix C, D in [8]) to rescaled balls, we derive the higher order estimate on $\{1\lesssim|\vec{x}|<R/2\}$:

[TABLE]

Using the topological energy formula similar to (11), the $L^{2}$-curvature inside the unit ball is controlled. All estimates are uniform in $R$. Taking a subsequential limit as $R\to\infty$, we obtain the HYM connection $H$ with estimates (12). Since the deviation is so small that the leading order asymptotic geometry at infinity is unchanged, and in particular $|F_{H}|=O(\frac{|\vec{x}|}{(|x|^{2}+|y|^{2}+|z|)^{2}})$ as $|\vec{x}|\to\infty$, we see the tangent cone at infinity is flat.

To see the uniqueness, notice if $H^{\prime}$ is another HYM metric satifying (12), then both $\log\frac{\text{Tr}(H^{\prime}H^{-1})}{2}$ and $\log\frac{\text{Tr}(HH^{\prime-1})}{2}$ are subharmonic in the distributional sense, and are asymptotic to zero at infinity. ∎

The tangent cone connection at the origin is determined a priori by complex geometry of the reflexive sheaf $E\simeq\ker(\mathbb{C}^{3}\xrightarrow{(x,y,z)}\mathbb{C})$ (cf. [2][9]). Complex geometrically, it is isomorphic to $E$, or equivalently the pullback of the cotangent bundle on $\mathbb{CP}^{2}$. The Levi-Civita connection on $\Omega_{\mathbb{P}^{2}}$ is HYM with respect to the integral Fubini-Study metric $\omega_{FS}$:

[TABLE]

The tangent cone at the origin is the pullback of the Levi-Civita connection, up to a conformal change of the Hermitian structure by a factor $|\vec{x}|^{-3}$ which cancels out the Einstein constant. Equivalently, the tangent cone is the natural connection on the kernel of $\underline{\mathbb{C}^{3}}\xrightarrow{(x,y,z)}\underline{\mathbb{C}},$ but $\underline{\mathbb{C}^{3}}$ is equipped with the nonstandard Hermitian structure $\text{diag}(\frac{1}{|\vec{x}|},\frac{1}{|\vec{x}|},\frac{1}{|\vec{x}|})$. An application of [9] shows that our HYM connection on $E$ is asymptotic to the tangent cone at the origin with polynomial decay rate.

Some additional insights can be gained by studying the growth rate of holomorphic sections as in [2]. On a reflexive sheaf with a conical HYM connection, there is a convexity estimate (cf. Proposition 3.5 in [2])

[TABLE]

A basic heuristic in [2] is that if a singular HYM connection is sufficiently close to being conical on a certain scale, then the convexity behaviour transfers to the HYM connection, so that $\log\int_{B(r)}|s|^{2}/\log r$ has some monotonicity property, and one can define the local growth degree

[TABLE]

which induces a filtration on the germ of holomorphic sections, intimately related to a Harder-Narasimhan-Seshardri filtration.

In the setting of our example, this motivates us to define the growth degree at infinity

[TABLE]

inducing a filtration on the holomorphic sections with at most polynomial growth. It is then easy to check explicitly that there are holomorphic sections whose growth degree at infinity is larger than the growth degree at the origin, so the two filtration structures are different. We expect that there are large classes of examples generalizing our construction, and these filtration structures should play a major role in a more systematic theory. In particular, it would be interesting to relate the filtration and the tangent cone at infinity.