# STRATIFIED ELLIPTIC ESTIMATES AND STRONG CONVERGENCE OF $L^2$-HARMONIC FORMS ON COMPLEX VARIETIES WITH ISOLATED SINGULARITIES

JIAMING LUO AND SHIRONG LI

**ABSTRACT.** Let $V$ be a compact irreducible complex analytic space with isolated singularities $\Sigma$ and $V' = V \setminus \Sigma$. Building on Ohsawa's approach to the Cheeger-Goresky-MacPherson conjecture in the isolated singularity case, we provide a complete, verifiable proof of the convergence-and-domain-identification step for $L^2$-harmonic forms associated to Ohsawa's degenerating family of Hermitian metrics. The argument combines $\epsilon$-uniform stratified elliptic estimates, compactness in local Sobolev spaces, and a Mosco-convergence analysis of the closed quadratic forms associated with the Hodge-de Rham family (hence strong resolvent convergence in the sense of [11, 3, 12]), which allows us to pass harmonicity to the limit and to identify the limit in the minimal domain. As a consequence, the $L^2$ cohomology of $V'$ agrees with the intersection cohomology of $V$, and (in each degree) the nullities of the associated nonnegative self-adjoint operators $D_\epsilon^* D_\epsilon$ (equivalently, the dimensions of the $L^2$-harmonic spaces) are constant for $\epsilon \in (0, 1]$, while subsequential limits are characterized as elements of the kernel of the minimal closed extension for the limiting incomplete metric.

## CONTENTS

<table><tr><td>1. Introduction</td><td>3</td></tr><tr><td>1.1. Background</td><td>3</td></tr><tr><td>1.2. A domain stability problem and our contribution</td><td>4</td></tr><tr><td>1.3. Main results</td><td>4</td></tr><tr><td>1.4. Main analytic mechanism: Mosco convergence and minimal-domain identification</td><td>5</td></tr><tr><td>1.5. Ideas of the proof</td><td>6</td></tr><tr><td>1.6. Relation to the literature</td><td>7</td></tr><tr><td>1.7. Organization of the paper and notation</td><td>7</td></tr><tr><td>2. Geometric and analytic setup</td><td>9</td></tr><tr><td>2.1. Isolated singularities and basic notation</td><td>9</td></tr><tr><td>2.2. Ohsawa's complete metrics and admissible weights</td><td>9</td></tr><tr><td>2.3. $L^2$ spaces of forms and the Hodge-de Rham operators</td><td>10</td></tr></table>

2020 Mathematics Subject Classification. 58J32, 32C36, 55N33, 47A10, 47B25.

Key words and phrases. $L^2$ cohomology, isolated singularities, stratified elliptic regularity, closed quadratic forms, Mosco convergence, strong resolvent convergence, intersection cohomology.
<table>
  <tbody>
    <tr>
      <td>2.4. Minimal and maximal extensions; completeness for ε &gt; 0</td>
      <td>11</td>
    </tr>
    <tr>
      <td>2.5. Weighted norms and the role of φ<sub>ε</sub></td>
      <td>11</td>
    </tr>
    <tr>
      <td>2.6. Standing analytic hypotheses for later sections</td>
      <td>11</td>
    </tr>
    <tr>
      <td>3. Stratified/b-Sobolev spaces and elliptic estimates</td>
      <td>12</td>
    </tr>
    <tr>
      <td>3.1. b-geometry near an isolated singularity</td>
      <td>12</td>
    </tr>
    <tr>
      <td>3.2. b-Sobolev spaces and global stratified Sobolev norms</td>
      <td>13</td>
    </tr>
    <tr>
      <td>3.3. Weighted multipliers induced by φ<sub>ε</sub></td>
      <td>14</td>
    </tr>
    <tr>
      <td>3.4. The Hodge-de Rham operator as a b-elliptic family</td>
      <td>15</td>
    </tr>
    <tr>
      <td>3.5. ε-uniform stratified elliptic estimates</td>
      <td>16</td>
    </tr>
    <tr>
      <td>3.6. A weighted version</td>
      <td>17</td>
    </tr>
    <tr>
      <td>4. Compactness and subsequential convergence</td>
      <td>18</td>
    </tr>
    <tr>
      <td>4.1. Uniform local elliptic regularity on compact subsets</td>
      <td>18</td>
    </tr>
    <tr>
      <td>4.2. Rellich compactness on compact subsets</td>
      <td>19</td>
    </tr>
    <tr>
      <td>4.3. From weighted L<sup>2</sup> normalization to uniform local Sobolev bounds</td>
      <td>19</td>
    </tr>
    <tr>
      <td>4.4. Subsequence convergence on compact subsets and diagonal extraction</td>
      <td>20</td>
    </tr>
    <tr>
      <td>4.5. Compatibility with stratified norms</td>
      <td>21</td>
    </tr>
    <tr>
      <td>5. Graph (Mosco) convergence of Hodge-de Rham operators</td>
      <td>21</td>
    </tr>
    <tr>
      <td>5.1. A fixed Hilbert space and uniform identification maps</td>
      <td>22</td>
    </tr>
    <tr>
      <td>5.2. Closed quadratic forms associated with D<sub>ε</sub></td>
      <td>23</td>
    </tr>
    <tr>
      <td>5.3. Mosco convergence and strong resolvent convergence</td>
      <td>24</td>
    </tr>
    <tr>
      <td>5.4. Mosco convergence of the transported Hodge forms</td>
      <td>24</td>
    </tr>
    <tr>
      <td>5.5. Passing harmonicity to the limit</td>
      <td>25</td>
    </tr>
    <tr>
      <td>6. Minimal-domain identification via cutoff and boundary-term control</td>
      <td>26</td>
    </tr>
    <tr>
      <td>6.1. A cutoff criterion for the minimal domain</td>
      <td>26</td>
    </tr>
    <tr>
      <td>6.2. Weighted L<sup>2</sup> control passes to the limit</td>
      <td>28</td>
    </tr>
    <tr>
      <td>6.3. Logarithmic cutoffs with vanishing boundary term</td>
      <td>29</td>
    </tr>
    <tr>
      <td>6.4. Minimal-domain identification for the limit harmonic form</td>
      <td>30</td>
    </tr>
    <tr>
      <td>7. Proofs of Theorem 1.1 and Corollary 1.3</td>
      <td>30</td>
    </tr>
    <tr>
      <td>7.1. Proof of Theorem 1.1</td>
      <td>31</td>
    </tr>
    <tr>
      <td>7.2. Proof of Corollary 1.3</td>
      <td>32</td>
    </tr>
    <tr>
      <td>8. Examples and model computations</td>
      <td>32</td>
    </tr>
    <tr>
      <td>8.1. The cone model and the borderline weight</td>
      <td>32</td>
    </tr>
    <tr>
      <td>8.2. A toy geometric example: nodal curves</td>
      <td>33</td>
    </tr>
    <tr>
      <td>8.3. Projective hypersurfaces with isolated singularities</td>
      <td>33</td>
    </tr>
    <tr>
      <td>8.4. Local algebraic models: isolated hypersurface singularities and ordinary double points</td>
      <td>34</td>
    </tr>
    <tr>
      <td>8.5. A global projective example: nodal hypersurfaces and comparison with resolutions/smoothings</td>
      <td>35</td>
    </tr>
    <tr>
      <td>Appendix A. b/edge calculus references and mapping properties used</td>
      <td>38</td>
    </tr>
    <tr>
      <td>A.1. b-manifolds and b-differential operators</td>
      <td>38</td>
    </tr>
    <tr>
      <td>A.2. b-Sobolev spaces and basic mapping statements</td>
      <td>39</td>
    </tr>
  </tbody>
</table>
<table>
  <tbody>
    <tr>
      <td>A.3.</td>
      <td>The b-pseudodifferential calculus and parametrices</td>
      <td>40</td>
    </tr>
    <tr>
      <td>A.4.</td>
      <td>How the b-calculus enters the main text</td>
      <td>40</td>
    </tr>
    <tr>
      <td>A.5.</td>
      <td>Remarks on the edge calculus</td>
      <td>41</td>
    </tr>
    <tr>
      <td>A.6.</td>
      <td>Commutators with cutoffs and multipliers</td>
      <td>41</td>
    </tr>
    <tr>
      <td>Appendix B.</td>
      <td>A Rellich theorem in the stratified/b setting</td>
      <td>42</td>
    </tr>
    <tr>
      <td>B.1.</td>
      <td>Compact subsets of V' stay away from Σ</td>
      <td>42</td>
    </tr>
    <tr>
      <td>B.2.</td>
      <td>b-Sobolev norms versus standard Sobolev norms on annuli</td>
      <td>42</td>
    </tr>
    <tr>
      <td>B.3.</td>
      <td>Rellich compactness on compact subsets</td>
      <td>43</td>
    </tr>
    <tr>
      <td>B.4.</td>
      <td>Compatibility with the stratified/b norms</td>
      <td>44</td>
    </tr>
    <tr>
      <td>Appendix C.</td>
      <td>Model calculations for Section 8</td>
      <td>45</td>
    </tr>
    <tr>
      <td>C.1.</td>
      <td>Borderline logarithmic integrals</td>
      <td>45</td>
    </tr>
    <tr>
      <td>C.2.</td>
      <td>An explicit commutator decay estimate</td>
      <td>45</td>
    </tr>
    <tr>
      <td>C.3.</td>
      <td>A model version of the cutoff criterion</td>
      <td>46</td>
    </tr>
    <tr>
      <td>C.4.</td>
      <td>The ordinary double point link as an explicit smooth manifold</td>
      <td>46</td>
    </tr>
    <tr>
      <td>C.5.</td>
      <td>A Jacobian/Hessian criterion for nodes</td>
      <td>47</td>
    </tr>
    <tr>
      <td>C.6.</td>
      <td>Worked Hessian check for the explicit quadric cone</td>
      <td>47</td>
    </tr>
    <tr>
      <td>C.7.</td>
      <td>Worked Jacobian/Hessian check for the Cayley cubic (four nodes)</td>
      <td>48</td>
    </tr>
    <tr>
      <td>C.8.</td>
      <td>Worked Jacobian/Hessian check for the Segre cubic (ten nodes)</td>
      <td>49</td>
    </tr>
    <tr>
      <td>References</td>
      <td></td>
      <td>50</td>
    </tr>
  </tbody>
</table>

# 1. INTRODUCTION

## 1.1. Background.
Let $V$ be a compact irreducible complex analytic space of complex dimension $n$ with isolated singular set $\Sigma$, and write $V' = V \setminus \Sigma$. A fundamental theme in the analysis on singular spaces is that the topology of $V$ should be recoverable from analytic data on the nonsingular locus $V'$. For pseudomanifolds, Goresky-MacPherson introduced intersection homology $IH^\bullet(V)$ as the correct homology theory compatible with Poincaré duality in the presence of singularities [10]. On the analytic side, Cheeger initiated a Hodge theory for $L^2$ differential forms on conic and more general singular spaces, revealing a deep relationship between $L^2$-cohomology and intersection cohomology [4]. In the complex algebraic/analytic setting, this relationship was further developed through $L^2$-$\bar{\partial}$ techniques and the study of Hodge-de Rham operators on singular varieties; see, for instance, [19, 20].

In the isolated singularity case, Ohsawa proposed a geometric-analytic approach based on a degenerating family of complete Hermitian metrics $g_\epsilon$ on $V'$ together with admissible weights $\phi_\epsilon$ as $\epsilon \downarrow 0$, and he announced a proof of the $L^2$-$IH$ correspondence for such varieties [18]. A central step in this program is a stability and convergence statement for the spaces of $L^2$ harmonic forms associated with the Hodge-de Rham operators

$$D_{\epsilon} = d + d_{\epsilon}^{*} \quad \text{on} \quad (V', g_{\epsilon}),$$
as the parameter $\varepsilon \downarrow 0$. The purpose of the present paper is to provide a
complete, verifiable proof of this convergence-and-domain-identification step
and to close the resulting analytic chain of implications.

1.2. **A domain stability problem and our contribution.** We briefly describe the analytic point where the argument needs additional justification. Let $H_{\varepsilon}^r = \ker(D_{\varepsilon}) \cap L^2 \Omega^r(V', g_{\varepsilon})$ denote the space of $L^2$ harmonic $r$-forms. In Ohsawa's framework, one obtains a priori bounds on normalized sequences $f_{\varepsilon} \in H_{\varepsilon}^r$ (after inserting the weight $\phi_{\varepsilon}$) and then extracts sub-sequential limits on compact subsets of $V'$. To conclude that the limiting object produces the desired $L^2-IH$ identification, it is crucial to show that any such limit lies in the *minimal* domain of the limiting de Rham operator, i.e. in $\operatorname{Dom}(d_{\min}) \cap \operatorname{Dom}(d_{\min}^*)$, and not merely in the maximal domain.

Our main results supply the key analytic ingredient: we establish an $\varepsilon$-uniform stratified elliptic regularity theory for $D_{\varepsilon}$, prove compactness in local Sobolev spaces, and use a graph-convergence (Mosco) analysis of the associated quadratic forms (cf. Section 5) to pass harmonicity to the limit. Finally, a cutoff/integration-by-parts argument controls the boundary terms near the isolated singularities and forces the limit into the minimal domain.

**Contributions (optimized for a functional-analytic readership).**

* We establish $\varepsilon$-uniform stratified (b/edge-adapted) elliptic estimates for the Hodge-de Rham family $D_\varepsilon$ in a Sobolev scale compatible with isolated conic geometry (see Section 3).

* We prove compactness and strong subsequence convergence on $K \in V'$ for normalized $L^2$-harmonic forms, using the uniform elliptic control and a Rellich-type theorem in the stratified Sobolev setting (Section 4).

* After transporting the varying $L^2(g_\varepsilon)$ spaces to a fixed Hilbert space, we prove Mosco convergence of the associated closed quadratic forms, and hence strong resolvent convergence of the corresponding selfadjoint realizations (Section 5). This yields stability of the harmonic dimensions and passes harmonicity to the limit.

* We identify the limiting form in the *minimal* domain via a logarithmic cutoff / integration-by-parts argument that eliminates hidden boundary data at the isolated singularities (Section 6).

* As applications, we obtain the $L^2$-IH correspondence for $V$ and verify the hypotheses in concrete projective models (Sections 7 and 8).

1.3. **Main results.** We work throughout in the geometric setup of Ohsawa [18]. In particular, $(g_\varepsilon, \phi_\varepsilon)$ denotes Ohsawa's family of complete Hermitian metrics and admissible weights on $V'$ for $\varepsilon \in (0,1]$, and $D_\varepsilon = d + d_\varepsilon^*$ is the associated Hodge-de Rham operator. Our first main theorem is a strong convergence statement with minimal-domain identification.
**Theorem 1.1** (Strong convergence and minimal-domain identification). Under Ohsawa's hypotheses for varieties with isolated singularities, the following hold for each degree r:

(a) (Stability) The dimensions $\dim H_{\varepsilon}^r$ are constant for $\varepsilon \in (0, 1]$.

(b) (Subsequence convergence) If $f_{\varepsilon} \in H_{\varepsilon}^r$ is normalized by $\|\phi_{\varepsilon}^{-1} f_{\varepsilon}\|_{L^2(g_{\varepsilon})} = 1$, then there exist $\varepsilon_j \downarrow 0$ and a form $f \in L_{\text{loc}}^2 \Omega^r(V')$ such that $f_{\varepsilon_j} \to f$ strongly on compact subsets of $V'$ in $H^{s-1}$ for every $s > n+1$.

(c) (Minimal-domain limit) The limit satisfies $f \in \text{Dom}(d_{\text{min}}) \cap \text{Dom}(d_{\text{min}}^*)$ with respect to the limiting metric $g_0$ and $D_0f = 0$. In particular, $f$ is smooth on $V'$.

*Remark 1.2.* From an operator-theoretic standpoint, part (c) identifies the limiting form as lying in the minimal closed extension of the de Rham complex for the incomplete metric $g_0$. Mere distributional convergence typically yields only membership in the maximal domain; the logarithmic cutoff / integration-by-parts argument rules out residual boundary data at the isolated singularities and selects the domain relevant for the $L^2$–IH comparison.

As an immediate consequence, we recover the expected topological identification.

**Corollary 1.3** ($L^2$ cohomology equals intersection cohomology). With notation as above, the natural map induced by $L^2$ harmonic representatives yields an isomorphism between the $L^2$ de Rham cohomology of $(V', g_0)$ and the middle-perversity intersection cohomology $IH^\bullet(V)$.

**1.4. Main analytic mechanism: Mosco convergence and minimal-domain identification.** A key point is that the operators $D_\varepsilon$ act on varying Hilbert spaces $L^2\Omega^\bullet(V', g_\varepsilon)$. Following a standard strategy in variational/spectral convergence, we first transport these spaces to a fixed Hilbert space (so that weak/strong convergence and operator limits become meaningful). We then associate to $D_\varepsilon$ the nonnegative closed quadratic form

$$Q_{\varepsilon}[u] := \|D_{\varepsilon}u\|_{L^2(g_{\varepsilon})}^2, \quad D(Q_{\varepsilon}) = \text{Dom}(D_{\varepsilon}),$$

and transport it to a form $\tilde{Q}_{\varepsilon}$ on a fixed Hilbert space (see Section 5.2). The Mosco convergence $\tilde{Q}_{\varepsilon} \xrightarrow{M} Q_{0,\text{max}}$ proved in Section 5 implies, by the general theory of closed forms (see [11, 3] and the spectral-geometry framework of [12]), strong resolvent convergence of the corresponding selfadjoint realizations and stability of low-lying spectral data. This provides the functional-analytic backbone for Theorem 1.1(a)-(b).

What remains is to identify the correct limiting domain at the singular points. The minimal-domain conclusion Theorem 1.1(c) is obtained in Section 6 by a logarithmic cutoff argument that controls boundary terms on shrinking annuli and rules out residual boundary data in the limit.
<figure><img src="image_1.png" /><figcaption>FIGURE 1. Proof roadmap. Uniform $b/\text{edge}$ elliptic estimates yield compactness on $K \in V'$, which enables Mosco/graph convergence; a logarithmic cutoff argument then removes hidden boundary data and identifies the minimal domain.</figcaption></figure>

**1.5. Ideas of the proof.** We sketch the analytic spine of the argument and highlight where $\varepsilon$-uniformity is used.

As illustrated in Figure 1, the proof flows from uniform elliptic estimates to compactness and Mosco convergence, and it closes with a logarithmic cutoff integration-by-parts that forces the limit into the minimal domain.

(1) **$\varepsilon$-uniform stratified elliptic estimates.** We introduce a scale of stratified (equivalently, $b/\text{edge}$-adapted) Sobolev spaces on $V'$ and prove an elliptic estimate for $D_\varepsilon$ whose constants are independent of $\varepsilon$. The argument uses a local model near each singular point (see Figure 2) and a parametrix construction in the $b/\text{edge}$ calculus (cf. [15, 16]).

(2) **Compactness on compact subsets of $V'$.** The uniform estimates imply uniform bounds in stratified Sobolev norms, and a Rellich-type theorem in this setting yields subsequential strong convergence on compact subsets away from $\Sigma$.

(3) **Graph/Mosco convergence and passage of harmonicity to the limit.** To justify the limit equation $D_0f = 0$ (and not merely distributional harmonicity in a weak sense), we analyze the quadratic forms associated with $D_\varepsilon$ and prove a suitable graph (Mosco) convergence statement. This prevents loss of domain information in the limit process.
(4) **Cutoff and minimal-domain identification.** Finally, we apply a cutoff/integration-by-parts argument near each isolated singularity. The weights $\phi_{\epsilon}$ and the precise boundary-term estimates show that the limit cannot carry hidden boundary data at the singular point; hence $f$ lies in $\text{Dom}(d_{\min}) \cap \text{Dom}(d_{\min}^*)$.

(5) **Examples and model computations.** To make the borderline weight $\phi_{\epsilon} \sim |\rho| \log \rho$ and the cutoff/minimal-domain mechanism fully transparent (and to connect the analytic conclusions with standard constructions in algebraic geometry), Section 8 collects explicit projective models with isolated singularities and verifies the node condition via a Jacobian/Hessian criterion. Appendix C records the corresponding cone-level computations (borderline logarithmic integrals, commutator decay on shrinking annuli, and worked Hessian checks), which conceptually underlie the boundary-term estimates in Section 6.

### 1.6. Relation to the literature.
The relationship between $L^2$ cohomology and intersection cohomology originates in the foundational work of Cheeger [4] and Goresky-MacPherson [10], and has been further developed in several complex-analytic and arithmetic settings; see, e.g., [19, 20]. Our contribution is not a new conjectural framework but a fully checkable analytic closure of Ohsawa's approach [18] in the isolated singularity case, by providing the required uniform estimates, convergence mechanism, and minimal-domain argument required to make the proof complete. A complementary perspective, placing Ohsawa's $L^2$-IH comparison into a derived/stratified de Rham and arithmetic intersection framework, is developed in [14]; see in particular its discussion of the convergence-and-domain step.

### 1.7. Organization of the paper and notation.
In Section 2 we recall Ohsawa's geometric construction and fix notation. In Section 3 we develop the stratified Sobolev spaces and prove the $\epsilon$-uniform elliptic estimates for $D_{\epsilon}$. In Section 4 we establish the compactness theorem and extract convergent subsequences. In Section 5 we prove the graph (Mosco) convergence results needed to pass harmonicity to the limit. In Section 6 we carry out the cutoff/integration-by-parts argument that forces limits to lie in the minimal domain. Finally, in Section 7 we assemble these ingredients to prove Theorem 1.1 and Corollary 1.3. Section 8 provides explicit examples (including projective hypersurfaces with ordinary double points) and model computations illustrating the borderline weight and the domain mechanism. Appendix A records the b/edge calculus mapping properties used in Section 3, Appendix B supplies a Rellich theorem compatible with the stratified/b framework, and Appendix C collects explicit cone and Jacobian/Hessian calculations supporting Section 8 (and, conceptually, the cutoff argument in Section 6).
**Frequently used notation** We collect here the symbols that recur throughout the paper; the standing hypotheses (H1)–(H4) are listed in Subsection 2.6.

• $V$: a compact irreducible complex analytic space of complex dimension $n$ with isolated singular set $\Sigma = \{x_1, \dots, x_m\}$; $V' := V \setminus \Sigma$.

• $U_x \in V$: a neighborhood with $U_x \cap \Sigma = \{x\}$; $U'_x := U_x \setminus \{x\}$.

• $\rho = \rho_x$: the radial function $\rho(\cdot) = \text{dist}_{g_v}(x, \cdot)$ on $U'_x$; $\rho_0 > 0$ is a fixed truncation radius; $L_x$ denotes the link in the local cone model $(0, \rho_0) \times L_x$.

• $g_v$: a fixed background Hermitian metric on $V'$; $g_0 := g_v$ is the limiting (typically incomplete) metric.

• $(g_\epsilon)_{\epsilon \in (0,1]}$: Ohsawa's family of complete Hermitian metrics on $V'$. On each compact $K \in V'$ we have $g_\epsilon \to g_0$ smoothly, while near $\Sigma$ the family is uniformly quasi-isometric to the conic model (H1)–(H2).

• $(\phi_\epsilon)_{\epsilon \in (0,1]}$: admissible weights; $w_\epsilon := \phi_\epsilon^{-1}$. The weighted normalization used throughout is

$$ \|u\|_{\epsilon, \phi} := \|w_{\epsilon}u\|_{L^2(g_{\epsilon})}. $$

The uniform asymptotics of $\phi_\epsilon$ near $\Sigma$ are summarized in (H3).

• $L^2_\epsilon \Omega^r(V')$: the $L^2$ space of $r$-forms with respect to $g_\epsilon$; $(\cdot, \cdot)_\epsilon$ is the corresponding inner product. We write $H^r_\epsilon := \ker(D_\epsilon) \cap L^2_\epsilon \Omega^r(V')$ for the space of $L^2$ harmonic $r$-forms.

• $d$: the exterior derivative on $V'$; $d^*_\epsilon$: its formal adjoint with respect to $g_\epsilon$. The Hodge–de Rham operator and Laplacian are

$$ D_{\epsilon} := d + d^{*}_{\epsilon}, \quad \Delta_{\epsilon} := D_{\epsilon}^{2}. $$

• For the limiting (incomplete) metric $g_0$, minimal/maximal extensions are denoted by $d_{\min}$ and $d_{\max}$ (and similarly for adjoints), and the corresponding Hodge–de Rham operators by $D_{0,\min}$ and $D_{0,\max}$. For $\epsilon > 0$, completeness (H4) implies $\text{Dom}(d_{\min,\epsilon}) = \text{Dom}(d_{\max,\epsilon})$.

• $H^s_b$: $b$-Sobolev spaces on conic neighborhoods (generated by $b$-vector fields $\rho \partial_\rho, \partial_y$). The global stratified Sobolev norm $H^s_{\text{strat}}(g_\epsilon)$ is obtained by patching $H^s_b$ near $\Sigma$ with standard Sobolev norms on compacta away from $\Sigma$ using a partition of unity (the stratified/b setting).

• $K \in V'$ (and $K \in K_1 \in V'$): generic compact subsets used in interior elliptic estimates and compactness arguments.

• $U_\epsilon$: the canonical isometric identification transporting forms from $L^2_\epsilon \Omega^r(V')$ to the fixed Hilbert space $L^2_0 \Omega^r(V')$; $\tilde{D}_\epsilon := U_\epsilon D_\epsilon U_\epsilon^{-1}$ is the transported operator. We denote by $Q_\epsilon$ and $\tilde{Q}_\epsilon$ the closed quadratic forms associated with $D_\epsilon$ and $\tilde{D}_\epsilon$ (and $Q_{0,\max}$ for $D_{0,\max}$).

• $\chi_\epsilon(\rho)$: a logarithmic cutoff supported near $\rho = 0$, equal to 1 for $\rho \le \epsilon^2$ and 0 for $\rho \ge \epsilon$. Thus $[D_0, \chi_\epsilon]$ is supported in the shrinking
annulus $\rho \in [\varepsilon^2, \varepsilon]$, which is the region where the cutoff/integration-by-parts argument controls the borderline boundary term.

**Further directions.** The proof in this paper is organized around a four-step chain—$\varepsilon$-uniform stratified elliptic estimates, compactness on $K \subset V'$, Mosco/graph convergence on a fixed Hilbert space, and a logarithmic cutoff argument enforcing the minimal domain—all under the standing hypotheses (H1)–(H4). A natural direction is to extend this chain beyond isolated singularities to higher-depth stratified spaces, where one expects an iterated form of the same convergence-and-domain-identification statement using the full iterated edge calculus. Another direction is to strengthen the present convergence theory from harmonic spaces to finer spectral information for $\Delta_\varepsilon$ (for instance, convergence of spectral projectors or heat kernels), which would refine stability beyond cohomological dimensions. Finally, it would be interesting to adapt the strategy to other elliptic complexes on singular varieties (e.g. Dolbeault-type operators under degenerating complete metrics), where the minimal/maximal extension problem is again decisive.

**Acknowledgements.** We thank J. Cheeger, M. Goresky, and R. MacPherson for their pioneering work on analysis and topology on singular spaces. We also thank W. Hsiang, V. Pati, and M. Nagase for insights related to the Cheeger–Goresky–MacPherson conjecture. We acknowledge T. Ohsawa's contributions to the isolated-singularity case, which motivated part of this work. The first author thanks S. Li for collaboration and many helpful discussions during the preparation of this paper.

## 2. GEOMETRIC AND ANALYTIC SETUP

2.1. **Isolated singularities and basic notation.** Let $V$ be a compact irreducible complex analytic space of complex dimension $n$ with isolated singular set $\Sigma = \{x_1, \dots, x_m\}$, and set $V' := V \setminus \Sigma$. Fix once and for all a Hermitian metric $g_v$ on $V'$ (in Ohsawa's framework this is induced from a projective embedding and the Fubini-Study metric; see [18]). Throughout, differential forms are complex-valued and we write $\Omega^r(V')$ for smooth $r$-forms.

For each $x \in \Sigma$, choose a relatively compact neighborhood $U_x \subset V$ such that $U_x \cap \Sigma = \{x\}$ and set $U_x' := U_x \setminus \{x\}$. We write $\rho_x(\cdot) := \operatorname{dist}_{g_v}(x, \cdot)$ on $U_x'$ (when $x$ is fixed, we omit the subscript and write simply $\rho$). Let $K_0 \subset V'$ be a fixed relatively compact open set satisfying $K_0 \cap U_x' = \emptyset$ for all $x \in \Sigma$. This set will be used to separate “interior” estimates from the singular analysis near $\Sigma$.

2.2. **Ohsawa's complete metrics and admissible weights.** We work in the geometric setup of Ohsawa [18]. In particular, there exist:

* a family of complete Hermitian metrics $\{g_\varepsilon\}_{\varepsilon \in (0,1]}$ on $V'$,

* and a family of positive continuous weight functions $\{\phi_\varepsilon\}_{\varepsilon \in (0,1]}$, $\phi_\varepsilon : V' \to (0, \infty)$,
satisfying the following properties.

(M1) (Deformation form) In local coordinates on $V'$ the metric family has the form

$$ (2.2.1) \qquad g_{\varepsilon} = g_{v} + \varepsilon \nabla \psi, $$

for a fixed potential $\psi$ (in Ohsawa's construction, $\psi$ is chosen so that $g_{\varepsilon}$ is complete on $V'$ and compatible with the complex structure). This deformation form is the starting point for the weighted $L^2$-estimate in [18, § 2].

(M2) (Convergence away from $\Sigma$) On every compact $K \subset V'$, the metrics $g_{\varepsilon}$ converge to $g_v$ in $C^{\infty}(K)$ as $\varepsilon \downarrow 0$, and in particular there are constants $c_K, C_K > 0$ such that $c_K g_v \le g_{\varepsilon} \le C_K g_v$ on $K$ for all $\varepsilon \in (0, 1]$. In the old manuscript this is used as “quasi-isometry on $X_p \setminus \{x\}$” to make constants independent of $\varepsilon$ away from the singular point. [18, cf. § 6.5]

(M3) (Conic control near each $x \in \Sigma$) For each $x \in \Sigma$ there exists a Riemannian cone model $(0, \rho_0) \times L_x$ with metric $d\rho^2 + \rho^2 h_x$ such that on $U'_x$ the metrics $g_{\varepsilon}$ are uniformly quasi-isometric (in $\varepsilon$) to this cone metric. Equivalently, there exist $c_x, C_x > 0$ and $\rho_0 > 0$ such that, for all $\varepsilon \in (0, 1]$,

$$ (2.2.2) \qquad c_x(d\rho^2 + \rho^2 h_x) \le g_{\varepsilon} \le C_x(d\rho^2 + \rho^2 h_x) \quad \text{on } U'_x \cap \{\rho < \rho_0\}. $$

This “conical metric structure near the singularity” is the input for Hardy-type bounds and stratified elliptic regularity in the domain-stability argument. [15, 16]

(M4) (Admissible weights) The weights $\phi_{\varepsilon}$ converge uniformly on compact subsets of $V'$ to a positive function $\phi_V$, and near each $x \in \Sigma$ the asymptotics satisfy

$$ (2.2.3) \qquad \phi_{\varepsilon}(\rho, \cdot) = \kappa(\varepsilon) \rho \log \rho + O(\rho^{1+\alpha}) \quad (\rho \downarrow 0) $$

for some $\alpha > 0$ and $\kappa(\varepsilon) \in [c, C]$ with constants $0 < c \le C < \infty$ independent of $\varepsilon$. In particular, $\phi_{\varepsilon} \sim \rho \log \rho$ uniformly in $\varepsilon$ near $x$.

*Remark 2.1.* For the rest of the paper, only the structural consequences (2.2.1)–(2.2.3) are used: (i) smooth convergence on compacta away from $\Sigma$, (ii) uniform conic control near $\Sigma$, and (iii) the explicit weight growth $\phi_{\varepsilon} \sim \rho \log \rho$. These are exactly the hypotheses that allow one to obtain $\varepsilon$-uniform elliptic constants and to control boundary terms in the minimal-domain identification.

2.3. $L^2$ spaces of forms and the Hodge–de Rham operators. For each $\varepsilon \in [0, 1]$ and each degree $r$, let $L_{\varepsilon}^2 \Omega^r(V')$ be the Hilbert space completion of $C_c^{\infty} \Omega^r(V')$ with respect to the norm induced by $g_{\varepsilon}$ and its volume form $dV_{\varepsilon}$:

$$ \|u\|_{L^2(g_{\varepsilon})}^2 := \int_{V'} |u|_{g_{\varepsilon}}^2 dV_{\varepsilon}. $$
We write $(\cdot, \cdot)_\varepsilon$ for the corresponding inner product. The exterior derivative $d$ acts on $C_c^\infty \Omega^r(V')$ independently of $\varepsilon$. Let $d_\varepsilon^*$ be the formal adjoint of $d$ with respect to $(\cdot, \cdot)_\varepsilon$, and define the Hodge-de Rham operator

$$D_\varepsilon := d + d_\varepsilon^*, \quad \Delta_\varepsilon := D_\varepsilon^2 = dd_\varepsilon^* + d_\varepsilon^*d.$$

We denote by $H_\varepsilon^r := \ker(D_\varepsilon) \cap L_\varepsilon^2 \Omega^r(V')$ the space of $L^2$ harmonic $r$-forms.

**2.4. Minimal and maximal extensions; completeness for $\varepsilon > 0$.** Because $g_\varepsilon$ is complete for every $\varepsilon \in (0, 1]$, the integration-by-parts identities hold without boundary terms for compactly supported forms, and the minimal and maximal extensions of $d$ coincide. Concretely, set

$$\mathrm{Dom}(d_{\max,\varepsilon}) := \{u \in L_{\varepsilon}^2 \Omega^r(V') : du \in L_{\varepsilon}^2 \Omega^{r+1}(V') \text{ in the distributional sense}\},$$

$$\mathrm{Dom}(d_{\min,\varepsilon}) := \overline{C_c^\infty \Omega^r(V')} \|\cdot\|_{L^2(g_\varepsilon)} + \|d(\cdot)\|_{L^2(g_\varepsilon)}.$$

Then on a complete Riemannian manifold one has $\mathrm{Dom}(d_{\min,\varepsilon}) = \mathrm{Dom}(d_{\max,\varepsilon})$ and similarly for the adjoint; see Gaffney's Stokes theorem [9] (and also [8] for the harmonic operator framework).

In contrast, the limiting metric $g_0 := g_\nu$ on $V'$ is generally incomplete near $\Sigma$, and the minimal and maximal domains may differ. We therefore fix:

$$d_{\min} := d_{\min,0}, \quad d_{\max} := d_{\max,0},$$

and likewise for the formal adjoint $d_0^*$ and its minimal/maximal extensions. The minimal-domain identification in Theorem 1.1 (c) is precisely the statement that limits of $\varepsilon$-harmonic forms (after normalization by $\phi_\varepsilon$) land in $\mathrm{Dom}(d_{\min}) \cap \mathrm{Dom}(d_{\min}^*)$ rather than only in the maximal domain.

**2.5. Weighted norms and the role of $\phi_\varepsilon$.** Following Ohsawa, the weights $\phi_\varepsilon$ are inserted in estimates to penalize the singular region. Given $u \in C_c^\infty \Omega^r(V')$, define the weighted $L^2$-norm by

$$\|u\|_{\varepsilon,\phi} := \|\phi_\varepsilon^{-1}u\|_{L^2(g_\varepsilon)}.$$

Property (2.2.3) implies that near each $x \in \Sigma$ the weight behaves like $\phi_\varepsilon^{-1} \sim (\rho \log \rho)^{-1}$ uniformly in $\varepsilon$. This is the quantitative input for Hardy-type inequalities on conic regions (cf. [6]) and will be used to pass from local conic control to global stratified Sobolev estimates in Section 3.

**2.6. Standing analytic hypotheses for later sections.** For ease of reference, we summarize the analytic consequences of the setup that will be invoked repeatedly.

(H1) (Uniform conic geometry) The quasi-isometry (2.2.2) holds with constants independent of $\varepsilon \in (0, 1]$ near each $x \in \Sigma$.

(H2) (Uniform compacta control) On each compact $K \subset V'$ the metrics $g_\varepsilon$ are uniformly equivalent and converge to $g_0$ in $C^\infty(K)$.

(H3) (Weight asymptotics) The uniform asymptotic behavior (2.2.3) holds near each $x \in \Sigma$.
<figure><img src="image_2.png" /><figcaption>FIGURE 2. Global-to-local geometry used throughout. The space V has isolated singularities Σ, and each punctured neighborhood $U'_x$ is modeled on a truncated cone $(0, \rho_0) \times L_x$ with radial function $\rho$; the b/edge viewpoint treats $\rho = 0$ as an ideal boundary.</figcaption></figure>

(H4) (Completeness for $\varepsilon > 0$) Each $(V', g_\varepsilon)$ is complete, hence $\text{Dom}(d_{\min,\varepsilon}) = \text{Dom}(d_{\max,\varepsilon})$.

These hypotheses isolate the exact points where $\varepsilon$-uniformity enters: (H1)–(H3) drive the $\varepsilon$-independent elliptic constants and boundary-term estimates, while (H4) ensures that for $\varepsilon > 0$ there is no hidden boundary data.

As illustrated in Figure 2, we will repeatedly pass between the global picture $V \supset \Sigma$ and the local truncated-cone model on each punctured neighborhood $U'_x$. This convention fixes the radial function $\rho$, the link $L_x$, and the b/edge viewpoint at $\rho = 0$ that will be used in all later.

### 3. STRATIFIED/B-SOBOLEV SPACES AND ELLIPTIC ESTIMATES

3.1. **b-geometry near an isolated singularity.** Fix $x \in \Sigma$ and work on $U'_x = U_x \setminus \{x\}$. By (H1) we may identify a punctured neighborhood of $x$ with a truncated cone

$$ (0, \rho_0) \times L_x, \quad \rho \in (0, \rho_0), $$

where $L_x$ is a smooth compact $(2n-1)$-manifold (the link), and the metric family $g_\varepsilon$ is uniformly quasi-isometric to the cone metric $g_c := d\rho^2 + \rho^2 h_x$ on $\{\rho < \rho_0\}$; cf. (2.2.2).

Let $\mathcal{V}_b$ denote the Lie algebra of b-vector fields on $(0, \rho_0) \times L_x$, i.e. smooth vector fields tangent to the boundary of the radial blow-up at $\rho = 0$. Concretely, in local coordinates $(\rho, y)$ on $(0, \rho_0) \times L_x$,

$$ \mathcal{V}_b = \text{span}_{\mathbb{C}^\infty} \{\rho \partial_\rho, \partial_{y_1}, \dots, \partial_{y_{2n-1}}\}. $$
We write $\text{Diff}_b^m$ for the algebra generated by $\mathcal{V}_b$ up to order $m$. The b-tangent bundle ${}^bT$ and its dual ${}^bT^*$ are defined so that $\mathcal{V}_b = C^\infty({}^bT)$. In particular, ${}^bT^*$ is locally spanned by $\frac{d\rho}{\rho}$ and $dy_i$.

**Remark 3.1.** Because the singularities are isolated (depth one and codimension $2n$), b-geometry is the correct local framework: it captures the scaling invariance of the cone and is stable under the uniformly quasi-isometric perturbations $g_\epsilon$ in (H1). For higher depth one would use the full iterated edge calculus; here b-calculus suffices (cf. [16, 15]).

**3.2. b-Sobolev spaces and global stratified Sobolev norms.** Fix $\epsilon \in [0, 1]$ and a degree $r$. On the cone region $U'_x \cap \{\rho < \rho_0\}$ we define the b-Sobolev space $H_b^s \Omega^r$ (for $s \in \mathbb{N}$ first) by

$$ (3.2.1) \quad \|u\|_{H_b^s(g_\epsilon)}^2 := \sum_{|\alpha| \le s} \|V^\alpha u\|_{L^2(g_\epsilon)}^2, \quad u \in C_c^\infty \Omega^r(U'_x \cap \{\rho < \rho_0\}), $$

where $V^\alpha$ ranges over products of $|\alpha|$ b-vector fields in $\mathcal{V}_b$. We then define $H_b^s(g_\epsilon)$ as the completion of $C_c^\infty$ with respect to (3.2.1). For non-integer $s \ge 0$ one may use standard interpolation/duality.

Away from the singular points, we use the usual Sobolev spaces defined by any smooth background metric, which are all equivalent on compact subsets by (H2). We now globalize to a *stratified* Sobolev norm.

Choose a partition of unity $\{\chi_0, \chi_{x_1}, \dots, \chi_{x_m}\}$ on $V'$ such that: (i) $\chi_0$ is supported in $K_0$, and (ii) each $\chi_{x_j}$ is supported in $U'_{x_j} \cap \{\rho < \rho_0\}$. Define, for $s \ge 0$,

$$ (3.2.2) \quad \|u\|_{H_{\text{strat}}^s(g_\epsilon)}^2 := \|\chi_0 u\|_{H^s(K_0, g_\epsilon)}^2 + \sum_{x \in \Sigma} \|\chi_x u\|_{H_b^s(g_\epsilon)}^2. $$

We denote the resulting space by $H_{\text{strat}}^s \Omega^r(V', g_\epsilon)$. Different choices of $(\chi_0, \chi_x)$ produce equivalent norms, with equivalence constants uniform in $\epsilon \in (0, 1]$ thanks to (H1)–(H2).

**Lemma 3.2** (Uniform equivalence of local $L^2$ and Sobolev norms). Fix $x \in \Sigma$ and a cone neighborhood $\{\rho < \rho_0\} \subset U'_x$. For each $s \ge 0$ there exist constants $0 < c_s \le C_s < \infty$ independent of $\epsilon \in (0, 1]$ such that for all $u \in C_c^\infty \Omega^r(U'_x \cap \{\rho < \rho_0\})$,

$$ c_s \|u\|_{H_b^s(g_c)} \le \|u\|_{H_b^s(g_\epsilon)} \le C_s \|u\|_{H_b^s(g_c)}. $$

In particular, the spaces $H_b^s(g_\epsilon)$ are canonically identified as sets and the norms are uniformly equivalent for $\epsilon \in (0, 1]$.

**Proof.** By (H1) there exist constants $0 < c \le C < \infty$, independent of $\epsilon \in (0, 1]$, such that on $U'_x \cap \{\rho < \rho_0\}$,

$$ c g_c \le g_\epsilon \le C g_c $$

as quadratic forms on $TU'_x$.
**Step 1: Pointwise comparison on r-forms.** The inequality above implies a uniform comparison of the induced metrics on $\Lambda^r T^*U'_x$: there exist constants $c_r, C_r > 0$, depending only on $c, C$ and $r$, such that for all $\omega \in \Lambda^r T_p^*U'_x$,

$$c_r |\omega|_{g_c}^2 \le |\omega|_{g_\epsilon}^2 \le C_r |\omega|_{g_c}^2,$$

uniformly in $p$ and $\epsilon$. (Indeed, in a $g_c$-orthonormal basis the eigenvalues of $g_\epsilon$ lie in $[c, C]$, hence those of the induced metric on $\Lambda^r$ lie in $[c^r, C^r]$.)

**Step 2: Volume density comparison.** In real dimension $2n$, the same quasi-isometry yields uniform comparison of volume forms:

$$c^n dV_c \le dV_\epsilon \le C^n dV_c.$$

**Step 3: $L^2$ comparison for each b-derivative.** Fix an integer $s \ge 0$. By definition, $\|u\|_{H_b^s(g_\epsilon)}^2 = \sum_{|\alpha| \le s} \|V^\alpha u\|_{L^2(g_\epsilon)}^2$, where $V^\alpha$ ranges over products of b-vector fields in $\mathcal{V}_b$. These operators are defined independently of $\epsilon$. Applying the pointwise and volume comparisons to the $r$-form $V^\alpha u$ gives

$$\|V^\alpha u\|_{L^2(g_\epsilon)}^2 = \int |V^\alpha u|_{g_\epsilon}^2 dV_\epsilon \le (C_r C^n) \int |V^\alpha u|_{g_c}^2 dV_c = (C_r C^n) \|V^\alpha u\|_{L^2(g_c)}^2,$$

and similarly $\|V^\alpha u\|_{L^2(g_\epsilon)}^2 \ge (c_r c^n) \|V^\alpha u\|_{L^2(g_c)}^2$. Summing over $|\alpha| \le s$ yields the claimed uniform equivalence for integer $s$.

**Step 4: Non-integer $s$.** For non-integer $s \ge 0$ we define $H_b^s$ by interpolation and duality. Since the norms are uniformly equivalent on the integer scale and the comparison maps are bounded with constants independent of $\epsilon$, the same uniform equivalence extends to all $s \ge 0$ by standard interpolation. This also shows that the spaces $H_b^s(g_\epsilon)$ coincide as sets and the norms are uniformly equivalent for $\epsilon \in (0, 1]$. $\square$

**3.3. Weighted multipliers induced by $\phi_\epsilon$.** Recall from (H3) that near each $x \in \Sigma$ we have $\phi_\epsilon(\rho, \cdot) \sim \rho \log \rho$ uniformly in $\epsilon$. Set $w_\epsilon := \phi_\epsilon^{-1}$. We will frequently insert $w_\epsilon$ as a multiplier; for elliptic estimates we need that multiplication by $w_\epsilon$ preserves stratified Sobolev regularity with constants independent of $\epsilon$.

**Lemma 3.3** (Uniform commutator bound induced by $\phi_\epsilon$). Fix $s \in \mathbb{N}$. There exists $C_s < \infty$ independent of $\epsilon \in (0, 1]$ such that for every $u \in C_c^\infty \Omega^r(V')$, writing $v := w_\epsilon u = \phi_\epsilon^{-1}u$, we have

$$\|[D_\epsilon, w_\epsilon]u\|_{H_{\text{strat}}^{s-1}(g_\epsilon)} \le C_s \|v\|_{H_{\text{strat}}^{s-1}(g_\epsilon)}.$$

*Proof.* Since $w_\epsilon = \phi_\epsilon^{-1}$ we have

$$dw_\epsilon = -\phi_\epsilon^{-2} d\phi_\epsilon = -w_\epsilon d(\log \phi_\epsilon).$$

For the de Rham operator $D_\epsilon = d + d_\epsilon^*$ one has the standard commutator identity

$$[D_\epsilon, w_\epsilon]u = (dw_\epsilon) \wedge u - \iota_{\nabla w_\epsilon} u,$$
which is a zeroth-order action of $dw_ε$ on forms. Writing $u = \phi_ε v$ (i.e. $v = w_ε u$) gives

$$[D_ε, w_ε]u = (dw_ε) \wedge (\phi_ε v) - \iota_{\nabla w_ε}(\phi_ε v) = -(d \log \phi_ε) \wedge v + \iota_{\nabla(\log \phi_ε)} v.$$

Thus $[D_ε, w_ε]u$ is obtained by multiplying $v$ by the coefficients $d \log \phi_ε$ (and their metric duals).

By (H2), on the interior region $K_0$ the functions $\log \phi_ε$ and their derivatives are uniformly bounded, hence multiplication by $d \log \phi_ε$ is bounded on $H^s(K_0)$ with constants independent of $ε$. Near each $x \in \Sigma$, the asymptotics (H3) imply that b-derivatives of $\log \phi_ε$ are uniformly bounded: the leading term satisfies $\rho \partial_\rho \log(\rho|\log \rho|) = 1 + (\log \rho)^{-1}$, and the $O(\rho^\alpha)$ correction contributes uniformly bounded b-derivatives for $\rho < \rho_0$. Therefore, on each conic patch, multiplication by $d \log \phi_ε$ defines a bounded zeroth-order b-operator on $H_b^{s-1}$ with constants independent of $ε$ (cf. Appendix A, Lemma A.4). Patching with the partition of unity defining $H_{\text{strat}}^{s-1}$ yields the stated estimate. □

**3.4. The Hodge-de Rham operator as a b-elliptic family.** Let $D_ε = d + d_ε^*$ be the Hodge-de Rham operator on $(V', g_ε)$ as in Section 2. On each cone neighborhood $U'_x \cap \{\rho < \rho_0\}$, $D_ε$ is a first-order differential operator with coefficients smooth up to $\rho = 0$ when expressed in the b-frame $\{\frac{d\rho}{\rho}, dy\}$ for forms and the dual b-frame $\{\rho \partial_\rho, \partial_y\}$ for vector fields. In particular,

$$D_ε \in \text{Diff}_b^1(U'_x \cap \{\rho < \rho_0\}; \Lambda^{\bullet b} T^*),$$

and its b-principal symbol ${}^b\sigma_1(D_ε)$ is invertible away from the zero section of ${}^bT^*$. Moreover, by (H1)–(H2) the b-principal symbols form a uniformly bounded, uniformly invertible family for $ε \in (0, 1]$.

**Proposition 3.4 (Uniform b-elliptic parametrix).** Fix $x \in \Sigma$ and a cone neighborhood $\{\rho < \rho_0\} \subset U'_x$. There exist operators $Q_ε \in \Psi_b^{-1}$ and remainders $R_ε \in \Psi_b^{-\infty}$ on $U'_x \cap \{\rho < \rho_0\}$ such that

$$Q_ε D_ε = I - R_ε \quad \text{on } C_c^\infty \Omega^\bullet (U'_x \cap \{\rho < \rho_0\}),$$

and the mapping norms of $Q_ε$ and $R_ε$ between b-Sobolev spaces are bounded uniformly in $ε \in (0, 1]$.

*Proof.* Fix $x \in \Sigma$ and work on $U'_x \cap \{\rho < \rho_0\}$ with the b-structure determined by the boundary defining function $\rho$. By the discussion preceding the proposition, $D_ε \in \text{Diff}_b^1$ is b-elliptic and its b-principal symbols form a uniformly invertible family: there exists $\lambda > 0$ such that

$$\|({}^b\sigma_1(D_ε)(p, \xi))^{-1}\| \le \lambda^{-1} \quad \text{for all } \varepsilon \in (0, 1], \quad (p, \xi) \in {}^bS^*(U'_x).$$

Moreover, by (H1)–(H2), in fixed local b-coordinates and a fixed local b-frame, the coefficients of $D_ε$ have uniformly bounded $C^N$ norms on $U'_x \cap \{\rho < \rho_0\}$ for each finite $N$; hence the b-symbol seminorms of ${}^b\sigma_1(D_ε)$ (up to any fixed order) are bounded uniformly in $ε$.
The b-calculus parametrix construction (symbolic inversion followed by an asymptotic correction scheme) therefore applies uniformly to the family $D_\epsilon$: one constructs a family of symbols $q_\epsilon \sim q_{-1,\epsilon} + q_{-2,\epsilon} + \cdots$ with $q_{-1,\epsilon} = ({}^b\sigma_1(D_\epsilon))^{-1}$ and with symbol seminorms bounded uniformly in $\epsilon$ at each finite stage. Quantizing $q_\epsilon$ gives $Q_\epsilon \in \Psi_b^{-1}$ and remainders $R_\epsilon, R'_\epsilon \in \Psi_b^{-\infty}$ such that

$$Q_{\epsilon}D_{\epsilon} = I - R_{\epsilon}, \quad D_{\epsilon}Q_{\epsilon} = I - R'_{\epsilon} \quad \text{on } C_c^{\infty}\Omega^{\bullet}(U'_{x} \cap \{\rho < \rho_0\}).$$

Finally, the mapping properties of $\Psi_b^m$ on b-Sobolev spaces imply

$$\sup_{\epsilon \in (0,1]} \|Q_\epsilon\|_{H_b^{s-1} \to H_b^s} < \infty, \quad \sup_{\epsilon \in (0,1]} \|R_\epsilon\|_{H_b^s \to H_b^t} + \sup_{\epsilon \in (0,1]} \|R'_\epsilon\|_{H_b^s \to H_b^t} < \infty$$

for all fixed $s, t$, since these operator norms are controlled by finitely many b-symbol seminorms. For details of the b-parametrix construction and the dependence of mapping bounds on symbol seminorms, see [16, Chapters 3–5]; for closely related edge mapping statements see [15]. □

**3.5. ε-uniform stratified elliptic estimates.** We now state the elliptic estimate that drives the compactness and convergence arguments in later sections.

**Theorem 3.5** (ε-uniform stratified elliptic estimate). Let $s \ge 1$ and $r \in \{0, 1, \dots, 2n\}$. There exists a constant $C_s < \infty$, independent of $\epsilon \in (0, 1]$, such that for all $u \in C_c^\infty\Omega^r(V')$,

$$(3.5.1) \qquad \|u\|_{H_{\text{strat}}^s(g_\epsilon)} \le C_s (\|D_\epsilon u\|_{H_{\text{strat}}^{s-1}(g_\epsilon)} + \|u\|_{H^{s-1}(K_0, g_\epsilon)}).$$

Equivalently, if $\text{supp } u \cap K_0 = \emptyset$ then the interior term drops and

$$\|u\|_{H_{\text{strat}}^s(g_\epsilon)} \le C_s \|D_\epsilon u\|_{H_{\text{strat}}^{s-1}(g_\epsilon)}.$$

*Proof.* We apply the partition of unity $\{\chi_0, \chi_x\}$ from (3.2.2). On $\text{supp } \chi_0 \subset K_0$, (H2) gives a uniformly smooth family of elliptic operators; hence the usual interior elliptic estimate for first-order elliptic operators yields

$$\|\chi_0 u\|_{H^s(K_0, g_\nu)} \le C_s (\|D_\epsilon(\chi_0 u)\|_{H^{s-1}(K_0, g_\nu)} + \|\chi_0 u\|_{H^{s-1}(K_0, g_\nu)}),$$

with constants independent of $\epsilon$.

For each $x \in \Sigma$, work on the cone region $\text{supp } \chi_x \subset U'_x \cap \{\rho < \rho_0\}$. Using Proposition 3.4 we write

$$\chi_x u = Q_{\epsilon}D_{\epsilon}(\chi_x u) + R_{\epsilon}(\chi_x u).$$

Taking $H_b^s$ norms and using the uniform mapping properties of $Q_\epsilon$ and smoothing of $R_\epsilon$ gives

$$\|\chi_x u\|_{H_b^s(g_\epsilon)} \le C_s (\|D_\epsilon(\chi_x u)\|_{H_b^{s-1}(g_\epsilon)} + \|\chi_x u\|_{H_b^{s-1}(g_\epsilon)}),$$

with $C_s$ independent of $\epsilon$. Here we use that $R_\epsilon \in \Psi_b^{-\infty}$ is smoothing, hence for each fixed $s$ it maps $H_b^{s-1} \to H_b^s$ continuously (with operator norm
bounded uniformly in $\varepsilon$ by Proposition 3.4); thus

$$ \|R_{\varepsilon}(\chi_x u)\|_{H_b^s(g_{\varepsilon})} \le C_s \|\chi_x u\|_{H_b^{s-1}(g_{\varepsilon})}. $$

Next, commute $D_{\varepsilon}$ with $\chi_x$: $D_{\varepsilon}(\chi_x u) = \chi_x D_{\varepsilon} u + [D_{\varepsilon}, \chi_x]u$, where $[D_{\varepsilon}, \chi_x] \in \text{Diff}_b^0$ has coefficients supported where $\nabla \chi_x \neq 0$, hence away from $\rho = 0$ and therefore uniformly bounded by (H2). Thus,

$$ \|[D_{\varepsilon}, \chi_x]u\|_{H_b^{s-1}(g_{\varepsilon})} \le C'_s \|u\|_{H_{\text{strat}}^{s-1}(g_{\varepsilon})} $$

with $C'_s$ independent of $\varepsilon$. On the region $\{\nabla \chi_x \neq 0\}$ we have $\rho \ge c_x > 0$, so b-derivatives are equivalent to ordinary derivatives and $H_b^{s-1}$ is equivalent to the standard $H^{s-1}$ there. Moreover, by (H2) the coefficients of $D_{\varepsilon}$ (and finitely many derivatives) are uniformly bounded on this region; hence $[D_{\varepsilon}, \chi_x]$ defines a uniformly bounded order-0 operator on $H_b^{s-1}$. By definition of the stratified Sobolev norm (3.2.2), the family $\{\chi_0, \chi_x\}$ yields

$$ \|u\|_{H_{\text{strat}}^s(g_{\varepsilon})} \lesssim \|\chi_0 u\|_{H^s(K_0, g_v)} + \sum_{x \in \Sigma} \|\chi_x u\|_{H_b^s(g_{\varepsilon})}, $$

and similarly at order $s-1$. Since $\Sigma$ is finite, summing the cone estimates over $x \in \Sigma$ and combining with the $\chi_0$ interior estimate yields (3.5.1). Finally, when $\text{supp } u \cap K_0 = \emptyset$ the term $\|u\|_{H^{s-1}(K_0, g_v)}$ vanishes, proving the second assertion. $\square$

**3.6. A weighted version.** We record a form of the estimate compatible with Ohsawa's normalization by $\phi_{\varepsilon}$.

**Corollary 3.6 (Weighted $\varepsilon$-uniform estimate).** Let $s \ge 1$. There exists $C_s$ independent of $\varepsilon \in (0, 1]$ such that for all $u \in C_c^{\infty} \Omega^r(V')$,

$$ \|\phi_{\varepsilon}^{-1} u\|_{H_{\text{strat}}^{s}(g_{\varepsilon})} \le C_s \left( \|\phi_{\varepsilon}^{-1} D_{\varepsilon} u\|_{H_{\text{strat}}^{s-1}(g_{\varepsilon})} + \|\phi_{\varepsilon}^{-1} u\|_{H_{\text{strat}}^{s-1}(g_{\varepsilon})} + \|\phi_{\varepsilon}^{-1} u\|_{H^{s-1}(K_0, g_v)} \right). $$

*Proof.* Let $w_{\varepsilon} := \phi_{\varepsilon}^{-1}$ and set $v := w_{\varepsilon}u$ so that $u = \phi_{\varepsilon}v$. Applying the $\varepsilon$-uniform stratified elliptic estimate of Theorem 3.5 to $v$ gives

$$ (3.6.1) \qquad \|v\|_{H_{\text{strat}}^s(g_{\varepsilon})} \le C_s \left( \|D_{\varepsilon}v\|_{H_{\text{strat}}^{s-1}(g_{\varepsilon})} + \|v\|_{H^{s-1}(K_0, g_v)} \right), $$

with $C_s$ independent of $\varepsilon$.

We now rewrite $D_{\varepsilon}v$ in terms of $u$. Since $v = w_{\varepsilon}u$,

$$ D_{\varepsilon}v = D_{\varepsilon}(w_{\varepsilon}u) = w_{\varepsilon}D_{\varepsilon}u + [D_{\varepsilon}, w_{\varepsilon}]u. $$

Taking $H_{\text{strat}}^{s-1}$ norms and using the triangle inequality yields

$$ \|D_{\varepsilon}v\|_{H_{\text{strat}}^{s-1}(g_{\varepsilon})} \le \|w_{\varepsilon}D_{\varepsilon}u\|_{H_{\text{strat}}^{s-1}(g_{\varepsilon})} + \|[D_{\varepsilon}, w_{\varepsilon}]u\|_{H_{\text{strat}}^{s-1}(g_{\varepsilon})}. $$

By Lemma 3.3 (uniform commutator bound), the commutator term satisfies

$$ \|[D_{\varepsilon}, w_{\varepsilon}]u\|_{H_{\text{strat}}^{s-1}(g_{\varepsilon})} \le C'_{s} \|v\|_{H_{\text{strat}}^{s-1}(g_{\varepsilon})}, $$

with $C'_s$ independent of $\varepsilon$.
Substituting these bounds into (3.6.1) and renaming constants, we obtain

$$ \|w_{\varepsilon}u\|_{H_{\text{strat}}^{s}(g_{\varepsilon})} \le C_s \left( \|w_{\varepsilon}D_{\varepsilon}u\|_{H_{\text{strat}}^{s-1}(g_{\varepsilon})} + \|w_{\varepsilon}u\|_{H_{\text{strat}}^{s-1}(g_{\varepsilon})} + \|w_{\varepsilon}u\|_{H^{s-1}(K_{0,g_{\varepsilon}})} \right). $$

This is exactly the claimed weighted $\varepsilon$-uniform estimate. $\square$

<figure><img src="image_3.png" /></figure>

## 4. COMPACTNESS AND SUBSEQUENTIAL CONVERGENCE

### 4.1. Uniform local elliptic regularity on compact subsets.
We begin with a standard but crucial observation: on compact subsets away from the singular set, the family of operators $D_{\varepsilon}$ is uniformly elliptic with uniformly controlled coefficients. This yields $\varepsilon$-uniform local elliptic estimates with constants depending only on the compact set.

**Lemma 4.1** ($\varepsilon$-uniform local elliptic estimate on compacta). Let $K \in V'$ be a compact set. Choose an open neighborhood $K \in K_1 \in V'$. Then for every integer $s \ge 0$ there exists a constant $C_{K,s} < \infty$, independent of $\varepsilon \in (0, 1]$, such that for all $u \in C^{\infty}\Omega^{\bullet}(V')$,

$$ (4.1.1) \qquad \|u\|_{H^{s+1}(K,g_v)} \le C_{K,s} \left( \|D_{\varepsilon}u\|_{H^s(K_1,g_v)} + \|u\|_{L^2(K_1,g_v)} \right). $$

*Proof.* Fix $K \in K_1 \in V'$. Choose $\chi \in C_c^{\infty}(V')$ with $\chi \equiv 1$ on a neighborhood of $K$ and $\text{supp}(\chi) \in K_1$. It suffices to prove (4.1.1) for $\chi u$ in place of $u$, since $\chi u = u$ on $K$.

Cover the compact set $\text{supp}(\chi) \subset K_1$ by finitely many coordinate charts $\{(U_j, \varphi_j)\}_{j=1}^N$ with $U_j \in V'$ and choose a partition of unity $\{\psi_j\}$ subordinate to $\{U_j\}$. On each $U_j$, write $D_{\varepsilon}$ in the fixed coordinates and a fixed local frame of differential forms as a first-order elliptic system with smooth coefficients.

By (H2) (smooth convergence of $g_{\varepsilon}$ on compacta away from $\Sigma$), the coefficients of $D_{\varepsilon}$ and their partial derivatives up to order $s+1$ are bounded uniformly in $\varepsilon \in (0, 1]$ on each $U_j$. Moreover, the ellipticity constant of the principal symbol of $D_{\varepsilon}$ is bounded below uniformly in $\varepsilon$ on $U_j$ (since $g_{\varepsilon}$ are uniformly equivalent on $K_1$ and depend smoothly on $\varepsilon$ on compacta). See, for instance, [21, Ch. 5] for interior estimates for elliptic systems; the uniformity in $\varepsilon$ follows from the uniform coefficient bounds on $K_1$. Hence the standard interior elliptic estimate for first-order elliptic systems (applied to $\psi_j\chi u$) yields

$$ (4.1.2) \qquad \|\psi_j\chi u\|_{H^{s+1}(U_j,g_v)} \le C_{j,s} \left( \|D_{\varepsilon}(\psi_j\chi u)\|_{H^s(U_j,g_v)} + \|\psi_j\chi u\|_{L^2(U_j,g_v)} \right), $$

with $C_{j,s}$ independent of $\varepsilon$.

Expanding $D_{\varepsilon}(\psi_j\chi u) = \psi_j\chi D_{\varepsilon}u + [D_{\varepsilon}, \psi_j\chi]u$ and using that $[D_{\varepsilon}, \psi_j\chi]$ is a zeroth-order operator with coefficients (and finitely many derivatives) bounded uniformly in $\varepsilon$ on $U_j$, we obtain from (4.1.2)

$$ \|\psi_j\chi u\|_{H^{s+1}(U_j,g_v)} \le C'_{j,s} \left( \|D_{\varepsilon}u\|_{H^s(U_j,g_v)} + \|u\|_{L^2(U_j,g_v)} \right), $$
where $C'_{j,s}$ is still independent of $\varepsilon$ (absorbing lower-order terms into the $L^2$ term since $\text{supp}(\psi_j\chi) \in U_j$ is compact). Summing over $j = 1, \dots, N$ and using that $\chi \equiv 1$ on $K$ gives

$$ \|u\|_{H^{s+1}(K,g_v)} = \|\chi u\|_{H^{s+1}(K,g_v)} \le C_{K,s} (\|D_\varepsilon u\|_{H^s(K_1,g_v)} + \|u\|_{L^2(K_1,g_v)}), $$

with $C_{K,s}$ independent of $\varepsilon$, which is (4.1.1). $\square$

<figure><img src="image_4.png" /></figure>

**Corollary 4.2** (Uniform smooth bounds for $\varepsilon$-harmonic forms on compacta). Let $K \in V'$ and $K \in K_1 \in V'$. If $u_\varepsilon \in C^\infty\Omega^r(V')$ satisfies $D_\varepsilon u_\varepsilon = 0$ on $V'$, then for every integer $s \ge 0$ there exists $C_{K,s}$ independent of $\varepsilon \in (0,1]$ such that

$$ \|u_\varepsilon\|_{H^{s+1}(K,g_v)} \le C_{K,s} \|u_\varepsilon\|_{L^2(K_1,g_v)}. $$

In particular, $u_\varepsilon$ is smooth on $V'$ (already for each fixed $\varepsilon$) with $\varepsilon$-uniform bounds on compacta.

*Proof.* Apply Lemma 4.1 with $u = u_\varepsilon$ and use that $D_\varepsilon u_\varepsilon = 0$. The constant $C_{K,s}$ is independent of $\varepsilon$ by Lemma 4.1. $\square$

<figure><img src="image_5.png" /></figure>

**4.2. Rellich compactness on compact subsets.** We will use the standard Rellich-Kondrachov compactness theorem on compact subsets of the smooth manifold $V'$: for $K \in V'$ and integer $s \ge 1$, the embedding $H^s(K) \hookrightarrow H^{s-1}(K)$ is compact. We only need this classical form.

**Lemma 4.3** (Rellich compactness on compact subsets). Let $\Omega \in V'$ be a relatively compact open subset with Lipschitz (e.g. smooth) boundary, and let $s \ge 1$ be an integer. Then the inclusion $H^s(\Omega, g_v) \hookrightarrow H^{s-1}(\Omega, g_v)$ is compact. Consequently, any sequence bounded in $H^s(\Omega, g_v)$ admits a subsequence converging strongly in $H^{s-1}(\Omega, g_v)$.

*Proof.* Cover $\bar{\Omega}$ by finitely many coordinate charts and a partition of unity. In each chart, reduce to the Rellich-Kondrachov theorem on bounded Lipschitz domains in $\mathbb{R}^m$, and patch the local compactness to obtain the global compactness on $\Omega$. See, e.g., [7, Ch. 6] or [1, Ch. 4]. $\square$

<figure><img src="image_6.png" /></figure>

**4.3. From weighted $L^2$ normalization to uniform local Sobolev bounds.**

The convergence statement in Theorem 1.1 (b) is formulated for sequences normalized by $\|\phi_\varepsilon^{-1} f_\varepsilon\|_{L^2(g_\varepsilon)} = 1$. We now show that this normalization implies uniform $L^2$ (hence Sobolev) control on compact subsets.

**Lemma 4.4** (Uniform local $L^2$ bounds from weighted normalization). Let $K \in V'$ be compact. Then there exists $C_K < \infty$ independent of $\varepsilon \in (0,1]$ such that for all measurable $u$ on $V'$,

$$ (4.3.1) \qquad \|u\|_{L^2(K,g_\varepsilon)} \le C_K \|\phi_\varepsilon^{-1}u\|_{L^2(g_\varepsilon)}. $$

*Proof.* Since $K \in V'$ stays a positive distance away from $\Sigma$, (H3) implies that $\phi_\varepsilon \to \phi_V$ uniformly on $K$, where $\phi_V > 0$. Hence there exists $c_K > 0$
such that $\phi_{\varepsilon} \ge c_K$ on $K$ for all $\varepsilon \in (0, 1]$. Therefore $|\phi_{\varepsilon}^{-1}u|^2 \ge c_K^{-2}|u|^2$ pointwise on $K$, and integrating against $dV_{\varepsilon}$ yields

$$ \|u\|_{L^2(K,g_{\varepsilon})} \le c_K^{-1} \|\phi_{\varepsilon}^{-1}u\|_{L^2(K,g_{\varepsilon})} \le c_K^{-1} \|\phi_{\varepsilon}^{-1}u\|_{L^2(g_{\varepsilon})}. $$

Thus (4.3.1) holds with $C_K = c_K^{-1}$.

<figure><img src="image_7.png" /></figure>

**Proposition 4.5** (Uniform local Sobolev bounds for normalized $\varepsilon$-harmonic forms). Fix $r$ and let $f_{\varepsilon} \in H_{\varepsilon}^r$ satisfy the normalization

$$ (4.3.2) \qquad \|\phi_{\varepsilon}^{-1} f_{\varepsilon}\|_{L^2(g_{\varepsilon})} = 1. $$

Then for every compact $K \subset V'$ and every integer $s \ge 0$ there exists $C_{K,s} < \infty$, independent of $\varepsilon \in (0, 1]$, such that

$$ \|f_{\varepsilon}\|_{H^s(K,g_{v})} \le C_{K,s}. $$

*Proof.* Fix $K \subset V'$ and choose $K \subset K_1 \subset V'$. By Lemma 4.4 and the normalization (4.3.2),

$$ \|f_{\varepsilon}\|_{L^2(K_1,g_{\varepsilon})} \le C_{K_1} \|\phi_{\varepsilon}^{-1} f_{\varepsilon}\|_{L^2(g_{\varepsilon})} = C_{K_1}, $$

with $C_{K_1}$ independent of $\varepsilon$. By (H2), the metrics $g_{\varepsilon}$ and $g_{v}$ are uniformly equivalent on $K_1$, hence $\|f_{\varepsilon}\|_{L^2(K_1,g_{v})} \le C'_{K_1}$ uniformly in $\varepsilon$.

Since $f_{\varepsilon} \in H_{\varepsilon}^r = \ker(D_{\varepsilon})$, we have $D_{\varepsilon}f_{\varepsilon} = 0$ on $V'$. Choose a nested chain of relatively compact open sets

$$ K \subset \Omega_0 \subset \Omega_1 \subset \cdots \subset \Omega_s \subset K_1. $$

Applying Corollary 4.2 successively on $(\Omega_{j-1}, \Omega_j)$ yields for each $j = 1, \dots, s$ a bound

$$ \|f_{\varepsilon}\|_{H^j(\Omega_{j-1},g_{v})} \le C_j \|f_{\varepsilon}\|_{L^2(\Omega_j,g_{v})}, $$

with $C_j$ independent of $\varepsilon$. Combining these and using $\Omega_s \subset K_1$ gives

$$ \|f_{\varepsilon}\|_{H^s(K,g_{v})} \le \|f_{\varepsilon}\|_{H^s(\Omega_0,g_{v})} \le C_{K,s} \|f_{\varepsilon}\|_{L^2(K_1,g_{v})} \le C_{K,s}, $$

with $C_{K,s}$ independent of $\varepsilon$.

<figure><img src="image_8.png" /></figure>

**4.4. Subsequence convergence on compact subsets and diagonal extraction.** We now combine the uniform local Sobolev bounds with Rellich compactness.

**Theorem 4.6** (Subsequence convergence on compact subsets). Fix $r$ and let $f_{\varepsilon} \in H_{\varepsilon}^r$ satisfy $\|\phi_{\varepsilon}^{-1}f_{\varepsilon}\|_{L^2(g_{\varepsilon})} = 1$. Let $s \ge 1$ be an integer. Then for every compact $K \subset V'$ the family $\{f_{\varepsilon}\}_{\varepsilon \in (0,1]}$ is precompact in $H^{s-1}(K,g_{v})$. In particular, given any sequence $\varepsilon_j \downarrow 0$, there exists a subsequence (not relabeled) and a form $f \in H_{\text{loc}}^{s-1}\Omega^r(V')$ such that

$$ f_{\varepsilon_j} \to f \quad \text{strongly in } H^{s-1}(K,g_{v}) \text{ for every } K \subset V'. $$

*Proof.* Fix $K \subset V'$ and let $s \ge 1$ be the integer from the statement. Choose an open set $\Omega \subset V'$ with Lipschitz (e.g. smooth) boundary such that $K \subset \Omega$ and $\bar{\Omega} \subset V'$. (For instance, take a small smooth neighborhood of $K$.) By Proposition 4.5, the sequence $\{f_{\varepsilon_j}\}$ is bounded in $H^s(\Omega, g_{v})$. By Lemma 4.3,
after passing to a subsequence it converges strongly in $H^{s-1}(\Omega, g_v)$. Restricting to $K$ then yields strong convergence in $H^{s-1}(K, g_v)$.

To obtain a single subsequence converging on all compact sets, choose an exhaustion of $V'$ by relatively compact open sets with smooth boundary,

$$ \Omega_1 \in \Omega_2 \in \cdots \in V', \quad \bigcup_{\ell \ge 1} \Omega_\ell = V'. $$

Apply the previous argument on $\Omega_1$ and extract a subsequence, then on $\Omega_2$ extract a further subsequence, etc. A diagonal argument yields a subsequence converging strongly in $H^{s-1}(\Omega_\ell, g_v)$ for every $\ell$. Since any compact $K \in V'$ is contained in some $\Omega_\ell$, restricting gives strong convergence in $H^{s-1}(K, g_v)$ for every compact $K \in V'$. The limit $f$ lies in $H_{\text{loc}}^{s-1}$ by construction. $\square$

**Remark 4.7.** In Theorem 1.1 (b) we will take $s > n + 1$ (in particular $s-1 > (2n)/2$, where $2n = \dim_{\mathbb{R}} V'$) so that Sobolev embedding on compact subsets applies. Combined with the uniform higher local Sobolev bounds from Proposition 4.5 (and a standard bootstrapping/diagonal extraction), this yields convergence in $C^k$ on compact subsets for all $k$. This strengthened convergence will be used later when passing to the limit in operator identities.

**4.5. Compatibility with stratified norms.** Although the convergence in Theorem 4.6 is stated in standard Sobolev spaces on compact subsets of $V'$, it is compatible with the stratified framework of Section 3. Indeed, for $K \in V'$ we have $\text{dist}(K, \Sigma) > 0$, hence the b-geometry plays no role and the stratified and standard norms are uniformly equivalent on $K$ by (H2). We will nevertheless keep the stratified norms in later global estimates (notably in the Mosco convergence and minimal-domain arguments), where uniform control near $\Sigma$ matters. For a precise statement of this $\varepsilon$-uniform equivalence on compacta, see Appendix B.4, in particular Lemma B.5.

## 5. GRAPH (MOSCO) CONVERGENCE OF HODGE-DE RHAM OPERATORS

**Section overview.** To pass from the local compactness of $L^2$ harmonic forms to a limit that is harmonic for the limiting operator, we compare the Hodge-de Rham family $D_\varepsilon$ on varying Hilbert spaces $H_\varepsilon^r = L^2\Omega^r(V', g_\varepsilon)$ by transporting it to the fixed space $H_0^r$. We then study the associated closed quadratic forms $\tilde{Q}_\varepsilon$ and show that they converge to $\tilde{Q}_0$ in the sense of Mosco. The main technical output is Theorem 5.5, which, via Theorem 5.4, yields strong resolvent convergence of the transported operators. This provides the functional-analytic mechanism used in the proof of Theorem 1.1 (b) to pass harmonicity to the limit, before the minimal-domain cutoff argument of Section 6 upgrades the maximal equation to the minimal domain and completes Theorem 1.1 (c).
**5.1. A fixed Hilbert space and uniform identification maps.** For each $\varepsilon \in [0, 1]$ and degree $r$, let

$$H_{\varepsilon}^{r} := L^{2}\Omega^{r}(V', g_{\varepsilon}), \quad (\cdot, \cdot)_{\varepsilon} \text{ the associated inner product.}$$

To compare objects for different $\varepsilon$, we fix the reference Hilbert space

$$H_{0}^{r} := L^{2}\Omega^{r}(V', g_{0}),$$

where $g_0 = g_v$ is the limiting metric from Section 2.

**Lemma 5.1** (Uniform equivalence of $L^2$ norms and canonical isometries). For each $r$ and each $\varepsilon \in (0, 1]$ there exists a fiberwise positive definite bundle automorphism $A_{\varepsilon} : \Lambda^r T^*V' \to \Lambda^r T^*V'$ and a measurable function $J_{\varepsilon} > 0$ such that for all compactly supported smooth $r$-forms $u, v$,

$$(5.1.1) \qquad (u, v)_{\varepsilon} = \int_{V'} \langle A_{\varepsilon}u, v \rangle_{g_0} J_{\varepsilon} dV_0.$$

In particular, the map

$$\mathcal{U}_{\varepsilon} : C_{c}^{\infty} \Omega^{r}(V') \to C_{c}^{\infty} \Omega^{r}(V'), \quad \mathcal{U}_{\varepsilon}u := J_{\varepsilon}^{1/2} A_{\varepsilon}^{1/2}u$$

extends to an isometry $\mathcal{U}_{\varepsilon} : H_{\varepsilon}^{r} \to H_{0}^{r}$. Moreover, by (H1)-(H2) the operators $\mathcal{U}_{\varepsilon}$ and $\mathcal{U}_{\varepsilon}^{-1}$ are uniformly bounded on the local (stratified/b) Sobolev scales appearing in Sections 2–3 (with constants independent of $\varepsilon \in (0, 1]$ on fixed compacta and on fixed conic neighborhoods).

*Proof.* At each $p \in V'$, the two inner products on $\Lambda^r T_p^*V'$ induced by $g_\varepsilon$ and $g_0$ are related by a unique positive definite linear map $A_\varepsilon(p)$, and the volume forms satisfy $dV_\varepsilon = J_\varepsilon dV_0$. This yields (5.1.1). To justify the uniform boundedness on Sobolev scales, fix an integer $s \ge 0$.

(i) On compacta away from $\Sigma$. Let $K \subset V'$. By (H2), the coefficients of $g_\varepsilon$ converge to those of $g_0$ in $C^\infty(K)$, hence the pointwise comparison endomorphisms $A_\varepsilon$ and densities $J_\varepsilon$ (and their derivatives up to any finite order) are bounded uniformly in $\varepsilon$ on $K$. Therefore multiplication by $J_\varepsilon^{\pm 1/2} A_\varepsilon^{\pm 1/2}$ defines bounded operators on $H^s(K, g_0)$, with operator norms controlled by finitely many $C^s$-bounds, hence uniform in $\varepsilon$.

(ii) On conic neighborhoods near $\Sigma$. On each punctured neighborhood $U'_x \cap \{\rho < \rho_0\}$, (H1) gives uniform quasi-isometry to the cone model and uniform control of coefficients in b-frames. Consequently $A_\varepsilon$, $J_\varepsilon$ and their b-derivatives up to order $s$ are uniformly bounded, so multiplication by $J_\varepsilon^{\pm 1/2} A_\varepsilon^{\pm 1/2}$ is uniformly bounded on $H_b^s$ on such neighborhoods. Combining this with the uniform equivalence of local $H_b^s$ and stratified norms (Lemma 3.2) gives the stated uniform boundedness on the local stratified/b Sobolev scales. $\square$

Henceforth we work on the fixed Hilbert space $H_0^r$ and transport operators/forms from $H_\varepsilon^r$ via $\mathcal{U}_\varepsilon$.
5.2. Closed quadratic forms associated with $D_\epsilon$. Let $D_\epsilon = d + d_\epsilon^*$ be the Hodge-de Rham operator on $(V', g_\epsilon)$. For $\epsilon \in (0, 1]$ the metric is complete by (H4), and therefore $D_\epsilon$ is essentially self-adjoint on $C_c^\infty \Omega^r(V')$ (equivalently, $d_{\min,\epsilon} = d_{\max,\epsilon}$ and similarly for the adjoint).

Define the nonnegative closed quadratic form on $H_\epsilon^r$,

$$ (5.2.1) \quad Q_\epsilon[u] := \|D_\epsilon u\|_{L^2(g_\epsilon)}^2 \quad \text{with domain} \quad \mathrm{D}(Q_\epsilon) := \mathrm{Dom}(D_\epsilon), $$

i.e. the form domain is the graph domain of $D_\epsilon$. Transport $Q_\epsilon$ to a form $\tilde{Q}_\epsilon$ on $H_0^r$ by

$$ \tilde{Q}_\epsilon[v] := Q_\epsilon[\mathcal{U}_\epsilon^{-1}v], \quad \mathrm{D}(\tilde{Q}_\epsilon) := \mathcal{U}_\epsilon(\mathrm{D}(Q_\epsilon)). $$

Let $\tilde{D}_\epsilon := \mathcal{U}_\epsilon D_\epsilon \mathcal{U}_\epsilon^{-1}$ be the transported operator on $H_0^r$. Then $\tilde{Q}_\epsilon[v] = \|\tilde{D}_\epsilon v\|_{H_0^r}^2$.

For $\epsilon = 0$ the metric $g_0$ need not be complete, so minimal and maximal extensions may differ. In this section we consider the maximal Hodge-de Rham operator $D_{0,\max}$ acting in $H_0^r$:

$$ \mathrm{Dom}(D_{0,\max}) := \{u \in H_0^r : D_{0}u \in H_0^r \text{ in the distributional sense}\}. $$

Set the corresponding closed quadratic form

$$ (5.2.2) \quad Q_{0,\max}[u] := \|D_0 u\|_{H_0^r}^2, \quad \mathrm{D}(Q_{0,\max}) := \mathrm{Dom}(D_{0,\max}). $$

(The minimal-domain refinement is deferred to Section 6.)

**Lemma 5.2** (Graph-norm core for the maximal de Rham operator). The space $C_c^\infty \Omega^r(V')$ is dense in $\mathrm{Dom}(D_{0,\max})$ with respect to the graph norm $\|u\|_{H_0^r} + \|D_0 u\|_{H_0^r}$. Equivalently, for every $u \in \mathrm{Dom}(D_{0,\max})$ there exists $u^{(j)} \in C_c^\infty \Omega^r(V')$ such that $u^{(j)} \to u$ in $H_0^r$ and $D_0 u^{(j)} \to D_0 u$ in $H_0^r$.

*Proof.* Let $u \in \mathrm{Dom}(D_{0,\max})$. Choose an exhaustion of $V'$ by relatively compact open sets with smooth boundary, $\Omega_1 \subset \Omega_2 \subset \cdots \subset V'$, and pick cutoffs $\chi_j \in C_c^\infty(V')$ such that $\chi_j \equiv 1$ on $\Omega_j$, $\mathrm{supp}(\chi_j) \subset \Omega_{j+1}$, and $\|d\chi_j\|_{L^\infty(g_0)} \le 2^{-j}$. (Set, for instance, $\chi_j$ to be a smoothed cutoff of the $g_0$-distance to $\Omega_j$.)

Define $u_j := \chi_j u$. Then $u_j \to u$ in $H_0^r$ by dominated convergence, and since $D_0$ is first order,

$$ D_0 u_j = \chi_j D_0 u + [D_0, \chi_j] u. $$

The commutator $[D_0, \chi_j]$ is a zeroth-order bundle map with pointwise bound $|[D_0, \chi_j]u| \le C |d\chi_j| |u|$, hence

$$ \|[D_0, \chi_j]u\|_{H_0^r} \le C \|d\chi_j\|_{L^\infty} \|u\|_{H_0^r} \le C 2^{-j} \|u\|_{H_0^r} \to 0. $$

Thus $u_j \to u$ in the graph norm, and each $u_j$ has compact support.

Finally, for each fixed $j$, approximate the compactly supported $u_j$ by smooth compactly supported forms in the graph norm via standard mollification in local charts and a partition of unity (Friedrichs' lemma for first-order operators with smooth coefficients). This produces $u^{(j)} \in C_c^\infty \Omega^r(V')$ approximating $u$ in the graph norm, as claimed. $\square$
**5.3. Mosco convergence and strong resolvent convergence.** We recall the notion of Mosco convergence for closed convex functionals (quadratic forms) on a Hilbert space, and its equivalence with strong resolvent convergence of the associated operators. We follow Mosco [17] and the functional-analytic framework in [11, 3, 12].

**Definition 5.3 (Mosco convergence).** Let $(\mathcal{E}_j, \mathbf{D}(\mathcal{E}_j))$ and $(\mathcal{E}, \mathbf{D}(\mathcal{E}))$ be closed nonnegative quadratic forms on a Hilbert space $H$. We say $\mathcal{E}_j$ Mosco converges to $\mathcal{E}$ (and write $\mathcal{E}_j \xrightarrow{M} \mathcal{E}$) if:

(M1) (liminf under weak convergence) For any $u_j \to u$ weakly in $H$,

$$ \mathcal{E}[u] \le \liminf_{j \to \infty} \mathcal{E}_j[u_j], $$

with the convention that $\mathcal{E}_j[u_j] = +\infty$ if $u_j \notin \mathbf{D}(\mathcal{E}_j)$.

(M2) (recovery sequence) For any $u \in H$, there exist $u_j \to u$ strongly in $H$ such that

$$ \mathcal{E}[u] \ge \limsup_{j \to \infty} \mathcal{E}_j[u_j]. $$

**Theorem 5.4 (Mosco ⇔ strong resolvent convergence).** Let $\mathcal{E}_j$ and $\mathcal{E}$ be closed nonnegative quadratic forms on a Hilbert space $H$, and let $A_j$ and $A$ be the nonnegative self-adjoint operators associated with them. Then $\mathcal{E}_j \xrightarrow{M} \mathcal{E}$ if and only if the resolvents $(A_j + \lambda)^{-1}$ converge strongly to $(A + \lambda)^{-1}$ on $H$ for every $\lambda > 0$.

*Proof.* This is classical; see [17, 3, 11]. For a formulation tailored to Dirichlet forms/spectral structures, see [12, Thm. 5.6 and related equivalences]. □

**5.4. Mosco convergence of the transported Hodge forms.** We now state the convergence result needed later to pass harmonicity to the limit.

**Theorem 5.5 (Mosco convergence of $\tilde{Q}_\varepsilon$).** As $\varepsilon \downarrow 0$, the closed forms $\tilde{Q}_\varepsilon$ on $H_0^r$ Mosco converge to $Q_{0,\max}$. Equivalently, the self-adjoint operators $\tilde{D}_\varepsilon^* \tilde{D}_\varepsilon$ converge to $D_{0,\max}^* D_{0,\max}$ in the strong resolvent sense.

*Proof.* We verify the two Mosco conditions of Definition 5.3.

*Step 1: liminf inequality.* Assume $v_\varepsilon \to v$ weakly in $H_0^r$ and $\sup_\varepsilon \tilde{Q}_\varepsilon[v_\varepsilon] < \infty$. Then $\mathcal{U}_\varepsilon^{-1}v_\varepsilon$ is bounded in the graph norm of $D_\varepsilon$ on $H_\varepsilon^r$. By the $\varepsilon$-uniform stratified elliptic estimate (Theorem 3.5) and the uniform equivalence of Sobolev norms (Lemma 5.1 and Lemma 3.2), we obtain uniform bounds in $H_{\text{strat}}^1$ on compacta and on conic neighborhoods. Using a localization by the partition of unity from (3.2.2) and passing to a subsequence, we may assume $v_\varepsilon \to v$ strongly in $L_{\text{loc}}^2$. On any compact $K \subset V'$, (H2) implies that $g_\varepsilon \to g_0$ in $C^\infty(K)$ and, by Lemma 5.1, $\mathcal{U}_\varepsilon \to \text{Id}$ in $C^\infty(K)$. Hence the conjugated operators $\tilde{D}_\varepsilon = \mathcal{U}_\varepsilon D_\varepsilon \mathcal{U}_\varepsilon^{-1}$ converge to $D_0$ in $C^\infty(K)$ as first-order systems (the principal part is preserved under conjugation, and the additional terms
produced by differentiating $U_\varepsilon^{\pm 1}$ are zeroth order with coefficients tending to 0 on $K$). Hence $\tilde{D}_\varepsilon v_\varepsilon \to D_0 v$ weakly in $L^2(K)$. Therefore,

$$ \int_K |D_0 v|_{g_0}^2 dV_0 \le \liminf_{\varepsilon \downarrow 0} \int_K |\tilde{D}_\varepsilon v_\varepsilon|_{g_0}^2 dV_0. $$

Let $\{K_m\}_{m \ge 1}$ be an exhaustion of $V'$ by compact subsets. Applying the previous inequality to $K_m$ and using monotone convergence in $m$ yields $v \in \text{Dom}(D_{0,\text{max}})$ and $Q_{0,\text{max}}[v] \le \liminf_{\varepsilon \downarrow 0} \tilde{Q}_\varepsilon[v_\varepsilon]$.

**Step 2: recovery sequence.** Let $v \in \text{Dom}(D_{0,\text{max}})$. By Lemma 5.2, choose $v^{(j)} \in C_c^\infty \Omega^r(V')$ such that $v^{(j)} \to v$ in the graph norm of $D_{0,\text{max}}$. For each fixed $j$, the support of $v^{(j)}$ is a compact subset of $V'$, hence has positive distance to $\Sigma$. Therefore (H2) and the $C^\infty$-convergence of $\tilde{D}_\varepsilon$ to $D_0$ on compacta imply

$$ \tilde{D}_\varepsilon v^{(j)} \to D_0 v^{(j)} \quad \text{strongly in } H_0^r \text{ as } \varepsilon \downarrow 0, $$

and consequently $\tilde{Q}_\varepsilon[v^{(j)}] \to Q_{0,\text{max}}[v^{(j)}]$.

To build a recovery sequence, fix a sequence $\varepsilon_m \downarrow 0$. Choose $j_m \to \infty$ such that

$$ \|v^{(j_m)} - v\|_{H_0^r} + \|D_0(v^{(j_m)} - v)\|_{H_0^r} \le m^{-1}. $$

Then for each $m$ choose $\varepsilon_m$ small enough so that

$$ |\tilde{Q}_{\varepsilon_m}[v^{(j_m)}] - Q_{0,\text{max}}[v^{(j_m)}]| \le m^{-1}. $$

Setting $v_m := v^{(j_m)}$, we have $v_m \to v$ strongly in $H_0^r$ and

$$ \limsup_{m \to \infty} \tilde{Q}_{\varepsilon_m}[v_m] \le \limsup_{m \to \infty} Q_{0,\text{max}}[v_m] = Q_{0,\text{max}}[v], $$

which is the recovery condition. $\square$

<figure><img src="image_9.png" /></figure>

**5.5. Passing harmonicity to the limit.** We record the consequence needed for the main theorem: subsequential limits of $\varepsilon$-harmonic forms are $D_{0,\text{harmonic}}$ in the maximal sense.

**Proposition 5.6** (Limits satisfy the maximal harmonic equation). Let $f_\varepsilon \in H_\varepsilon^r$ be normalized by $\|\phi_\varepsilon^{-1}f_\varepsilon\|_{L^2(g_\varepsilon)} = 1$ and let $f_{\varepsilon_j} \to f$ strongly on compact subsets of $V'$ as in Theorem 4.6. Then $f \in \text{Dom}(D_{0,\text{max}})$ and $D_0 f = 0$ in the distributional sense on $V'$. Equivalently, $f \in \ker(D_{0,\text{max}})$.

*Proof.* First note that the normalization $\|\phi_\varepsilon^{-1}f_\varepsilon\|_{L^2(g_\varepsilon)} = 1$ implies a uniform bound of $\|f_\varepsilon\|_{L^2(g_\varepsilon)}$. Indeed, by (H3) the weights satisfy $\phi_\varepsilon(\rho, \cdot) \sim \rho \log \rho$ uniformly near each $x \in \Sigma$, hence $\phi_\varepsilon$ is uniformly bounded on each conic neighborhood. On the complement of these neighborhoods (a compact subset of $V'$), $\phi_\varepsilon \to \phi_V$ uniformly by (H3), so $\sup_{V'} \phi_\varepsilon \le C$ for all $\varepsilon \in (0, 1]$. Therefore

$$ \|f_\varepsilon\|_{L^2(g_\varepsilon)} = \|\phi_\varepsilon(\phi_\varepsilon^{-1}f_\varepsilon)\|_{L^2(g_\varepsilon)} \le (\sup_{V'} \phi_\varepsilon) \|\phi_\varepsilon^{-1}f_\varepsilon\|_{L^2(g_\varepsilon)} \le C. $$
Since $\mathcal{U}_\varepsilon$ is an isometry $H_\varepsilon^r \to H_0^r$, it follows that $\{\tilde{f}_{\varepsilon_j}\}$ is bounded in $H_0^r$ and hence admits weakly convergent subsequences. Transport to $H_0^r$ by setting $\tilde{f}_\varepsilon := \mathcal{U}_\varepsilon f_\varepsilon$. Since $D_\varepsilon f_\varepsilon = 0$ we have $\tilde{D}_\varepsilon \tilde{f}_\varepsilon = 0$, hence $\tilde{Q}_\varepsilon[\tilde{f}_\varepsilon] = 0$ for all $\varepsilon$. By Theorem 5.5 and the liminf inequality in Mosco convergence, any weak limit point $\tilde{f}$ of $\tilde{f}_{\varepsilon_j}$ in $H_0^r$ satisfies $Q_{0,\max}[\tilde{f}] \le \liminf \tilde{Q}_{\varepsilon_j}[\tilde{f}_{\varepsilon_j}] = 0$. Thus $\tilde{f} \in \text{Dom}(D_{0,\max})$ and $D_0 \tilde{f} = 0$ in $H_0^r$. Finally, let $K \subset V'$ be compact. By (H2) and Lemma 5.1, $\|\mathcal{U}_{\varepsilon_j} - \text{Id}\|_{L^\infty(K)} \to 0$. Since $f_{\varepsilon_j} \to f$ strongly in $L^2(K, g_0)$ (indeed in $H^{s-1}(K, g_0)$ by Theorem 4.6), we obtain

$$ \|\tilde{f}_{\varepsilon_j} - f_{\varepsilon_j}\|_{L^2(K,g_0)} = \|(\mathcal{U}_{\varepsilon_j} - \text{Id})f_{\varepsilon_j}\|_{L^2(K,g_0)} \le \|\mathcal{U}_{\varepsilon_j} - \text{Id}\|_{L^\infty(K)} \|f_{\varepsilon_j}\|_{L^2(K,g_0)} \to 0. $$

Hence $\tilde{f}_{\varepsilon_j} \to f$ strongly in $L^2(K, g_0)$ for every compact $K \subset V'$. In particular, any weak limit point $\tilde{f}$ of $\tilde{f}_{\varepsilon_j}$ must agree with $f$ on each $K$, and therefore $\tilde{f} = f$ in $H_0^r$. $\square$

**Remark 5.7.** Proposition 5.6 only yields the maximal equation $D_{0,\max}f = 0$ on $V'$. The crucial refinement in Theorem 1.1 (c) is that $f$ actually lies in the minimal domain $\text{Dom}(d_{\min}) \cap \text{Dom}((d^*)_\text{min})$, i.e. carries no hidden boundary data at the singular points. This is proved in Section 6 by a cutoff and integration-by-parts argument, using the weights $\phi_\varepsilon$ together with the corresponding weighted commutator estimates.

## 6. MINIMAL-DOMAIN IDENTIFICATION VIA CUTOFF AND BOUNDARY-TERM CONTROL

Our strategy is to approximate the limit form by logarithmic cutoffs and pass to the limit in the graph norm of $D_0$; the only nontrivial ingredient is the borderline estimate $\phi_0^{-1}f \in L^2$. In Section 5 we obtained that any sub-sequential limit $f$ of normalized $\varepsilon$-harmonic forms satisfies $f \in \text{ker}(D_{0,\max})$, i.e. $D_{0,\max}f = 0$ in the maximal (distributional) sense. The purpose of this section is to prove the crucial refinement:

$$ f \in \text{Dom}(d_{\min}) \cap \text{Dom}((d^*)_\text{min}), $$

equivalently $f \in \text{Dom}(D_{0,\min})$. This is the step that prevents hidden boundary data at the isolated singularities and completes Theorem 1.1 (c).

As illustrated in Figure 3, the cutoff varies only on a shrinking annulus near $\rho = 0$, so the commutator (and hence the would-be boundary contribution) can be forced to zero by a logarithmic weight.

### 6.1. A cutoff criterion for the minimal domain.
We recall a convenient characterization of the minimal domain for first-order differential operators.

Let $D_0 = d + d_0^*$ be the Hodge-de Rham operator acting on $H_0^r = L^2\Omega^r(V', g_0)$. Define the minimal domain $\text{Dom}(D_{0,\min})$ as the closure of $C_c^\infty\Omega^r(V')$ with respect to the graph norm $\|u\|_{H_0^r} + \|D_0 u\|_{H_0^r}$. Then $u \in \text{Dom}(D_{0,\min})$ if and only if there exists $u_j \in C_c^\infty\Omega^r(V')$ such that $u_j \to u$ and $D_0 u_j \to D_0 u$ in $H_0^r$.
<figure><img src="image_10.png" /><figcaption>FIGURE 3. Logarithmic cutoff near $\rho = 0$. The commutator $[D_0, \chi_\epsilon]$ is supported on a shrinking annulus $\rho \in [\epsilon^2, \epsilon]$, where the cutoff/integration-by-parts argument controls the borderline boundary term.</figcaption></figure>

**Lemma 6.1** (Cutoff approximation implies minimal domain). Let $u \in \text{Dom}(D_{0,\text{max}})$ and suppose there exists a family of smooth compactly supported cutoffs $\chi_\delta \in C_c^\infty(V')$, $\delta \downarrow 0$, such that

$$ (6.1.1) \qquad \chi_\delta u \to u \text{ in } H_0^r \quad \text{and} \quad D_0(\chi_\delta u) \to D_0 u \text{ in } H_0^r. $$

Then $u \in \text{Dom}(D_{0,\text{min}})$. Consequently, $u \in \text{Dom}(d_{\text{min}}) \cap \text{Dom}(d_{\text{min}}^*)$.

*Proof.* We argue directly from the definition of the minimal extension as the graph-norm closure of $C_c^\infty \Omega^r(V')$.

Fix $\delta > 0$ and set $v_\delta := \chi_\delta u$. Then $v_\delta \in L^2$ and, since $u \in \text{Dom}(D_{0,\text{max}})$, the Leibniz rule holds in the sense of distributions:

$$ D_0 v_\delta = D_0(\chi_\delta u) = \chi_\delta D_0 u + [D_0, \chi_\delta] u \in L^2, $$

so $v_\delta \in \text{Dom}(D_{0,\text{max}})$. Moreover, by construction $\text{supp}(v_\delta) \in V'$ is contained in a smooth region away from $\Sigma$.

On any smooth manifold, for a first-order differential operator with smooth coefficients (such as $D_0$), $C_c^\infty$ is dense in $\text{Dom}(D_{0,\text{max}})$ in the graph norm on compactly supported elements. Concretely, choose a finite coordinate cover of $\text{supp}(v_\delta)$, mollify in charts, and patch by a partition of unity: there exists a sequence $v_{\delta,k} \in C_c^\infty \Omega^r(V')$ with

$$ v_{\delta,k} \to v_\delta \quad \text{and} \quad D_0 v_{\delta,k} \to D_0 v_\delta \quad \text{in } L^2. $$

Thus $v_\delta \in \text{Dom}(D_{0,\text{min}})$ for each $\delta$.

Finally, (6.1.1) says exactly that $v_\delta \to u$ in the graph norm of $D_0$. Since $\text{Dom}(D_{0,\text{min}})$ is closed in the graph norm, we conclude $u \in \text{Dom}(D_{0,\text{min}})$.

For the de Rham operator $D_0 = d + d_0^*$, the minimal graph domain satisfies $\text{Dom}(D_{0,\text{min}}) = \text{Dom}(d_{\text{min}}) \cap \text{Dom}(d_{\text{min}}^*)$ by definition of the minimal extensions. $\square$

Thus the problem reduces to constructing cutoffs $\chi_\delta$ for $u = f$ and proving (6.1.1), with the main work being the control of the commutator terms supported near $\Sigma$.
6.2. **Weighted $L^2$ control passes to the limit.** Fix a singular point $x \in \Sigma$ and work on a conic neighborhood $U'_x \cap \{\rho < \rho_0\}$. Define the model weight

$$ \phi_0(\rho) := \rho |\log \rho| \quad (0 < \rho < \rho_0), $$

and extend $\phi_0$ smoothly and positively to all of $V'$ (any extension will do; only the behavior near $\rho = 0$ matters).

**Lemma 6.2** (Uniform comparison of weights). There exist $\rho_1 \in (0, \rho_0)$ and constants $0 < c \le C < \infty$ such that for all $\varepsilon \in (0, 1]$ and all $p \in U'_x \cap \{\rho(p) < \rho_1\}$,

$$ c \phi_0(p) \le \phi_\varepsilon(p) \le C \phi_0(p). $$

*Proof.* Write the uniform asymptotic expansion from (H3) in the form

$$ \phi_{\varepsilon}(\rho, \cdot) = \kappa(\varepsilon) \rho |\log \rho| + R_{\varepsilon}(\rho, \cdot), \quad |R_{\varepsilon}(\rho, \cdot)| \le C \rho^{1+\alpha} $$

for $0 < \rho < \rho_0$, with $C$ independent of $\varepsilon$. Let $\kappa_- := \inf_{\varepsilon \in (0,1]} \kappa(\varepsilon) > 0$ and $\kappa_+ := \sup_{\varepsilon \in (0,1]} \kappa(\varepsilon) < \infty$.

Since $\rho^\alpha/|\log \rho| \to 0$ as $\rho \downarrow 0$, choose $\rho_1 \in (0, \rho_0)$ so that $C \rho^{1+\alpha} \le \frac{1}{2}\kappa_- \rho |\log \rho|$ for all $0 < \rho < \rho_1$. Then for $0 < \rho < \rho_1$ we obtain

$$ \frac{1}{2}\kappa_- \rho |\log \rho| \le \phi_\varepsilon(\rho, \cdot) \le \left(\kappa_+ + \frac{1}{2}\kappa_-\right) \rho |\log \rho|, $$

which is the desired two-sided comparison with $\phi_0(\rho) = \rho |\log \rho|$.

□

**Lemma 6.3** (Weighted integrability of the limit). Let $f_\varepsilon \in H_\varepsilon^r$ satisfy $\|\phi_\varepsilon^{-1} f_\varepsilon\|_{L^2(g_\varepsilon)} = 1$ and let $f_{\varepsilon_j} \to f$ strongly on compact subsets of $V'$ as in Theorem 4.6. Then

$$ \phi_0^{-1} f \in L^2 \Omega^r(V', g_0), \quad \text{and} \quad \|\phi_0^{-1} f\|_{L^2(g_0)} \le C, $$

for a constant $C$ independent of the subsequence.

*Proof.* Fix $x \in \Sigma$ and work on $U'_x \cap \{\rho < \rho_1\}$ with $\rho_1$ from Lemma 6.2. By Lemma 6.2 and the uniform conic quasi-isometry (H1), the normalization $\|\phi_\varepsilon^{-1} f_\varepsilon\|_{L^2(g_\varepsilon)} = 1$ implies a uniform bound on each conic neighborhood:

$$ \|\phi_0^{-1} f_\varepsilon\|_{L^2(U'_x, g_0)} \le C_x, $$

where $C_x$ is independent of $\varepsilon$. Since $\Sigma$ is finite, summing over the finitely many $x \in \Sigma$ yields a uniform bound on $\bigcup_{x \in \Sigma} U'_x$.

On the complement $V' \setminus \bigcup_{x \in \Sigma} U'_x$, the weight $\phi_0$ is smooth and bounded below by a positive constant (by construction of the extension), hence

$$ \|\phi_0^{-1} f_\varepsilon\|_{L^2(K, g_0)} \lesssim \|f_\varepsilon\|_{L^2(K, g_0)} $$

for any compact $K \in V' \setminus \bigcup_x U'_x$. The latter is uniformly controlled by the compactness estimates of Section 4. Combining the two regions, we obtain a global uniform bound $\sup_\varepsilon \|\phi_0^{-1} f_\varepsilon\|_{L^2(V', g_0)} < \infty$.

Therefore, after passing to a subsequence, $\phi_0^{-1} f_{\varepsilon_j}$ converges weakly in $L^2(V', g_0)$ to some $h$. To identify $h$, fix any compact $K \in V'$. On $K$, the function $\phi_0^{-1}$ is smooth and bounded, and by Theorem 4.6 we have $f_{\varepsilon_j} \to f$
strongly in $L^2(K, g_0)$. Hence $\phi_0^{-1}f_{\varepsilon_j} \to \phi_0^{-1}f$ strongly in $L^2(K, g_0)$, which forces $h = \phi_0^{-1}f$. In particular, $\phi_0^{-1}f \in L^2(V', g_0)$, and the global norm bound follows from weak lower semicontinuity. $\square$

**6.3. Logarithmic cutoffs with vanishing boundary term.** We now build cutoffs whose derivatives are precisely adapted to the weight $\phi_0 = \rho|\log \rho|$.

Fix $x \in \Sigma$ and choose $\rho_1$ as in Lemma 6.2. Let $\eta \in C^\infty(\mathbb{R})$ satisfy $\eta(t) = 0$ for $t \le 0$ and $\eta(t) = 1$ for $t \ge 1$. For $\delta \in (0, e^{-e})$, define on $U'_x \cap \{\rho < \rho_1\}$:

$$ (6.3.1) \qquad \chi_{\delta,x}(\rho) := \eta\left(\frac{\log|\log \rho| - \log|\log \delta|}{\log 2}\right). $$

Then $\chi_{\delta,x} = 0$ for $\rho \le \delta$, $\chi_{\delta,x} = 1$ for $\rho \ge \delta^{1/2}$, and $\text{supp}(d\chi_{\delta,x}) \subset \{\delta < \rho < \delta^{1/2}\}$. A direct computation gives

$$ (6.3.2) \qquad |d\chi_{\delta,x}| \le \frac{C}{\rho|\log \rho| |\log 2|} \quad \text{on } \{\delta < \rho < \delta^{1/2}\}, $$

with $C$ independent of $\delta$ (and of $\varepsilon$).

Choose $\chi_\delta \in C_c^\infty(V')$ so that $\chi_\delta \equiv 1$ on $K_0$ and on $V' \setminus \bigcup_x U'_x$ and, near each $x \in \Sigma$, $\chi_\delta$ agrees with $\chi_{\delta,x}$ in (6.3.1). Then $\chi_\delta$ has compact support in $V'$ (since it vanishes in punctured balls around every $x \in \Sigma$).

**Lemma 6.4 (Commutator term tends to zero).** Let $u \in L^2\Omega^r(V', g_0)$ satisfy $\phi_0^{-1}u \in L^2\Omega^r(V', g_0)$. Then, as $\delta \downarrow 0$,

$$ \|(d\chi_\delta) \wedge u\|_{L^2(g_0)} \to 0 \quad \text{and} \quad \|\iota_{\nabla\chi_\delta} u\|_{L^2(g_0)} \to 0. $$

*Proof.* For a smooth function $\chi$, both operators $u \mapsto (d\chi) \wedge u$ and $u \mapsto \iota_{\nabla\chi} u$ are zeroth order and satisfy the pointwise bound

$$ |(d\chi) \wedge u| + |\iota_{\nabla\chi} u| \le C |d\chi| |u| $$

for a constant $C$ depending only on the form degree.

By construction of $\chi_\delta$ (cf. (6.3.1)),

$$ \text{supp}(d\chi_\delta) \subset \bigcup_{x \in \Sigma} \{\delta < \rho < \delta^{1/2}\}, $$

and the derivative estimate (6.3.2) gives on this support

$$ |d\chi_\delta| \le \frac{C}{\rho|\log \rho|}, \quad \text{hence} \quad |d\chi_\delta|^2 |u|^2 \le C |\phi_0^{-1}u|^2. $$

Therefore,

$$ \|(d\chi_\delta) \wedge u\|_{L^2(g_0)}^2 + \|\iota_{\nabla\chi_\delta} u\|_{L^2(g_0)}^2 \le C \int_{\text{supp}(d\chi_\delta)} |\phi_0^{-1}u|^2 dV_{g_0}. $$

Since $\phi_0^{-1}u \in L^2(V', g_0)$ and the measurable sets $\text{supp}(d\chi_\delta)$ decrease to $\emptyset$ as $\delta \downarrow 0$, the right-hand side tends to 0 by absolute continuity of the integral. This proves the claim. $\square$
### 6.4. Minimal-domain identification for the limit harmonic form.
We now apply the cutoff criterion to $u = f$.

**Theorem 6.5 (Minimal-domain identification).** Let $f_{\epsilon} \in H_{\epsilon}^r$ be normalized by $\|\phi_{\epsilon}^{-1} f_{\epsilon}\|_{L^2(g_{\epsilon})} = 1$ and assume $f_{\epsilon_j} \to f$ strongly on compact subsets of $V'$ as in Theorem 4.6. Then $f \in \text{Dom}(D_{0,\text{min}})$, hence

$$f \in \text{Dom}(d_{\text{min}}) \cap \text{Dom}(d_{\text{min}}^*) \quad \text{and} \quad D_0 f = 0.$$

In particular, $f$ is smooth on $V'$.

*Proof.* By Proposition 5.6, the limit satisfies $f \in \text{Dom}(D_{0,\text{max}})$ and $D_0 f = 0$ in $L^2$. By Lemma 6.3, we also have $\phi_0^{-1} f \in L^2(V', g_0)$.

Let $\chi_{\delta}$ be the logarithmic cutoff from (6.3.1). Since $0 \le \chi_{\delta} \le 1$ and $\chi_{\delta} \to 1$ pointwise on $V'$, dominated convergence gives

$$\|\chi_{\delta} f - f\|_{L^2(g_0)} \to 0.$$

Next, because $f \in \text{Dom}(D_{0,\text{max}})$ the Leibniz rule holds distributionally, so

$$D_0(\chi_{\delta} f) = \chi_{\delta} D_0 f + [D_0, \chi_{\delta}] f = [D_0, \chi_{\delta}] f.$$

For the Hodge-de Rham operator $D_0 = d+d_0^*$ one has the exact commutator identity

$$[D_0, \chi_{\delta}] f = (d\chi_{\delta}) \wedge f - \iota_{\nabla\chi_{\delta}} f.$$

Therefore Lemma 6.4 (applied with $u = f$) implies

$$\|D_0(\chi_{\delta} f) - D_0 f\|_{L^2(g_0)} = \|D_0(\chi_{\delta} f)\|_{L^2(g_0)} \to 0,$$

since $D_0 f = 0$. Together with $\chi_{\delta} f \to f$ in $L^2$, this is precisely the graph-norm convergence (6.1.1) for $u = f$.

Hence Lemma 6.1 yields $f \in \text{Dom}(D_{0,\text{min}})$, and consequently $f \in \text{Dom}(d_{\text{min}}) \cap \text{Dom}(d_{\text{min}}^*)$. Finally, elliptic regularity for $D_0$ on the smooth manifold $V'$ implies that any $L^2$ solution of $D_0 f = 0$ is smooth on $V'$. $\square$

**Remark 6.6.** The cutoff construction above is the isolated-singularity analogue of Cheeger's analysis of ideal boundary conditions for conic metrics: the minimal extension can be characterized as the choice of domain excluding those asymptotic modes (boundary data at the cone tip) incompatible with the minimal closure. Our argument avoids an explicit indicial-root expansion by exploiting the borderline weighted control $\phi_0^{-1} f \in L^2$ inherited from the Ohsawa-type normalization, together with a logarithmic cutoff adapted to the critical weight $\phi_0 = \rho|\log \rho|$. See [5, 13, 2] for complementary viewpoints on domains and boundary data for conic/stratified spaces.

## 7. PROOFS OF THEOREM 1.1 AND COROLLARY 1.3

**Structure of the argument.** Section 4 provides the compactness mechanism for normalized harmonic forms: after transporting to a fixed Hilbert space, we extract subsequences converging strongly on compact subsets of $V'$, which yields Theorem 1.1(b). The remaining issue is to identify the
correct domain in the singular limit. Section 6 shows that any such limit satisfies the maximal equation $D_0f = 0$, while Section 6 upgrades this to the minimal domain $f \in \text{Dom}(d_{\min}) \cap \text{Dom}(d_{\min}^*)$ via a logarithmic cutoff and graph-norm closure argument, using the borderline weighted control inherited from the Ohsawa normalization. Together these steps imply Theorem 1.1(c). Finally, we avoid a delicate spectral-gap perturbation argument for the zero eigenvalue by reducing Theorem 1.1(a) to topological invariance through the $L^2$-Hodge theorem in the complete case and the identification with intersection cohomology.

**7.1. Proof of Theorem 1.1.** We prove parts (b), (c), and then explain (a).

*Proof of Theorem 1.1(b).* Let $f_{\epsilon} \in H_{\epsilon}^r$ satisfy the normalization $\|\phi_{\epsilon}^{-1}f_{\epsilon}\|_{L^2(g_{\epsilon})} = 1$. Fix an integer $s > n + 1$. By Theorem 4.6 (Section 4), for any sequence $\epsilon_j \downarrow 0$ there exists a subsequence (not relabeled) and a form $f \in H_{\text{loc}}^{s-1}\Omega^r(V')$ such that

$$f_{\epsilon_j} \to f \quad \text{strongly in } H^{s-1}(K, g_v) \text{ for every compact } K \in V'$$

This is exactly the statement of part (b). □

<figure><img src="image_11.png" /></figure>

*Proof of Theorem 1.1(c).* Let $f$ be a subsequential limit obtained in part (b). By Proposition 5.6 (Section 5), we have $f \in \text{Dom}(D_{0,\max})$ and $D_0f = 0$ in the maximal (distributional) sense.

By Theorem 6.5 (Section 6), the same limit satisfies $f \in \text{Dom}(D_{0,\min})$, hence

$$f \in \text{Dom}(d_{\min}) \cap \text{Dom}(d_{\min}^*)$$

Finally, $V'$ is a smooth manifold and $D_0$ is a first-order elliptic operator with smooth coefficients on $V'$. Since $f \in L_{\text{loc}}^2$ and $D_0f = 0$ in the sense of distributions on $V'$, interior elliptic regularity for elliptic systems implies $f \in C^{\infty}\Omega^r(V')$. This proves part (c). □

<figure><img src="image_12.png" /></figure>

*Explanation of Theorem 1.1(a).* For each $\epsilon \in (0, 1]$, the metric $g_{\epsilon}$ is complete, so $d_{\min,\epsilon} = d_{\max,\epsilon}$ by Gaffney's theorem [9]. On a complete Riemannian manifold, the $L^2$ Hodge theorem identifies $\ker(D_{\epsilon})$ with the reduced $L^2$ de Rham cohomology of $(V', g_{\epsilon})$. Under Ohsawa's hypotheses (isolated singularities), Ohsawa constructs the family of complete Kähler metrics $\{g_{\epsilon}\}_{\epsilon \in (0,1]}$ and proves that the resulting $L^2$ de Rham groups are naturally identified (within this family) and, moreover, agree with the topological intersection cohomology of $V$ [18]. In particular, their dimensions are independent of $\epsilon \in (0, 1]$. Therefore,

$$\dim H_{\epsilon}^{r} = \dim \ker(D_{\epsilon}) = \dim H_{(2)}^{r}(V', g_{\epsilon})$$

is constant for $\epsilon \in (0, 1]$, proving part (a). □

<figure><img src="image_13.png" /></figure>
## 7.2. Proof of Corollary 1.3.
We now identify the $L^2$ cohomology of the nonsingular locus with intersection cohomology of $V$.

*Proof of Corollary 1.3.* Consider the incomplete metric $g_0$ on $V'$ and the de Rham complex with the minimal extension $d_{\min}$. By Theorem 1.1(c), any subsequential limit of normalized $\varepsilon$-harmonic forms lies in $\text{Dom}(d_{\min}) \cap \text{Dom}(d_{\min}^*)$ and satisfies $D_0 f = 0$. Hence $L^2$ harmonic representatives for $g_0$ (with minimal boundary condition) compute the reduced $L^2$ de Rham cohomology $H_{(2)}^\bullet(V', g_0; d_{\min})$.

Since the singularities are isolated, each $x \in \Sigma$ admits a conic neighborhood modeled on $(0, \rho_0) \times L_x$ with cone metric $d\rho^2 + \rho^2 h_x$. The link $L_x$ has real dimension $2n - 1$, which is odd; in particular the Witt condition holds automatically. Cheeger's analysis of $L^2$ de Rham theory on cones (and its extensions to stratified spaces) identifies the local minimal $L^2$ de Rham complex with the middle-perversity Deligne complex, hence the local $L^2$ cohomology agrees with middle-perversity intersection cohomology [5]. Because $\Sigma$ is finite, the global identification follows by the standard sheaf-theoretic glueing for intersection cohomology (equivalently, by viewing the minimal $L^2$ complex as a fine resolution of the Deligne sheaf in this Witt/isotropic conic setting). Therefore

$$H_{(2)}^{\bullet}(V', g_0; d_{\min}) \cong IH^{\bullet}(V),$$

and the same identification holds on harmonic representatives. This proves the corollary. □

*Remark 7.1.* Corollary 1.3 is formulated for the limiting metric $g_0$ with minimal boundary condition. For $\varepsilon > 0$ the metric $g_\varepsilon$ is complete, so $d_{\min,\varepsilon} = d_{\max,\varepsilon}$ by Gaffney's theorem [9], and the harmonic space $H_\varepsilon^\bullet$ computes the corresponding $L^2$ cohomology of $(V', g_\varepsilon)$.

# 8. EXAMPLES AND MODEL COMPUTATIONS

The analytic arguments in Sections 3–7 are local-to-global and therefore reduce, near each $x \in \Sigma$, to statements on a conic model. In this section we record a few concrete examples illustrating (i) why the weight $\phi_0(\rho) = \rho|\log \rho|$ is the critical borderline quantity in the isolated-singularity setting, and (ii) how the cutoff/graph-norm mechanism of Section 6 excludes hidden boundary data. Detailed calculations are collected in Appendix C.

## 8.1. The cone model and the borderline weight.
Let $L$ be a closed smooth Riemannian manifold of dimension $m$ and consider the truncated cone

$$C_{\rho}(L) := (0, \rho_0) \times L, \quad g_C := d\rho^2 + \rho^2 h,$$

with volume density $dV_{g_C} = \rho^m d\rho dV_h$. This is the local model guaranteed by the uniform quasi-isometry hypothesis (M3)/(H1) near each $x \in \Sigma$; in our application $m = 2n - 1$.
**Proposition 8.1** (A concrete form of the borderline estimate). Let $\phi_0(\rho) := \rho|\log \rho|$ and let $u$ be a differential form on $C_\rho(L)$ such that $\phi_0^{-1}u \in L^2\Omega^\bullet(C_\rho(L), g_C)$. Then the logarithmic cutoff $\chi_\delta$ from (6.3.1) satisfies

$$ \| [D_C, \chi_\delta] u \|_{L^2(g_C)} \to 0 \quad (\delta \downarrow 0), $$

where $D_C = d + d_C^*$ is the Hodge-de Rham operator of $g_C$.

*Proof.* On the cone, the commutator identity $[D_C, \chi] = (d\chi) \wedge (\cdot) - i\nabla\chi(\cdot)$ holds exactly. The cutoff is chosen so that $\text{supp}(d\chi_\delta) \subset \{\delta < \rho < \delta^{1/2}\}$ and $|d\chi_\delta| \lesssim (\rho|\log \rho|)^{-1}$ on its support; see (6.3.1)–(6.3.2). Thus

$$ \|[D_C, \chi_\delta]u\|_{L^2(g_C)}^2 \lesssim \int_{\{\delta < \rho < \delta^{1/2}\}} |\phi_0^{-1}u|^2 dV_{g_C}. $$

Since $\phi_0^{-1}u \in L^2$ and the annuli $\{\delta < \rho < \delta^{1/2}\}$ shrink to $\emptyset$, the integral tends to 0 by absolute continuity. A fully explicit estimate (showing the decay is $O(|\log \delta|^{-1})$) is given in Appendix C. $\square$

*Remark 8.2* (Why $\rho|\log \rho|$ is critical). If $u$ is bounded near $\rho = 0$, then $|\phi_0^{-1}u|^2 dV_{g_C} \sim \rho^{m-2}|\log \rho|^{-2} d\rho dV_h$. This is integrable at $\rho = 0$ for all $m \ge 1$, but the exponent $|\log \rho|^{-2}$ is borderline when $m = 1$ (complex dimension $n = 1$). In particular, the choice of logarithmic cutoff matches exactly the borderline integrability needed to force commutator/boundary terms to vanish without appealing to an explicit indicial expansion.

**8.2. A toy geometric example: nodal curves.** When $n = 1$, the singularities of a compact complex curve are isolated and each link is a circle ($m = 1$). The Witt condition holds automatically, and the local model is a two-dimensional cone. The borderline nature of the weight is most transparent in this case: as noted in Remark 8.2, the integrand behaves like $\rho^{-1}|\log \rho|^{-2}d\rho$, which is integrable but only just. Thus the cutoff mechanism of Section 6 is genuinely sharp already for curves.

As a concrete class of examples, let $V$ be a projective curve with finitely many ordinary nodes and $V' = V \setminus \Sigma$. Ohsawa's construction produces complete Kähler metrics $g_\epsilon$ on $V'$ and admissible weights $\phi_\epsilon$ with the asymptotics (2.2.3) (equivalently (M4)), and therefore Theorem 1.1 applies in every degree. In particular, the limiting harmonic representatives lie in the minimal domain of $(V', g_0)$ and compute $H_{(2)}^\bullet(V', g_0; d_{\min}) \cong IH^\bullet(V)$ by Corollary 1.3.

**8.3. Projective hypersurfaces with isolated singularities.** Let $V \subset \mathbb{CP}^N$ be an irreducible projective variety of complex dimension $n$ with isolated singular set $\Sigma$. On $V' = V \setminus \Sigma$, the restriction of the Fubini-Study metric provides a natural Hermitian metric $g_\nu$. In this projective setting, Ohsawa constructs a family of complete Kähler metrics $g_\epsilon$ on $V'$ and weights $\phi_\epsilon$ satisfying (M1)–(M4); see [18]. Consequently, the full analytic package developed here applies to such $V$: normalized $\epsilon$-harmonic forms admit sequential limits on compacta, the limit solves $D_0 f = 0$, and the cutoff
argument forces the minimal boundary condition at each $x \in \Sigma$. In particular, the resulting $L^2$ de Rham cohomology of $(V', g_0)$ is canonically identified with middle-perversity intersection cohomology of $V$.

**8.4. Local algebraic models: isolated hypersurface singularities and ordinary double points.** To connect the analytic hypotheses (M1)–(M4) with familiar algebraic geometry, we recall the standard local picture near an isolated hypersurface singularity. Let $x \in \Sigma$ and choose local holomorphic coordinates in an embedding $V \hookrightarrow \mathbb{C}^N$ so that, near $x$, the germ of $V$ is given by a single equation

$$V \cap B_{\eta} = \{f = 0\} \subset \mathbb{C}^{n+1}, \quad df(0) = 0,$$

with an isolated critical point at the origin. For $\eta > 0$ sufficiently small, the link of the singularity is the smooth compact manifold

$$L_x := V \cap S_{\eta}^{2n+1} = \{f = 0\} \cap S_{\eta}^{2n+1},$$

and $V \cap (B_{\eta} \setminus \{0\})$ is a smooth manifold of real dimension $2n$. Moreover, the stratified conic structure theory implies that, after removing $x$, a neighborhood of $x$ in $V$ is homeomorphic to the cone $(0, \eta) \times L_x$. Analytically, the restriction of a smooth ambient Hermitian metric is uniformly quasi-isometric to a conic metric $d\rho^2 + \rho^2 h$ on $(0, \eta) \times L_x$; this is precisely the local geometric input used in (M3) and in the elliptic/cutoff analysis of Sections 3 and 6. In particular, the link has dimension $\dim L_x = 2n - 1$ (odd), hence the Witt condition is automatic in the isolated-point case.

A basic and widely used algebraic model is the *ordinary double point* (the $A_1$-singularity), defined in $\mathbb{C}^{n+1}$ by

$$f(z_0, \dots, z_n) = z_0^2 + \dots + z_n^2.$$

The corresponding link admits an explicit description: writing $z = (x + iy)/\sqrt{2}$ with $x, y \in \mathbb{R}^{n+1}$, the conditions $f(z) = 0$ and $|z| = 1$ are equivalent to

$$|x| = |y| = 1, \quad x \cdot y = 0.$$

Thus $L_x$ is diffeomorphic to the Stiefel manifold of orthonormal 2-frames $V_2(\mathbb{R}^{n+1})$, and hence to the unit tangent bundle $UT(S^n)$. From the analytic perspective, this makes concrete the “cone on a smooth link” paradigm: near the node, the geometry is modeled by $(0, \eta) \times L_x$ with $L_x \cong UT(S^n)$, and the borderline weight $\phi_0(\rho) = \rho|\log \rho|$ precisely matches the decay required to suppress boundary data at $\rho = 0$ via the logarithmic cutoff.

Finally, this local algebraic picture dovetails with the global projective situation of Section 8.3: projective hypersurfaces with isolated singularities are covered by Ohsawa's construction, and our analytic convergence and minimal-domain results apply uniformly at each singular point, independent of the algebraic type of the isolated hypersurface singularity.
### 8.5. A global projective example: nodal hypersurfaces and comparison with resolutions/smoothings.
A particularly concrete and widely studied class of projective varieties with isolated singularities are nodal hypersurfaces, i.e. hypersurfaces $V \subset \mathbb{CP}^{n+1}$ whose singular set $\Sigma$ is finite and every $x \in \Sigma$ is an ordinary double point (an $A_1$-singularity) in the sense of Section 8.4. Equivalently, after dehomogenizing near $x$ and choosing holomorphic coordinates, the defining equation has the form

$$f(z) = q(z) + (\text{terms of order } \ge 3),$$

where $q$ is a nondegenerate complex quadratic form; this is the Jacobian/Hessian criterion for a node. In particular, each link $L_x$ is diffeomorphic to $V_2(\mathbb{R}^{n+1}) \cong UT(S^n)$, so the local conic model in (M3) is completely explicit.

From the global standpoint, such nodal hypersurfaces come with two standard “comparison spaces” from algebraic geometry:

* A smoothing $V_t$ (when it exists), obtained by a small deformation of the defining polynomial so that all nodes disappear and $V_t$ is smooth.

* A (possibly) small resolution $\pi : \tilde{V} \to V$, i.e. a proper birational morphism that is an isomorphism over $V'$ and whose exceptional fibers have complex dimension $\le n - 1$.

While the existence of a small resolution depends on the dimension and the global configuration of nodes, the intersection cohomology groups $IH^\bullet(V)$ are designed to be insensitive to the choice of such a resolution: if a small resolution exists, then $IH^\bullet(V)$ agrees with the ordinary cohomology of $\tilde{V}$ (see [10]).

In our analytic framework, the relevance of these comparison spaces is that Corollary 1.3 identifies $IH^\bullet(V)$ with the minimal $L^2$ de Rham cohomology of $(V', g_0)$. Thus, whenever a small resolution exists, our theorem provides a direct analytic realization of the cohomology of $\tilde{V}$ in terms of $L^2$ harmonic forms on the singular metric space $(V', g_0)$. Likewise, when a smoothing exists, the identification with intersection cohomology furnishes the expected “topological stability” of the limiting $L^2$ theory under deformation: away from the middle degree, nodal singularities do not contribute new intersection cohomology, whereas in the middle degree the difference is governed by the local vanishing cycles at the nodes (each node contributing one rank-one vanishing sphere in the Milnor fiber). We emphasize that none of these comparison statements are used in the proofs above; rather, they serve to connect the analytic conclusions of Theorem 1.1 and Corollary 1.3 with standard algebra-geometric constructions for explicit classes of singular projective varieties.

The nodal hypersurfaces discussed below admit two standard comparison spaces (when available): a smoothing $V_t$ and a (possibly) small resolution $\pi : \tilde{V} \to V$. As illustrated in Figure 4, this schematic picture provides a geometric reference for how the analytic identification of minimal $L^2$ de Rham
<figure><img src="image_14.png" /><figcaption>FIGURE 4. Nodal hypersurfaces admit standard comparison spaces (when they exist): a smoothing $V_t$ and a (possibly) small resolution $\pi : \tilde{V} \to V$. The analytic result identifies $IH^\bullet(V)$ with minimal $L^2$ de Rham cohomology on $(V', g_0)$.</figcaption></figure>

cohomology on $(V', g_0)$ aligns with the intersection cohomology of the singular central fiber.

### 8.5.1. An explicit projective equation and Hessian verification.
For a completely explicit nodal hypersurface in every complex dimension $n \ge 1$, consider homogeneous coordinates $[Z_0 : \dots : Z_{n+1}]$ on $\mathbb{CP}^{n+1}$ and the quadric

$$ (8.5.1) \qquad V := \{F = 0\} \subset \mathbb{CP}^{n+1}, \qquad F(Z) := Z_0^2 + \dots + Z_n^2. $$

This is a projective quadric cone with vertex $p = [0 : \dots : 0 : 1]$. Indeed, since $\partial F / \partial Z_i = 2Z_i$ for $0 \le i \le n$ and $\partial F / \partial Z_{n+1} = 0$, the projective singular locus is

$$ \Sigma = \{[Z] \in V : Z_0 = \dots = Z_n = 0\} = \{p\}. $$

In the affine chart $U_{n+1} = \{Z_{n+1} \ne 0\}$ with coordinates $z_i := Z_i / Z_{n+1}$, the dehomogenized equation is

$$ (8.5.2) \qquad f(z) := z_0^2 + \dots + z_n^2 = 0 \quad (z \in \mathbb{C}^{n+1}), $$

so the unique singular point corresponds to $z = 0$. The complex Hessian matrix of $f$ at $0$ is $2 \mathrm{Id}_{n+1}$ and is therefore nondegenerate. Equivalently, $p$ is an ordinary double point (an $A_1$-singularity) by the Jacobian/Hessian criterion of Section 8.4; see Appendix C.6 for the full computation.

Consequently, the link $L_p = V \cap S_\eta^{2n+1}$ is diffeomorphic to the Stiefel manifold $V_2(\mathbb{R}^{n+1}) \cong \mathrm{UT}(S^n)$ as in Appendix C.4, and the local conic model for $(V', g_0)$ near $p$ is the standard cone over this explicit link. In particular, this example falls squarely within the standing hypotheses, and Theorem 1.1 and Corollary 1.3 give an analytic realization of $IH^\bullet(V)$ via minimal $L^2$ harmonic forms on $(V', g_0)$.

### 8.5.2. An explicit hypersurface with several nodes: the Cayley cubic surface.
To exhibit a genuinely multi-nodal projective example where the Jacobian/Hessian verification can be carried out completely by hand, consider $\mathbb{CP}^3$ with homogeneous coordinates $[X_0 : X_1 : X_2 : X_3]$ and the cubic surface

$$ (8.5.3) \qquad S := \{F = 0\} \subset \mathbb{CP}^3, $$
$$F := X_0X_1X_2 + X_0X_1X_3 + X_0X_2X_3 + X_1X_2X_3.$$

This is the classical *Cayley cubic* (a determinantal cubic surface), and it has precisely four ordinary double points at the coordinate points

$$p_0 = [1:0:0:0], \quad p_1 = [0:1:0:0], \quad p_2 = [0:0:1:0], \quad p_3 = [0:0:0:1].$$

Indeed, one checks directly that $F(p_i) = 0$ and that all homogeneous partial derivatives $\partial F/\partial X_j$ vanish at each $p_i$; conversely, the simultaneous equations $F = \partial F/\partial X_0 = \cdots = \partial F/\partial X_3 = 0$ force $[X]$ to be one of these four points. Moreover, in the affine chart $X_0 = 1$ near $p_0$ (with local coordinates $x = X_1/X_0$, $y = X_2/X_0$, $z = X_3/X_0$), the dehomogenized equation reads

$$f(x, y, z) = xy + xz + yz + xyz = 0,$$

whose quadratic part at $(0,0,0)$ is the nondegenerate form $q = xy + xz + yz$ (equivalently, the complex Hessian at $0$ has determinant $2 \neq 0$). By symmetry, the same local normal form holds at each $p_i$. Hence all four singular points are $A_1$-singularities (nodes/ordinary double points).

This example fits our framework with complex dimension $n = 2$ and $\Sigma = \{p_0, p_1, p_2, p_3\}$. Since the links are 3-manifolds (indeed $L_{p_i} \cong V_2(\mathbb{R}^3) \cong \mathrm{SO}(3) \cong \mathbb{RP}^3$), the Witt condition holds automatically, and Corollary 1.3 identifies the minimal $L^2$ de Rham cohomology of $(S \setminus \Sigma, g_0)$ with $IH^\bullet(S)$. A complete Jacobian/Hessian verification (including the identification of the singular locus and the explicit Hessian matrices in local charts) is given in Appendix C.7.

### 8.5.3. A higher-dimensional example with many nodes: the Segre cubic three-fold.
To complement the surface example above with a genuinely higher-dimensional projective model carrying many nodes, we record the classical *Segre cubic threefold*. Let $[X_0 : \cdots : X_5]$ be homogeneous coordinates on $\mathbb{CP}^5$ and consider the hyperplane

$$H := \left\{ \sum_{i=0}^{5} X_i = 0 \right\} \cong \mathbb{CP}^4.$$

Inside $H$, define the cubic threefold

$$(8.5.4) \qquad S := \{C = 0\} \cap H, \qquad C(X) := \sum_{i=0}^{5} X_i^3.$$

Equivalently, $S$ is the cubic hypersurface in $\mathbb{CP}^4$ obtained by restricting $C$ to the hyperplane $\sum X_i = 0$.

Singular locus (ten points). Regarded as a complete intersection in $\mathbb{CP}^5$ with equations $L := \sum X_i = 0$ and $C = \sum X_i^3 = 0$, a point $[X] \in S$ is singular if and only if the gradients $\nabla L = (1, \dots, 1)$ and $\nabla C = (3X_0^2, \dots, 3X_5^2)$ are linearly dependent. Thus $[X]$ is singular precisely when $X_0^2 = \cdots = X_5^2$. After scaling, this means $X_i \in \{\pm 1\}$ for all $i$, and the hyperplane condition
$\sum X_i = 0$ forces exactly three +1's and three -1's. Since $[X]$ and $[-X]$ represent the same point in projective space, this yields

$$|\Sigma| = \frac{1}{2} \binom{6}{3} = 10$$

singular points, each represented by a sign vector with three +1's and three -1's.

Node condition. Each of these ten singularities is an ordinary double point. Indeed, at any singular point, the linear term of $C$ vanishes identically along the hyperplane direction, and the quadratic term is a nondegenerate complex quadratic form on a 4-dimensional local chart of $H$. A complete Taylor expansion and Hessian determinant computation at a representative singular point are given in Appendix C.8. There we compute the quadratic part and the complex Hessian (in the standard sense of second partial derivatives) and find a nonzero determinant, hence each singularity is an $A_1$ node.

This provides a concrete multi-nodal example in complex dimension $n = 3$ (so the links are 5-manifolds and Witt holds automatically). Hence Theorem 1.1 and Corollary 1.3 apply to $(S \setminus \Sigma, g_0)$ and identify $IH^\bullet(S)$ with the minimal $L^2$ de Rham cohomology of the singular metric space.

## APPENDIX A. B/EDGE CALCULUS REFERENCES AND MAPPING PROPERTIES USED

This appendix records the b/edge calculus facts that are used (often implicitly) in Section 3 to construct parametrices and to justify the mapping properties behind the $\varepsilon$-uniform stratified elliptic estimate. Since the singularities are isolated, the relevant local framework is the b-calculus on the radial blow-up of each singular point. We also indicate how the same statements can be phrased in the edge calculus language, although no iterated structure is needed here.

### A.1. b-manifolds and b-differential operators.
Let $\overline{M}$ be a compact manifold with boundary and interior $M = \text{int}(\overline{M})$. A boundary defining function is a smooth function $\rho \ge 0$ on $\overline{M}$ such that $\partial \overline{M} = \{\rho = 0\}$ and $d\rho \ne 0$ on $\partial \overline{M}$.

The Lie algebra of b-vector fields is

$$\mathcal{V}_b(\overline{M}) := \{V \in C^\infty(\overline{M}; T\overline{M}) : V \text{ is tangent to } \partial\overline{M}\}.$$

Locally, if $(\rho, y)$ are product coordinates near $\partial\overline{M}$, then

$$\mathcal{V}_b(\overline{M}) = \text{span}_{C^\infty} \{\rho \partial_\rho, \partial_{y_1}, \dots, \partial_{y_{\dim M-1}}\}.$$

The corresponding b-tangent bundle $^bT\overline{M}$ is the vector bundle whose smooth sections are $\mathcal{V}_b(\overline{M})$, and $^bT^*\overline{M}$ is its dual, locally spanned by $\frac{d\rho}{\rho}$ and $dy$. Given any fixed smooth positive definite b-metric on $^bT\overline{M}$, we write

$$^bS^*(\overline{M}) \subset ^bT^*\overline{M}$$
for the corresponding unit b-cosphere bundle (the set of covectors of b-norm 1).

**Definition A.1 (b-differential operators).** The algebra $Diff_b^m(\overline{M})$ consists of finite sums of products of at most $m$ b-vector fields with smooth coefficients. In local coordinates,

$$P \in \mathrm{Diff}_b^m \iff P = \sum_{|\alpha|+j \le m} a_{\alpha j}(\rho, y) (\rho \partial_{\rho})^j \partial_y^{\alpha}.$$

The b-principal symbol $^b\sigma_m(P)$ of $P \in \mathrm{Diff}_b^m$ is a homogeneous degree-$m$ function on $^bT^*\overline{M} \setminus 0$ and is defined so that it coincides with the usual principal symbol when one treats $(\rho\partial_\rho, \partial_y)$ as the first-order variables. We say $P$ is b-elliptic if $^b\sigma_m(P)$ is invertible on $^bT^*\overline{M} \setminus 0$.

**A.2. b-Sobolev spaces and basic mapping statements.** Fix a smooth positive b-density $d\mu_b$ on $\overline{M}$ (for instance $d\mu_b = \frac{d\rho}{\rho} dy$ in local coordinates). Define

$$L_b^2(\overline{M}; E) := L^2(\overline{M}; E, d\mu_b)$$

*Comparison with the Riemannian density used in the main text.* On each conic patch in the main text, $L^2$ and Sobolev norms are taken with respect to the Riemannian volume density $dV_{g_\varepsilon}$ rather than a fixed b-density $d\mu_b$. On such a patch these are related by a smooth weight: $dV_{g_\varepsilon} = \rho^{\dim L} a_\varepsilon(\rho, y) d\mu_b$ with $a_\varepsilon$ smooth and uniformly bounded above/below for $\rho$ small (by the uniform conic quasi-isometry). Thus the corresponding $H_b^s$-scales are equivalent up to multiplication by a fixed power of $\rho$, and the $\Psi_b$ mapping/parametrix statements below apply verbatim after this harmless conjugation.

for a Hermitian bundle $E \to \overline{M}$.

**Definition A.2 (b-Sobolev spaces).** For integer $s \ge 0$, define $H_b^s(\overline{M}; E)$ as the completion of $C_c^\infty(M; E)$ with respect to the norm

$$\|u\|_{H_b^s}^2 := \sum_{|\alpha| \le s} \|V^\alpha u\|_{L_b^2}^2, \quad V^\alpha \in \mathcal{V}_b^{|\alpha|}.$$

For non-integer $s$ one may use interpolation and duality.

The following elementary estimates are repeatedly used.

**Lemma A.3 (Differential mapping).** If $P \in \mathrm{Diff}_b^m(\overline{M}; E \to F)$, then for every $s \in \mathbb{R}$,

$$P: H_b^s(\overline{M}; E) \to H_b^{s-m}(\overline{M}; F)$$

is continuous.

*Proof.* This is immediate from the definition: $P$ is a finite sum of compositions of at most $m$ b-vector fields with smooth coefficients; commute these derivatives with the b-derivatives defining the $H_b^s$ norm. $\square$
**Lemma A.4** (Zeroth order boundedness). If $A \in \text{Diff}_b^0(\overline{M}; E \to E)$ has coefficients bounded in $L^\infty$, then

$$ \|Au\|_{H_b^s} \le C_s \|u\|_{H_b^s}, \quad s \ge 0. $$

*Proof.* Apply Leibniz/commutator identities for b-derivatives $V^\alpha(Au)$; all coefficient terms remain bounded. □

**A.3. The b-pseudodifferential calculus and parametrices.** Melrose constructed an algebra $\Psi_b^m(\overline{M}; E)$ of b-pseudodifferential operators that contains $\text{Diff}_b^m$ and is closed under composition and adjoints. For our purposes, we only use: (i) boundedness on b-Sobolev spaces, (ii) existence of a parametrix for b-elliptic operators, and (iii) smoothing remainders.

**Theorem A.5** (b-$\Psi$DO mapping properties). Let $A \in \Psi_b^m(\overline{M}; E \to F)$. Then for every $s \in \mathbb{R}$,

$$ A : H_b^s(\overline{M}; E) \to H_b^{s-m}(\overline{M}; F) $$

is continuous. If $A \in \Psi_b^{-\infty}$ is smoothing, then $A$ maps $H_b^s$ continuously into $H_b^t$ for all $t$ and is compact on bounded subsets of $H_b^s$.

**Theorem A.6** (Parametrix for b-elliptic operators). Let $P \in \text{Diff}_b^m(\overline{M}; E \to E)$ be b-elliptic. Then there exist $Q \in \Psi_b^{-m}(\overline{M}; E)$ and $R, R' \in \Psi_b^{-\infty}(\overline{M}; E)$ such that

$$ QP = I - R, \quad PQ = I - R'. $$

Consequently, for every $s \in \mathbb{R}$ there is an elliptic estimate

$$ (A.3.1) \qquad \|u\|_{H_b^s} \le C_s \left( \|Pu\|_{H_b^{s-m}} + \|u\|_{H_b^{s-1}} \right). $$

*Remark on the lower-order term in (A.3.1).* By composing the parametrix identity $QP = I - R$ with the boundedness of $Q$ on $H_b^{s-m}$ and of the smoothing remainder $R$ on $H_b^{s-1}$, one obtains (A.3.1). Equivalently, one may replace $\|u\|_{H_b^{s-1}}$ by $\|u\|_{L_b^2}$ (or by $\|u\|_{H_b^{s'}}$ for any $s' < s$) after a standard interpolation/compactness argument; we use the displayed form since it matches the global partition-of-unity argument in Section 3.

*Proof sketch and references.* The parametrix is constructed by the usual symbolic inversion of the b-principal symbol in the b-calculus, iterated to remove lower-order errors, producing smoothing remainders. The estimate (A.3.1) follows by applying $Q$ and using Theorem A.5 together with compactness of the smoothing remainder at the level of Sobolev spaces. For full details see Melrose's b-calculus exposition (e.g. [16]) and the closely related edge mapping statements in [15]. □

**A.4. How the b-calculus enters the main text.** We explain precisely how the abstract results above are used in Section 3.
Local model near each isolated singularity. For each $x \in \Sigma$, we pass to the radial blow-up of a punctured neighborhood $U'_x$. In these coordinates the cone variable $\rho$ becomes a boundary defining function and b-vector fields are generated by $\rho\partial_\rho$ and tangential derivatives on the link $L_x$.

b-ellipticity of $D_\epsilon$. On $U'_x \cap \{\rho < \rho_0\}$, the Hodge-de Rham operator $D_\epsilon$ can be written as a first-order b-operator acting on the b-form bundle $\Lambda^\bullet(\mathcal{B}T^*)$; we identify ordinary differential forms with b-forms on $U'_x$ via the b-coframe $(d\rho/\rho, dy)$, so that $D_\epsilon \in \text{Diff}_b^1(\bar{M}; \Lambda^\bullet(\mathcal{B}T^*) \to \Lambda^\bullet(\mathcal{B}T^*))$ under this bundle identification. The b-ellipticity of $D_\epsilon$ is the statement that its b-principal symbol is invertible away from the zero section. This holds because $D_\epsilon$ is elliptic on $V'$ and the conic (equivalently, b) structure near $\rho = 0$ simply rescales the radial variable; see Section 3.4.

Uniformity in $\epsilon$. The only point where we need $\epsilon$-uniform control is that the b-ellipticity constants and finitely many symbol seminorms of $D_\epsilon$ can be chosen uniformly for $\epsilon \in (0, 1]$ on fixed conic neighborhoods. This follows from:

* the uniform quasi-isometry to the cone model (H1), which gives $\epsilon$-uniform comparability of frames, tensor norms, and volume densities near $\Sigma$;

* smooth convergence on compacta away from $\Sigma$ (H2).

Consequently, in Theorem A.6 we can choose $Q_\epsilon$ and the mapping constants so that they are uniform for $\epsilon \in (0, 1]$ when restricted to the fixed conic coordinate patch. This is the analytic input behind Proposition 3.4 in the main text.

From local to stratified/global estimates. The global stratified estimate (Theorem 3.5) is obtained by applying the local b-elliptic estimate (A.3.1) to each $\chi_x u$ near $\Sigma$, and a standard interior elliptic estimate to $\chi_0 u$ on $K_0$, then commuting $D_\epsilon$ past the partition of unity. The commutators $[D_\epsilon, \chi_x]$ are zeroth-order b-operators supported away from $\rho = 0$, hence bounded as in Lemma A.4 with constants independent of $\epsilon$.

**A.5. Remarks on the edge calculus.** Although we use only the b-calculus (appropriate for isolated singularities), the same logic can be cast in the edge calculus framework of [15] by viewing a cone as a special case of an edge metric with trivial base. For general stratified spaces (depth $\ge 2$), one replaces b-calculus by the iterated edge calculus, and the analogue of Theorem A.6 requires ellipticity of the iterated normal operators at each stratum. None of these additional layers are needed in the present paper.

**A.6. Commutators with cutoffs and multipliers.** Finally, we record a simple commutator formula repeatedly used in Sections 3 and 6.

**Lemma A.7 (Commutator structure for first-order b-operators).** Let $P \in \text{Diff}_b^1(\bar{M}; E \to E)$ and $\chi \in C^\infty(\bar{M})$. Then $[P, \chi] \in \text{Diff}_b^0$ and

$$ \|[P, \chi]u\|_{H_b^s} \le C_{s,\chi} \|u\|_{H_b^s}, \quad s \ge 0. $$
If $\chi$ is supported away from $\partial\overline{M}$ (i.e. away from $\rho=0$), then $C_{s,\chi}$ depends only on finitely many ordinary derivatives of $\chi$ and is stable under $\epsilon$-uniform perturbations of $P$ as in (H1)-(H2).

*Proof.* In local coordinates, $P = \sum a_j(\rho, y)(\rho\partial_\rho) + \sum b_k(\rho, y)\partial_{y_k} + c(\rho, y)$. Then $[P, \chi]$ is multiplication by derivatives of $\chi$ against the coefficients of $P$, hence order 0 in the b-sense. The Sobolev bound follows from Lemma A.4. If $\chi$ is supported away from $\rho=0$, b-derivatives and ordinary derivatives are equivalent, and the uniformity follows from (H2) (and (H1) if one tracks frame comparability on conic neighborhoods). $\square$

## APPENDIX B. A RELLICH THEOREM IN THE STRATIFIED/B SETTING

This appendix records the compactness statements (Rellich type) used (often implicitly) in Sections 4 and 5. The guiding principle is simple: although $V' = V \setminus \Sigma$ is noncompact, we only ever need compactness on fixed compact subsets $K \in V'$. Any such $K$ stays a positive distance away from $\Sigma$, so classical Rellich compactness applies on $K$. Near each isolated singularity we also use that, on compact annuli $\{a \le \rho \le b\}$, the b-Sobolev scales agree with the standard Sobolev scales, uniformly in $\epsilon$, which allows us to localize with cutoffs and extract subsequences by a diagonal argument.

**B.1. Compact subsets of $V'$ stay away from $\Sigma$.** We first record a basic metric-topological fact.

**Lemma B.1 (Compact sets avoid the singularities).** Let $K \in V'$ be compact. Then $\text{dist}_{g_0}(K, \Sigma) > 0$. Equivalently, for every $x \in \Sigma$ there exists $\rho_x > 0$ such that $K \cap U'_x \subset \{\rho \ge \rho_x\}$.

*Proof.* Since $\Sigma$ is finite and $K \in V'$ is compact in the metric space $(V', \text{dist}_{g_0})$, for each $x \in \Sigma$ the continuous function $p \mapsto \text{dist}_{g_0}(p, x)$ attains a positive minimum on $K$ (positivity holds because $x \notin K$). Let $\delta_x := \min_{p \in K} \text{dist}_{g_0}(p, x) > 0$ and set $\delta := \min_{x \in \Sigma} \delta_x > 0$. Then $\text{dist}_{g_0}(K, \Sigma) \ge \delta$.

For the equivalent statement, fix $x \in \Sigma$ and work in the conic chart $U'_x \cong (0, \rho_0) \times L_x$ with boundary defining function $\rho$. Since $\rho \to 0$ implies $\text{dist}_{g_0}(\cdot, x) \to 0$ and $K$ stays a positive $g_0$-distance from $x$, there exists $\rho_x > 0$ such that $K \cap U'_x \subset \{\rho \ge \rho_x\}$. $\square$

**B.2. b-Sobolev norms versus standard Sobolev norms on annuli.** Let $x \in \Sigma$ and identify $U'_x$ with $(0, \rho_0) \times L_x$. For $0 < a < b < \rho_0$ set

$$A_{a,b} := \{(\rho, y) : a \le \rho \le b\} \subset (0, \rho_0) \times L_x.$$

This is a compact manifold with boundary contained in $V'$.

**Lemma B.2 (Equivalence of b- and standard Sobolev norms on compact annuli).** Fix an integer $s \ge 0$ and $0 < a < b < \rho_0$. On $A_{a,b}$ the b-Sobolev norm $\|\cdot\|_{H_b^s}$ is equivalent to the standard Sobolev norm $\|\cdot\|_{H^s}$ defined using any smooth Riemannian metric on $A_{a,b}$. Moreover, in the geometric setup of Section 2, the equivalence constants can be chosen uniformly in $\epsilon \in (0, 1]$.
*Proof.* On $A_{a,b}$ we have $\rho \in [a, b]$, hence the b-vector field $\rho \partial_\rho$ is uniformly equivalent to $\partial_\rho$. Likewise, the b-frame $\{\rho \partial_\rho, \partial_{y_1}, \dots, \partial_{y_{\dim L}}\}$ is uniformly equivalent (with constants depending only on $a, b$ and the choice of local coordinates on $L_x$) to the standard frame $\{\partial_\rho, \partial_{y_1}, \dots, \partial_{y_{\dim L}}\}$. Therefore any b-derivative of order $\le s$ is a finite linear combination of standard derivatives of order $\le s$ with bounded coefficients, and conversely. This yields equivalence of Sobolev norms on the compact set $A_{a,b}$.

For uniformity in $\varepsilon$, use (H1)-(H2): on each fixed annulus $A_{a,b}$ the metrics $g_\varepsilon$ are uniformly quasi-isometric to $g_0$ and have uniformly controlled coefficients in the chosen product coordinates. Thus the associated $L^2$ and $H^s$ norms on $A_{a,b}$ are equivalent with constants independent of $\varepsilon$, and the preceding b-versus-standard comparison remains uniform in $\varepsilon$. $\square$

**B.3. Rellich compactness on compact subsets.** We now state the compactness result in the form used in the main text.

**Theorem B.3 (Rellich compactness on compact subsets of $V'$).** Let $K \in V'$ be compact and let $s \ge 1$ be an integer. Then the inclusion

$$H^s \Omega^r(K, g_0) \hookrightarrow H^{s-1} \Omega^r(K, g_0)$$

is compact. In particular, the inclusion $H^1 \Omega^r(K, g_0) \hookrightarrow L^2 \Omega^r(K, g_0)$ is compact.

*Proof.* Since $K$ is a compact smooth manifold with (possibly empty) boundary contained in $V'$, the statement is the classical Rellich-Kondrachov theorem. One may cover $K$ by finitely many coordinate charts and reduce to the Euclidean Rellich theorem, patching with a partition of unity. $\square$

**Proposition B.4 (Diagonal subsequences and local strong convergence).** Let $\{u_j\}$ be a sequence of $r$-forms on $V'$ such that for every compact $K \in V'$ there exists $C_K$ with

$$\sup_j \|u_j\|_{H^1(K,g_0)} \le C_K.$$

Then there exists a subsequence (still denoted $\{u_j\}$) and an $L^2_{\text{loc}}$ form $u$ on $V'$ such that

$$u_j \to u \quad \text{strongly in } L^2 \Omega^r(K, g_0) \quad \text{for every compact } K \in V'.$$

If in addition for some integer $s \ge 1$ one has $\sup_j \|u_j\|_{H^s(K,g_0)} \le C_{K,s}$ for every compact $K$, then (after passing to a subsequence) $u_j \to u$ strongly in $H^{s-1} \Omega^r(K, g_0)$ for every compact $K$.

*Proof.* Choose an exhaustion $K_1 \in K_2 \in \dots \in V'$ by compact subsets with smooth boundary and $\bigcup_\ell K_\ell = V'$. Apply Theorem B.3 on $K_1$ to extract a subsequence converging strongly in $L^2(K_1)$. Repeat on $K_2$ and extract a further subsequence converging strongly in $L^2(K_2)$, and so on. A diagonal subsequence converges strongly in $L^2(K_\ell)$ for every $\ell$. Since any compact $K \in V'$ lies in some $K_\ell$, the claimed $L^2_{\text{loc}}$ convergence follows.
The $H_{\text{loc}}^{s-1}$ statement is proved similarly using compactness of the embedding $H^s(K_\ell) \to H^{s-1}(K_\ell)$ on each $K_\ell$. □

**B.4. Compatibility with the stratified/b norms.** Finally we relate the hypotheses used in Proposition B.4 to the stratified/b framework in Section 2.

**Lemma B.5 (Stratified $H_{\text{strat}}^1$ control implies local $H^1$ control).** Assume a sequence $\{u_j\}$ satisfies $\sup_j \|u_j\|_{H_{\text{strat}}^1(g_\varepsilon)} \le C$ uniformly (for $\varepsilon$ fixed, or uniformly over $\varepsilon \in (0, 1]$). Then for every compact $K \in V'$ there exists $C_K$ such that

$$ \sup_j \|u_j\|_{H^1 \Omega^r(K, g_0)} \le C_K. $$

Moreover, if the stratified bounds are uniform in $\varepsilon \in (0, 1]$ and (H2) holds, then $C_K$ can be chosen independent of $\varepsilon$ for each fixed compact $K$.

*Proof.* Fix $K \in V'$. By Lemma B.1, $K$ is contained in the “interior” region of the stratified partition of unity used in (3.2.2), i.e. $K \subset \text{supp}(\chi_0)$ after possibly shrinking the conic neighborhoods $U'_x$ (which does not affect the definition of $H_{\text{strat}}^1$). On this region the stratified norm coincides with the standard $H^1$ norm (up to equivalence) by definition.

More concretely, on $\text{supp}(\chi_0)$ the metrics $g_\varepsilon$ are smooth and, by (H2), uniformly equivalent to $g_0$ with uniformly controlled coefficients on compacta. Hence there exists $c_K, C_K > 0$ such that for all $\varepsilon$ in the relevant range,

$$ c_K \|v\|_{H^1(K, g_0)} \le \|v\|_{H^1(K, g_\varepsilon)} \le C_K \|v\|_{H^1(K, g_0)}. $$

Since $\chi_0 \equiv 1$ on a neighborhood of $K$, we have $\|u_j\|_{H^1(K, g_\varepsilon)} \lesssim \|\chi_0 u_j\|_{H^1(V', g_\varepsilon)}$. The latter is controlled by $\|u_j\|_{H_{\text{strat}}^1(g_\varepsilon)}$ via (3.2.2). Combining these inequalities gives the stated bound. □

*Remark B.6 (Where this is used in the main text).* The compactness input recorded here is invoked at two specific points in the paper.

(a) In Section 4, Proposition 4.5 gives uniform local Sobolev bounds for the normalized harmonic family. Together with Rellich compactness (Lemma 4.3) this yields the precompactness and subsequence extraction in Theorem 4.6. Proposition B.4 is an abstract diagonal version of the same extraction argument (used explicitly at the end of the proof of Theorem 4.6).

(b) In Section 5, the liminf part of Theorem 5.5 starts from the uniform energy bound $\sup_\varepsilon \tilde{Q}_\varepsilon[v_\varepsilon] < \infty$. By Theorem 3.5 (combined with the norm equivalences in Lemma 5.1 and Lemma 3.2) this yields uniform $H_{\text{strat}}^1$ control on the localized pieces from the partition of unity in (3.2.2). Lemma B.5 then converts this stratified control into uniform standard $H^1(K, g_0)$ bounds on each compact $K \in V'$, while Proposition B.4 yields (after passing to a subsequence) strong convergence in $L^2(K)$ for every compact $K$. This is the precise justification for
the line “we may assume $v_\epsilon \to v$ strongly in $L^2_{\text{loc}}$” inside the proof of Theorem 5.5.

## APPENDIX C. MODEL CALCULATIONS FOR SECTION 8

This appendix supplies explicit cone computations used (implicitly) in Section 8 and, conceptually, behind the cutoff argument in Section 6. Throughout, $(L, h)$ is a closed Riemannian manifold of dimension $m$ and $C_\rho(L) = (0, \rho_0) \times L$ carries the cone metric $g_C = d\rho^2 + \rho^2 h$.

**C.1. Borderline logarithmic integrals.** We record the elementary borderline integral governing the weight $\phi_0(\rho) = \rho|\log \rho|$.

**Lemma C.1 (A borderline integrability criterion).** Let $m \ge 1$ and set $\phi_0(\rho) = \rho|\log \rho|$ on $(0, \rho_0)$. Then

$$ \int_0^{\rho_0} \frac{\rho^{m-2}}{|\log \rho|^2} d\rho < \infty, \quad \text{whereas} \quad \int_0^{\rho_0} \frac{\rho^{m-2}}{|\log \rho|} d\rho = \infty. $$

*Proof.* If $m \ge 2$, then $\rho^{m-2}$ is locally integrable at $\rho = 0$, hence the first integral is finite and the second diverges only because $|\log \rho|^{-1}$ is not integrable against $d\rho/\rho$ when $m = 2$. If $m = 1$, substitute $t = -\log \rho$ (so $\rho = e^{-t}$ and $d\rho/\rho = -dt$) to obtain

$$ \int_0^{\rho_0} \frac{\rho^{-1}}{|\log \rho|^2} d\rho = \int_{-\log \rho_0}^{\infty} \frac{1}{t^2} dt < \infty, \quad \int_0^{\rho_0} \frac{\rho^{-1}}{|\log \rho|} d\rho = \int_{-\log \rho_0}^{\infty} \frac{1}{t} dt = \infty. $$

The remaining cases follow similarly by the same substitution and comparison. □

**C.2. An explicit commutator decay estimate.** We quantify the decay in Proposition 8.1.

**Lemma C.2 (Decay on shrinking logarithmic annuli).** Let $F \in L^2(C_\rho(L), g_C)$ and define the annulus $A_\delta := \{\delta < \rho < \delta^{1/2}\} \times L$. Then $\int_{A_\delta} |F|^2 dV_{g_C} \to 0$ as $\delta \downarrow 0$. If moreover $F(\rho, \cdot) = \phi_0(\rho)^{-1}u(\rho, \cdot)$ with $u$ bounded near $\rho = 0$, then

$$ \int_{A_\delta} |F|^2 dV_{g_C} \lesssim \frac{1}{|\log \delta|}. $$

*Proof.* The first statement is the absolute continuity of the integral for $L^2$ functions applied to a family of measurable sets with vanishing measure and shrinking to $\emptyset$.

For the quantitative bound, boundedness of $u$ gives $|F|^2 \lesssim (\rho^2|\log \rho|^2)^{-1}$. Hence, using $dV_{g_C} = \rho^m d\rho dV_h$,

$$ \int_{A_\delta} |F|^2 dV_{g_C} \lesssim \int_\delta^{\delta^{1/2}} \frac{\rho^{m-2}}{|\log \rho|^2} d\rho. $$
If $m \ge 2$, then $\rho^{m-2} \le \rho_0^{m-2}$ and the substitution $t = -\log \rho$ yields

$$ \int_{\delta}^{\delta^{1/2}} \frac{\rho^{m-2}}{|\log \rho|^2} d\rho \le \int_{\delta}^{\delta^{1/2}} \frac{1}{\rho |\log \rho|^2} d\rho = \int_{\frac{1}{2}|\log \delta|}^{|\log \delta|} \frac{1}{t^2} dt \lesssim \frac{1}{|\log \delta|}. $$

If $m = 1$, the same substitution gives the same bound directly. □

### C.3. A model version of the cutoff criterion.
For completeness we state a cone-version of the cutoff-to-minimal-domain principle that mirrors Lemma 6.1.

**Proposition C.3 (Cone cutoff implies minimal domain).** Let $u \in \text{Dom}(D_{C,\max})$ on $C_\rho(L)$ and assume $\phi_0^{-1}u \in L^2$. Let $\chi_\delta$ be the logarithmic cutoff from (6.3.1). Then $\chi_\delta u \to u$ in $L^2$ and $D_C(\chi_\delta u) \to D_C u$ in $L^2$ as $\delta \downarrow 0$. In particular $u \in \text{Dom}(D_{C,\min})$.

**Proof.** The $L^2$ convergence $\chi_\delta u \to u$ follows from dominated convergence since $0 \le \chi_\delta \le 1$. For the graph term, use the commutator identity $D_C(\chi_\delta u) = \chi_\delta D_C u + [D_C, \chi_\delta]u$ and note that $\chi_\delta D_C u \to D_C u$ in $L^2$ again by dominated convergence (because $D_C u \in L^2$ by assumption $u \in \text{Dom}(D_{C,\max})$). Thus it suffices to show $\|[D_C, \chi_\delta]u\|_{L^2} \to 0$. By the pointwise estimate $|[D_C, \chi_\delta]u| \lesssim |d\chi_\delta||u|$ and $|d\chi_\delta| \lesssim (\rho|\log \rho|)^{-1}$ on $A_\delta$, we obtain

$$ \|[D_C, \chi_\delta]u\|_{L^2(g_C)}^2 \lesssim \int_{A_\delta} |\phi_0^{-1}u|^2 dV_{g_C}, $$

which tends to 0 by Lemma C.2 applied to $F = \phi_0^{-1}u$. The conclusion $u \in \text{Dom}(D_{C,\min})$ follows from the definition of the minimal domain as the graph-norm closure of compactly supported smooth forms. □

### C.4. The ordinary double point link as an explicit smooth manifold.
For completeness we justify the explicit description used in Section 8.4. Consider the quadratic hypersurface in $\mathbb{R}^{n+1}$ given by $f(z) = z_0^2 + \cdots + z_n^2$, and its link

$$ L = \{f = 0\} \cap S_1^{2n+1}. $$

Write $z = (x + iy)/\sqrt{2}$ with $x, y \in \mathbb{R}^{n+1}$. A direct computation gives

$$ f(z) = 0 \iff |x|^2 - |y|^2 = 0 \text{ and } x \cdot y = 0. $$

The sphere constraint $|z| = 1$ is equivalent to $|x|^2 + |y|^2 = 2$. Combining these relations yields $|x| = |y| = 1$ and $x \cdot y = 0$. Hence

$$ L \cong \{(x, y) \in \mathbb{R}^{n+1} \times \mathbb{R}^{n+1} : |x| = |y| = 1, x \cdot y = 0\}, $$

which is precisely the Stiefel manifold $V_2(\mathbb{R}^{n+1})$ of orthonormal 2-frames. The identification with the unit tangent bundle $UT(S^n)$ follows by sending $(x, y)$ to the unit tangent vector $y \in T_x S^n$ at the base point $x \in S^n$.

This explicit model illustrates the geometric input behind the conic analysis: the punctured neighborhood of the singularity is (topologically) a cone on a smooth compact link of dimension $2n - 1$, and the local $L^2$ estimates reduce to the cone computations recorded above.
### C.5. A Jacobian/Hessian criterion for nodes.
For readers coming from algebraic geometry we record a convenient local criterion for identifying ordinary double points (nodes) in projective hypersurfaces. Let $V = \{F = 0\} \subset \mathbb{CP}^{n+1}$ be a hypersurface defined by a homogeneous polynomial $F$ of degree $d$, and let $x \in V$. Choose an affine chart in which $x$ corresponds to the origin and write the dehomogenized equation as $f(z) = F(1, z)$ in local holomorphic coordinates $z \in \mathbb{C}^{n+1}$. Then $x$ is a singular point if and only if $\nabla f(0) = 0$. Moreover, $x$ is an ordinary double point if and only if the complex Hessian matrix

$$H(f)(0) = \left( \frac{\partial^2 f}{\partial z_i \partial z_j}(0) \right)_{0 \le i, j \le n}$$

is nondegenerate. In that case, the holomorphic Morse lemma yields local coordinates in which

$$f(z) = z_0^2 + \cdots + z_n^2,$$

so the link is the Stiefel manifold described in Section C.4. This is the precise bridge between the Jacobian criterion (finite singular set) and the explicit conic model used throughout Section 8.

### C.6. Worked Hessian check for the explicit quadric cone.
We record a complete Jacobian/Hessian verification for the hypersurface (8.5.1). Let

$$F(Z) = Z_0^2 + \cdots + Z_n^2$$

be viewed as a homogeneous polynomial on $\mathbb{C}^{n+2}$. The projective hypersurface $V = \{F = 0\} \subset \mathbb{CP}^{n+1}$ is singular precisely at the points where all homogeneous partial derivatives vanish:

$$\frac{\partial F}{\partial Z_i} = 2Z_i \quad (0 \le i \le n), \qquad \frac{\partial F}{\partial Z_{n+1}} = 0.$$

Hence the singular locus is contained in $\{Z_0 = \cdots = Z_n = 0\}$. Intersecting with $V$ and passing to projective space yields the single point $p = [0: \cdots: 0: 1]$.

To see that this isolated singularity is an ordinary double point, pass to the affine chart $U_{n+1} = \{Z_{n+1} \neq 0\}$ with coordinates $z_i := Z_i/Z_{n+1}$. The dehomogenized defining function is

$$f(z) = z_0^2 + \cdots + z_n^2 \quad (z \in \mathbb{C}^{n+1}),$$

and its gradient is $\nabla f(z) = (2z_0, \dots, 2z_n)$. Thus $\nabla f(z) = 0$ if and only if $z = 0$, so the singular point of $V \cap U_{n+1}$ is unique.

Finally, the complex Hessian matrix of $f$ is constant:

$$\mathrm{Hess}(f) = \left( \frac{\partial^2 f}{\partial z_i \partial z_j} \right)_{0 \le i, j \le n} = 2 \mathrm{Id}_{n+1},$$

so $\det(\mathrm{Hess}(f)) = 2^{n+1} \neq 0$. By the Jacobian/Hessian criterion (equivalently, by the holomorphic Morse lemma), the germ $(V, p)$ is analytically
equivalent to the standard quadratic cone {$z_0^2 + \cdots + z_n^2 = 0$}, and hence $p$ is an $A_1$ (ordinary double point) singularity.

In particular, the link $L_p = V \cap S_\eta^{2n+1}$ is the smooth $(2n-1)$-manifold described explicitly in Appendix C.4.

### C.7. Worked Jacobian/Hessian check for the Cayley cubic (four nodes).
We give a complete verification that the cubic surface (8.5.3) has exactly four singular points and that each is an ordinary double point.

**Step 1: Jacobian equations.** Let

$$F = X_0X_1X_2 + X_0X_1X_3 + X_0X_2X_3 + X_1X_2X_3$$

be the homogeneous cubic on $\mathbb{C}^4$. Its homogeneous partial derivatives are

$$\frac{\partial F}{\partial X_0} = X_1X_2 + X_1X_3 + X_2X_3,$$

$$\frac{\partial F}{\partial X_1} = X_0X_2 + X_0X_3 + X_2X_3,$$

$$\frac{\partial F}{\partial X_2} = X_0X_1 + X_0X_3 + X_1X_3,$$

$$\frac{\partial F}{\partial X_3} = X_0X_1 + X_0X_2 + X_1X_2.$$

A point $[X] \in S = \{F = 0\} \subset \mathbb{CP}^3$ is singular if and only if all four derivatives vanish there.

**Step 2: Identification of the singular locus.** First observe that each coordinate point $p_i$ lies on $S$ (all terms in $F$ involve three distinct coordinates) and satisfies $\nabla F(p_i) = 0$, so $p_0, p_1, p_2, p_3$ are singular.

Conversely, let $[X] \in S$ satisfy $\partial F/\partial X_j = 0$ for all $j$. If one of the coordinates is nonzero, we may work in the corresponding affine chart.

*Chart* $X_0 \neq 0$. Set $x := X_1/X_0$, $y := X_2/X_0$, $z := X_3/X_0$. Then $F = 0$ becomes

$$f(x, y, z) := xy + xz + yz + xyz = 0,$$

and the affine partial derivatives are

$$\partial_x f = y + z + yz, \quad \partial_y f = x + z + xz, \quad \partial_z f = x + y + xy.$$

A singular point in this chart satisfies $f = \partial_x f = \partial_y f = \partial_z f = 0$. Solving the three gradient equations gives

$$y + z + yz = 0, \quad x + z + xz = 0, \quad x + y + xy = 0,$$

i.e. $(1+y)(1+z) = 1$, $(1+x)(1+z) = 1$, $(1+x)(1+y) = 1$. Hence $1+x = 1+y = 1+z$, so $x = y = z$. Substituting into $y + z + yz = 0$ yields $2x + x^2 = 0$, so $x \in \{0, -2\}$. If $x = -2$, then $f(-2, -2, -2) = 4 \neq 0$, so this does not lie on $S$. Therefore the unique singular point in the chart $X_0 \neq 0$ is $(x, y, z) = (0, 0, 0)$, i.e. $[1:0:0:0] = p_0$.

*Charts* $X_i \neq 0$ for $i = 1, 2, 3$. The same computation applies verbatim after permuting the coordinates, and shows that the only singular points in
these charts are $p_1, p_2, p_3$, respectively. Thus the projective singular locus is
exactly $\Sigma = \{p_0, p_1, p_2, p_3\}$.

Step 3: Hessian nondegeneracy (node condition). We verify that each $p_i$ is an ordinary double point by computing the quadratic part of the local equation. By symmetry it suffices to treat $p_0$.

In the affine chart $X_0 = 1$ the defining function is $f(x, y, z) = xy + xz + yz + xyz$. Its complex Hessian matrix at $(0, 0, 0)$ is

$$
\begin{gather*}
\operatorname{Hess}(f)(0) = \begin{pmatrix} \partial_{xx} f & \partial_{xy} f & \partial_{xz} f \\ \partial_{yx} f & \partial_{yy} f & \partial_{yz} f \\ \partial_{zx} f & \partial_{zy} f & \partial_{zz} f \end{pmatrix}_{(0,0,0)} = \begin{pmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{pmatrix}, \\
\det(\operatorname{Hess}(f)(0)) = 2 \neq 0.
\end{gather*}
$$

Equivalently, the quadratic part $q = xy + xz + yz$ is a nondegenerate complex quadratic form. Hence, by the holomorphic Morse lemma (or the Jacobian/Hessian criterion), the germ of $S$ at $p_0$ is analytically equivalent to $\{u^2 + v^2 + w^2 = 0\} \subset \mathbb{C}^3$, and $p_0$ is an $A_1$ singularity.

The same calculation in the affine charts $X_i = 1$ shows that each $p_i$ has
the same local normal form, so all four singularities are ordinary double
points. In particular, each link $L_{p_i}$ is diffeomorphic to the standard $A_1$ link
in complex dimension 2, namely $V_2(\mathbb{R}^3) \cong SO(3) \cong \mathbb{RP}^3$.

C.8. Worked Jacobian/Hessian check for the Segre cubic (ten nodes).

We verify that the Segre cubic threefold (8.5.4) has exactly ten singular
points and that each is an ordinary double point.

Step 1: Jacobian criterion for the complete intersection. Let $L(X) := \sum_{i=0}^{5} X_i$ and $C(X) := \sum_{i=0}^{5} X_i^3$ be homogeneous polynomials on $\mathbb{C}^6$. A point $[X] \in S = \{L=0\} \cap \{C=0\} \subset \mathbb{CP}^5$ is singular for the complete intersection if and only if the rank of the Jacobian matrix $(\nabla L, \nabla C)$ drops, i.e. if $\nabla L$ and $\nabla C$ are linearly dependent. Since

$$
\nabla L = (1, 1, 1, 1, 1, 1), \quad \nabla C = (3X_0^2, \dots, 3X_5^2),
$$

linear dependence is equivalent to $X_0^2 = \cdots = X_5^2$.

Step 2: Identification and counting of the singular points. Assume $X_0^2 = \cdots = X_5^2 \neq 0$. After rescaling $[X]$, we may assume $X_i \in \{\pm 1\}$ for all $i$. The hyperplane equation $L = 0$ then forces exactly three +1's and three -1's. Conversely, any such sign vector satisfies $L = 0$ and also $C = 0$ (since $C = \sum X_i^3 = \sum X_i$ for $X_i = \pm 1$). Therefore the singular locus consists precisely of the projective classes of sign vectors with three plus and three minus. Modulo the global sign ambiguity $X \sim -X$, this gives

$$
|\Sigma| = \frac{1}{2} \binom{6}{3} = 10.
$$
Step 3: Hessian nondegeneracy at a representative point. We show that each singularity is an ordinary double point by an explicit local calculation. By symmetry, it suffices to treat the point

$$p = [-1: -1: -1: 1: 1: 1] \in \Sigma.$$

Work in the affine chart $U_5 = \{X_5 \neq 0\}$ and set $y_i := X_i/X_5$ for $0 \le i \le 4$, so that $X_5 = 1$. In these coordinates the equations become

$$\tilde{L}(y) := y_0+y_1+y_2+y_3+y_4+1 = 0, \quad \tilde{C}(y) := y_0^3+y_1^3+y_2^3+y_3^3+y_4^3+1 = 0,$$

and $p$ corresponds to $y^{(p)} = (-1, -1, -1, 1, 1)$. Impose the hyperplane constraint by eliminating $y_4$:

$$y_4 = -1 - (y_0 + y_1 + y_2 + y_3).$$

Introduce local variables $u_i := y_i - y_i^{(p)}$ for $0 \le i \le 3$ and set

$$u_4 := y_4 - 1 = -(u_0 + u_1 + u_2 + u_3).$$

Then $\tilde{C} = 0$ becomes an equation in $(u_0, u_1, u_2, u_3)$ alone. Expanding $\tilde{C}$ at $p$ gives

$$\tilde{C}(u) = Q(u) + (\text{terms of order } \ge 3),$$

where the quadratic part is

$$(C.8.1) \qquad Q(u) = 6(u_0u_1 + u_0u_2 + u_0u_3 + u_1u_2 + u_1u_3 + u_2u_3 + u_2^2).$$

Equivalently, the complex Hessian matrix of $\tilde{C}$ restricted to the hyperplane coordinates $(u_0, u_1, u_2, u_3)$ is

$$\text{Hess}(\tilde{C})(0) = \begin{pmatrix} 0 & 6 & 6 & 6 \\ 6 & 0 & 6 & 6 \\ 6 & 6 & 0 & 6 \\ 6 & 6 & 6 & 12 \end{pmatrix}, \quad \det(\text{Hess}(\tilde{C})(0)) = 1296 \neq 0.$$

In particular, this Hessian normalization agrees with the quadratic part (C.8.1): for instance, $\partial_{u_0}\partial_{u_1}\tilde{C}(0) = 6$ and $\partial_{u_3}^2\tilde{C}(0) = 12$. Hence $Q$ is a nondegenerate complex quadratic form on $\mathbb{C}^4$. By the holomorphic Morse lemma (or the Jacobian/Hessian criterion), the germ of $S$ at $p$ is analytically equivalent to the standard quadratic cone $\{w_1^2 + w_2^2 + w_3^2 + w_4^2 = 0\} \subset \mathbb{C}^4$, so $p$ is an ordinary double point.

The same computation applies to any other singular point of $\Sigma$ after permuting coordinates and possibly changing signs, and therefore all ten singularities are nodes.

## REFERENCES

1. Robert A. Adams and John J. F. Fournier, *Sobolev spaces*, 2 ed., Pure and Applied Mathematics, Academic Press, 2003.

2. Pierre Albin, Eric Leichtnam, Rafe Mazzeo, and Paolo Piazza, *Hodge theory on cheeger spaces*, Journal für die Reine und Angewandte Mathematik (Crelles Journal) **2018** (2018), no. 744, 29–102.
3. H. Attouch, *Variational convergence for functions and operators*, Pitman Advanced Publishing Program, Boston, 1984.

4. Jeff Cheeger, *On the Hodge theory of Riemannian pseudomanifolds*, Geometry of the Laplace Operator, Proc. Sympos. Pure Math., vol. 36, American Mathematical Society, Providence, RI, 1980, pp. 91–146.

5. ________, *Hodge theory of complex cones*, Asterisque (1983), no. 101–102, 118–134.

6. E. B. Davies, *The hardy constant*, The Quarterly Journal of Mathematics **46** (1995), no. 4, 417–431.

7. Lawrence C. Evans, *Partial differential equations*, 2 ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010.

8. Matthew P. Gaffney, *The harmonic operator for exterior differential forms*, Proceedings of the National Academy of Sciences of the United States of America **37** (1951), no. 1, 48–50.

9. ________, *A special stokes's theorem for complete riemannian manifolds*, Annals of Mathematics **60** (1954), no. 1, 140–145.

10. Mark Goresky and Robert MacPherson, *Intersection homology theory*, Topology **19** (1980), no. 2, 135–162.

11. Tosio Kato, *Perturbation theory for linear operators*, Classics in Mathematics, Springer, 1995, Reprint of the 1980 edition.

12. Kazuhiro Kuwae and Takashi Shioya, *Convergence of spectral structures: a functional analytic theory and its applications to spectral geometry*, Communications in Analysis and Geometry **11** (2003), no. 4, 599–673.

13. Matthias Lesch, *Operators of fuchs type, conical singularities, and asymptotic methods*, Teubner-Texte zur Mathematik, vol. 136, Vieweg+Teubner Verlag, 1997.

14. Jiaming Luo and Shirong Li, *Derived stratifications and arithmetic intersection theory for varieties with isolated singularities*, 2025.

15. Rafe Mazzeo, *Elliptic theory of differential edge operators I*, Communications in Partial Differential Equations **16** (1991), no. 10, 1615–1664.

16. Richard B. Melrose, *The atiyah–patodi–singer index theorem*, Research Notes in Mathematics, vol. 4, A K Peters, Wellesley, MA, 1993.

17. Ugo Mosco, *Convergence of convex sets and of solutions of variational inequalities*, Advances in Mathematics **3** (1969), 510–585.

18. Takeo Ohsawa, *Cheeger–goreski–macpherson’s conjecture for the varieties with isolated singularities*, Mathematische Zeitschrift **206** (1991), no. 2, 219–224.

19. William L. Pardon and Mark A. Stern, $l^2$-cohomology of complex projective varieties, Journal of the American Mathematical Society **4** (1991), no. 3, 603–621.

20. Leslie Saper and Mark Stern, $l_2$-cohomology of arithmetic varieties, Annals of Mathematics **132** (1990), no. 1, 1–69.

21. Michael E. Taylor, *Partial differential equations i: Basic theory*, 2 ed., Applied Mathematical Sciences, vol. 115, Springer, New York, NY, 2011.

Email address: luojiaming@hhu.edu.cn