# Integrable structures and finite-regularity coordinates This reading constructs compatible complex coordinates for an integrable almost-complex structure of finite regularity. Two polynomial coordinate changes, a weighted solution of the Cauchy–Riemann equations and finite elliptic regularity give the coordinate theorem. The final section retains the holomorphic parameters in [a Picard-zero deformation of the Fermat quartic](../../SH03-a-picard-zero-deformation-of-the-fermat-quartic.html). The construction uses no deformation-existence theorem and assumes no smooth bootstrap of the original almost-complex structure. *Original programme exposition by GPT-6 Astra (OpenAI), Ultra, October 2026. This independently expressed exposition is dedicated under CC0-1.0. Human sources retain their own terms.* ## 1. Statement and regularity budget Let \(M\) be a smooth real manifold of dimension \(d=2n\), and let \(J\in C^{5,1/2}(\operatorname{End}TM)\) satisfy \(J^2=-1\). Suppose that sections of \(T^{0,1}_J M\) are closed under the Lie bracket. Each point has a neighbourhood with a \(C^{2,1/4}\) diffeomorphism to an open set of \(\mathbb C^n\) whose differential intertwines \(J\) and multiplication by \(i\). The changes between these coordinates are holomorphic. When \(n=0\) the assertion is immediate on the discrete zero-dimensional manifold. Henceforth assume \(n\geq1\). Here and below a smaller Hölder exponent is harmless. The conclusion asserted is deliberately weaker than optimal finite-regularity Newlander–Nirenberg estimates. It is enough for the real dimension 44 used by the family construction. More generally, if \(h=(h_1,\ldots,h_b):M\to\mathbb C^b\) is \(C^{2,1/4}\), annihilated by \(T^{0,1}_J\), and has real rank \(2b\), then the coordinates can be chosen as \((w_1,\ldots,w_{n-b},h_1,\ldots,h_b)\). The proof uses only two spatial Taylor cancellations. The local Kähler metric constructed below has class \(C^{4,1/2}\), its connection has class at least \(C^{3,1/2}\), and its curvature has class at least \(C^{2,1/2}\). The final scalar elliptic argument uses coefficients of class \(C^{2,1/2}\), an \(L^2\) distributional solution and a finite sequence of \(L^p\) improvements. There is no Sobolev embedding from a fixed small \(H^s\) into \(C^1\) in real dimension 44. ## 2. Two polynomial changes of coordinates Fix a point, written as \(0\), and make a complex-linear choice of its real tangent coordinates. In a sufficiently small coordinate ball the distribution has a frame \[ L_{\bar j}=\partial_{\bar z_j}+\sum_a A_{\bar j}^{a}(z)\partial_{z_a}, \qquad A(0)=0. \tag{N1} \] The frame coefficients are \(C^{5,1/2}\). A bracket of two members of (N1) has no \(\partial_{\bar z}\) coefficient. Involutivity therefore makes that bracket zero, rather than just another linear combination of the frame. Its \(\partial_{z_a}\) coefficient is \[ \partial_{\bar z_j}A_{\bar k}^{a}-\partial_{\bar z_k}A_{\bar j}^{a} +A_{\bar j}^{b}\partial_{z_b}A_{\bar k}^{a} -A_{\bar k}^{b}\partial_{z_b}A_{\bar j}^{a}=0. \tag{N2} \] Suppose \(A=O(|z|^s)\), where \(s=1\) or \(2\), and let \(P\) be its homogeneous Taylor term of degree \(s\). In (N2) the nonlinear terms have order at least \(2s-1\). The terms of degree \(s-1\) consequently give \[ \partial_{\bar z_j}P_{\bar k}^{a} =\partial_{\bar z_k}P_{\bar j}^{a}. \tag{N3} \] Regard \(z\) and \(\bar z\) temporarily as independent polynomial variables and define \[ Q^a(z,\bar z)=\int_0^1\sum_k\bar z_k P_{\bar k}^{a}(z,t\bar z)\,dt. \tag{N4} \] Differentiation with respect to \(\bar z_j\), followed by (N3), makes the integrand \(\frac{d}{dt}[tP_{\bar j}^{a}(z,t\bar z)]\). Hence \(\partial_{\bar z_j}Q^a=P_{\bar j}^{a}\). The real polynomial coordinate change \(z\mapsto z-Q(z,\bar z)\) has derivative the identity at zero and kills this leading coefficient in (N1). Indeed \(L_{\bar j}(z_a-Q^a)=A_{\bar j}^a-P_{\bar j}^a-A_{\bar j}^b\partial_bQ^a\), whose last term has order \(2s\geq s+1\). Changing the normalized frame does not change this order. Apply this operation first for \(s=1\) and then for \(s=2\). In the resulting coordinates, \[ A=O(r^3),\quad DA=O(r^2),\quad D^2A=O(r), \qquad r=|z|. \tag{N5} \] The derivative bounds are Taylor bounds, not conclusions inferred from an order bound on the function alone. Only the two indicated finite jets were used. ## 3. Differential identities before holomorphic coordinates exist Involutivity of \(T^{0,1}\), and its conjugate, imply that \(d\) on complex forms has only its type \((1,0)\) and \((0,1)\) parts. Write them as \(\partial\) and \(\bar\partial\). Splitting \(d^2=0\) by type gives \[ \partial^2=\bar\partial^2=0,\qquad \partial\bar\partial+\bar\partial\partial=0. \tag{N6} \] These identities can first be checked on functions and a local frame by the bracket formula, then on exterior products by the derivation rule. Their distributional extension follows from the same coefficient identities and testing against compactly supported forms. The available regularity is more than sufficient for these second-order identities. Put \(\psi=|z|^2\) and \(\omega=i\partial\bar\partial\psi\). Equations (N5) give \[ \omega=\omega_0+O(r^3),\quad D\omega=D\omega_0+O(r^2), \qquad |\partial\psi|_\omega^2=\psi+O(r^5), \tag{N7} \] where \(\omega_0=i\sum dz_j\wedge d\bar z_j\), using its usual Hermitian norm convention. The first error includes the term \(r\,DJ\), which is also \(O(r^3)\). Shrinking the ball makes \(\omega\) positive. It is real and closed by (N6), and has class \(C^{4,1/2}\). The Kähler connection and its identities can be established here without presupposing the coordinates being constructed. For the real metric \(g\) of \(\omega\), let \(\nabla\) be its Levi-Civita connection and set \(F(X,Y,Z)=g((\nabla_XJ)Y,Z)\). Expanding the torsion-free bracket formula gives, with \(\omega(X,Y)=g(JX,Y)\), \[ 2F(X,Y,Z)=d\omega(X,Y,Z)-d\omega(X,JY,JZ) +g(N_J(Y,Z),JX). \tag{N8} \] For example, the two algebraic rules used in the expansion are \(F(X,Y,Z)=-F(X,Z,Y)\) and \(F(X,JY,Z)=F(X,Y,JZ)\). Both follow by differentiating \(J^2=-1\) and the metric compatibility of \(J\). Involutivity makes \(N_J=0\), so (N8) gives \(\nabla J=0\). Let \(E\) be a Hermitian line with an integrable \((0,1)\) connection, and let \(D=D'+D^{\prime\prime}\) be its metric connection. In a unitary frame parallel to first order at a point, exterior multiplication and contraction satisfy \(\iota_j\epsilon_k+\epsilon_k\iota_j=\delta_{jk}\), with the corresponding barred identities. The type-preserving Levi-Civita connection has no torsion terms. With \(\Lambda\) the adjoint of exterior multiplication by \(\omega\), direct substitution gives \[ [\Lambda,D']=i(D^{\prime\prime})^*,\qquad [\Lambda,D^{\prime\prime}]=-i(D')^*. \] Apply the graded product rule to these two identities and use \(D'D^{\prime\prime}+D^{\prime\prime}D'=\Theta(E)\). The resulting identity is \[ \Delta^{\prime\prime}=\Delta'+[i\Theta(E),\Lambda]. \tag{N9} \] Thus for compactly supported \(E\)-valued \((n,1)\)-forms \(v\), \[ \|D^{\prime\prime}v\|^2+\|(D^{\prime\prime})^*v\|^2 \geq ([i\Theta(E),\Lambda]v,v). \tag{N10} \] All these are first-order operator and integration-by-parts calculations with a twice differentiable metric and connection; they do not require a holomorphic frame. On \((n,1)\)-forms a positive curvature matrix \(C=(C_{j\bar k})\), expressed in a unitary coframe, acts in (N10) by that same positive matrix on the single antiholomorphic index. This also checks the sign in (N9). ## 4. A weighted scalar solution, with its domains specified Take \(\Omega=\{\psi0\), all later members have the uniform bound in the stronger norm with weight \(e^{-\Phi_{\varepsilon_0}}\). Weak lower semicontinuity in that space follows either by a second weak extraction or by its supremum-of-bounded-linear-functionals description. Its weak limit is the same distribution. Finally monotone convergence as \(\varepsilon_0\downarrow0\) gives \[ \bar\partial u_j=g_j,\qquad \int_\Omega |u_j|^2e^{-a\psi}r^{-2n-2}dV_\omega<\infty. \tag{N17} \] ## 6. Finite elliptic regularity in any dimension We give the particular rough-coefficient estimate needed for (N17), including its initial distributional hypothesis. **Elliptic lemma.** Let \(L\) be a scalar second-order elliptic differential operator on an open subset of \(\mathbb R^d\), with coefficients of class \(C^{2,1/2}\). If \(u\in L^2_{\rm loc}\), \(Lu=f\) distributionally, and \(f\in C^{2,1/2}_{\rm loc}\), then \(u\in C^{2,1/4}_{\rm loc}\). Write \(H^{s,p}\) for the Bessel-potential spaces. On a small ball extend the coefficients, using a cutoff and a frozen elliptic principal part, to globally bounded coefficients of the same class, still elliptic. The closeness of the principal coefficients on the small ball makes this extension elliptic. Divide the frequency space into smooth shells \(|\xi|\asymp2^k\), and on the \(k\)-th shell smooth the coefficients in \(x\) at scale \(2^{-k/2}\). Taylor's remainder estimate of order \(5/2\) then decomposes the symbol as \[ L=L^\#+L^b,\qquad L^\#\in S^2_{1,1/2},\qquad L^b\in C^{5/2}S^{3/4}_{1,1/2}. \tag{N18} \] For clarity, a symbol in \(C^rS^m_{1,\delta}\) has ordinary symbol bounds of order \(m\), and its \(C^r\) seminorm after \(\xi\)-differentiation of order \(\alpha\) is bounded by a constant times \(\langle\xi\rangle^{m-|\alpha|+\delta r}\); the corresponding bounds hold through the intermediate integer \(x\)-derivatives. The remainder order in (N18) is \(2-(5/2)(1/2)=3/4\). The Fourier estimates proved in the appendix give \[ L^b:H^{\sigma,p}\longrightarrow H^{\sigma-3/4,p} \quad\text{if}\quad -5/4<\sigma-3/4<5/2, \qquad 1p\) with \(1/q>1/p-5/(4d)\). Choose successively \(1/q=\max\{1/(8d),1/p-1/(2d)\}\) until \(p=8d\). There are finitely many steps; their cutoffs can be nested inside one fixed original ball. This gives \(u\in H^{5/4,8d}_{\rm loc}\). Apply (N19) again with \(\sigma=5/4\); now its target exponent is \(1/2\), in the stated range. Since a compactly supported \(C^{5/2}\) right-hand side belongs to \(H^{1/2,8d}\), (N20) gives \[ u\in H^{5/2,8d}_{\rm loc}\subset C^{2,1/4}_{\rm loc}. \tag{N22} \] The latter embedding has room to spare, since \(5/2-d/(8d)=19/8>2+1/4\). At no step were infinitely many derivatives of the coefficients required. This proves the lemma. For (N17) apply the **unweighted** formal adjoint of \(\bar\partial\) in the fixed metric \(\omega\): \[ L=\bar\partial^*\bar\partial, \qquad Lu_j=\bar\partial^*g_j. \tag{N23} \] The principal symbol of \(L\) is \(|\xi^{0,1}|_\omega^2\), positive for every nonzero real covector \(\xi\). Its coefficients and the right-hand side are at least \(C^{3,1/2}\), so they satisfy the weaker hypotheses of the lemma. The singular weight was used only to select the solution and is absent from (N23). We conclude \(u_j\in C^{2,1/4}\). ## 7. Vanishing of the first jet and coordinate compatibility Since \(dV_\omega\) is comparable to Euclidean volume near zero, (N17) and continuity force \(u_j(0)=0\). Otherwise its radial integral would dominate \(\int_0 r^{-3}dr\). If the real linear map \(du_j(0)\) were nonzero, it would be bounded away from zero on an open cone of unit directions. Differentiability would then give \(|u_j(r\theta)|\geq cr\) on a smaller cone for sufficiently small \(r\). This contradicts the divergent integral \(\int_0 r^{-1}dr\) in (N17). Therefore \[ u_j(0)=0,\qquad du_j(0)=0. \] Set \(F_j=\chi z_j-u_j\). Then \(\bar\partial F_j=0\), and \(dF(0)=dz(0)\). The real inverse-function theorem gives a \(C^{2,1/4}\) local diffeomorphism \(F=(F_1,\ldots,F_n)\). Its differential is complex-linear for \(J\), precisely because each component is annihilated by \(T^{0,1}_J\). If \(G\) is another such coordinate system, the \(C^1\) transition \(G\circ F^{-1}\) has complex-linear differential everywhere. Fix all but one complex variable in a coordinate polydisc. The ordinary one-variable Cauchy–Riemann equations and the real Green formula give zero integrals around every rectangle, hence the one-variable Cauchy integral formula. Applying that formula successively in all variables gives the polydisc Cauchy formula. It follows by differentiating its nonsingular boundary integral that the transition is holomorphic. This also shows that every \(C^1\) function annihilated by \(T^{0,1}_J\) becomes holomorphic in these coordinates. Thus the atlas is compatible and its induced complex structure is exactly \(J\). ## 8. Retaining the holomorphic parameters Apply the theorem on \(X_0\times B\), of complex dimension \(n=22\), to the distribution \[ W_{\bar j}=\partial_{\bar z_j}-\phi_{\bar j}^{a}(h)\partial_{z_a}, \qquad \partial_{\bar h_\nu},\quad 1\leq\nu\leq20. \tag{N24} \] Its plus-coefficient \(A\) in (N1) is \(-\phi\) in the vertical block and zero in the parameter block. The identity \(\bar\partial\phi=\tfrac12[\phi,\phi]\) makes the vertical brackets vanish with precisely this minus sign. Holomorphic dependence of \(\phi\) as an \(H^8\)-valued function makes the mixed brackets vanish. Small \(C^0\) norm gives complementarity with the conjugate distribution. The uniform spatial \(C^{5,1/2}\) bounds for every parameter derivative, obtained on smaller parameter balls from its normally convergent Banach-valued series, give the joint regularity used in Section 1. Each function \(h_\nu\) is annihilated by (N24). It is therefore holomorphic in the atlas just constructed. The real differential of \(h\) has rank 40 and is complex-linear, so its complex rank is 20. In one holomorphic coordinate chart choose two linear coordinate combinations whose differentials complement the independent \(dh_\nu\). The map \[ (F_1,\ldots,F_{22})\longmapsto (w_1,w_2,h_1,\ldots,h_{20}) \tag{N25} \] has invertible complex derivative. The real inverse-function theorem and the preceding transition argument show that its inverse is holomorphic. This proves the local product-coordinate assertion for the parameter map. Consequently the underlying product projection becomes a holomorphic submersion, with complex two-dimensional fibres. It is proper because \(X_0\) is compact. At \(h=0\), \(\phi(0)=0\), so the central fibre has its original complex structure. The new atlas is holomorphic and hence smooth in its own coordinates; no assertion that the original \(J\) has become smooth in the original product coordinates is needed. Smooth horizontal-lift arguments can now be made in this new smooth structure. None of these steps changes the integral marking, the rational coefficient field in the lesson, or the later arbitrary-rank argument. ## Appendix. The Fourier estimates used in the elliptic lemma This appendix specifies the analytic inputs in (N18)–(N22) and supplies their derivation. Ordinary Fourier inversion, Plancherel, the Lebesgue differentiation theorem, and the elementary Hilbert-space and integration results used above are the underlying analysis conventions. Choose smooth dyadic Fourier projections \(\Delta_k\), and enlarged projections equal to one on their supports. The two estimates needed are the square-function equivalence \[ \|v\|_{H^{s,p}}\asymp \left\|\left(\sum_k2^{2ks}|\Delta_kv|^2\right)^{1/2}\right\|_p, \quad10\). Here is one derivation, including the \(L^p\) ingredient. A smooth multiplier \(m(\xi)\), also allowed to have values in bounded operators between Hilbert spaces, whose derivatives satisfy \(\|D^\alpha m(\xi)\|\leq C_\alpha\langle\xi\rangle^{-|\alpha|}\), has uniformly \(L^2\)-bounded smooth truncations by Plancherel. Integration by parts in each frequency shell and summation give the kernel bounds \(|K(x)|\leq C|x|^{-d}\), \(|\nabla K(x)|\leq C|x|^{-d-1}\). For an \(L^1\) input, take maximal dyadic cubes where the average of its norm exceeds \(\lambda\). They are disjoint, their total volume is at most \(\|v\|_1/\lambda\), and their parent-cube averages bound their averages by \(2^d\lambda\). Replace \(v\) on each cube by its average and call the remaining, mean-zero, cube-supported parts \(b_Q\). Lebesgue differentiation gives a good part bounded by \(2^d\lambda\), with its squared \(L^2\) norm at most \(C\lambda\|v\|_1\). The \(L^2\) estimate handles this part. Outside a fixed dilation of \(Q\), subtract the kernel value at its centre; the gradient bound and the mean-zero property give \(\int_{(Q^*)^c}|m(D)b_Q|\leq C\|b_Q\|_1\). Summing proves the weak \((1,1)\) bound. Interpolation with the \(L^2\) bound follows directly by splitting \(v\) at size proportional to \(\lambda\) and integrating the distribution formula \(\|Tv\|_p^p=p\int_0^\infty\lambda^{p-1}|\{|Tv|>\lambda\}|d\lambda\): the two inner integrals have exponents \(p-2\) and \(p-3\), convergent on their respective ranges for \(12\). The argument uses norms, averages and Hilbert-space Plancherel, so it also proves the Hilbert-valued assertion. Apply this assertion to the operator-valued multiplier \(v\mapsto(\varphi_k(D)v)_k\), where \(\sum_k\varphi_k^2=1\), and its transpose. This gives (F1) for \(s=0\). Applying it also to the enlarged shell multipliers \(2^{ks}\langle\xi\rangle^{-s}\varphi_k(\xi)\) gives general \(s\). Annular synthesis follows by the transpose estimate and finite overlap. For ball synthesis, use the upper-triangular matrix of shell cutoffs with coefficients \(2^{-(k-j)s}\), \(k\geq j\). Its row and column sums, and their symbol derivatives, are bounded when \(s>0\). The same Hilbert-valued multiplier bound gives the asserted synthesis estimate. This accounts for the sign restriction in the ball version. It suffices first to take order zero. Choose a fixed annular cutoff and a larger compactly supported cutoff equal to one on its support. On frequency shell \(k\), rescale the frequency variable by \(2^{-k}\) and extend the resulting cutoff symbol periodically on one fixed cube. Write its Fourier series as \[ a_k(x,\xi)=\sum_{\ell\in\mathbb Z^d} Q_{k\ell}(x)e^{ic\,2^{-k}\ell\cdot\xi}\varphi_k(\xi), \] where the fixed constant \(c\) is determined by the cube's period. One can absorb a harmless enlargement of the shell cutoff into \(\varphi_k\). Integration by parts in the periodic variable gives, for every chosen integer \(M\), \[ \|Q_{k\ell}\|_\infty\le C_M(1+|\ell|)^{-M}, \qquad \|Q_{k\ell}\|_{C^r}\le C_M(1+|\ell|)^{-M}2^{kr}. \] Only finitely many frequency derivatives, and the stated \(r\) spatial Hölder derivatives, enter any fixed such bound. The phase factor must be retained: it is a translation of input \(k\) at a scale depending on \(k\). Put \[ v_{k\ell} =e^{ic\,2^{-k}\ell\cdot D}\varphi_k(D)v. \] For a fixed Sobolev exponent \(s\), the multiplier from scalar functions to \(\ell^2\)-valued functions with components \[ 2^{ks}\langle\xi\rangle^{-s} e^{ic\,2^{-k}\ell\cdot\xi}\varphi_k(\xi) \] has finite overlap at every nonzero frequency. Its operator-valued derivatives satisfy the hypotheses of the preceding Hilbert-valued multiplier argument, with constants bounded by a polynomial in \(1+|\ell|\). Indeed each derivative of the phase contributes \(2^{-k}\ell\); on the shell, \(2^{-k}\) is comparable to \(\langle\xi\rangle^{-1}\). Thus, for some finite \(N=N(d,p,s)\), \[ \left\|\left(\sum_k2^{2ks}|v_{k\ell}|^2\right)^{1/2}\right\|_p \le C(1+|\ell|)^N\|v\|_{H^{s,p}}. \] The finite value of \(N\) follows from the finite number of kernel derivatives used in the preceding Calderón–Zygmund proof; no spatial derivatives of the original coefficient beyond \(r\) have been added. For each fixed \(\ell\), split \(Q_{k\ell}\) into spatial frequencies below \(k-3\), within three of \(k\), and above \(k+3\). The low part has a uniformly bounded low-pass kernel and gives annular output. The middle part has finitely many uniformly bounded pieces and gives ball output. Their weighted square-function bounds are therefore at most \[ C_M(1+|\ell|)^{-M} \left\|\left(\sum_k2^{2ks}|v_{k\ell}|^2\right)^{1/2}\right\|_p, \] using annular synthesis for the first part and ball synthesis, hence \(s>0\), for the second. For the high part, the Hölder tail estimate gives \[ \|\Delta_jQ_{k\ell}\|_\infty \le C_M(1+|\ell|)^{-M}2^{-r(j-k)} \quad (j>k+3). \] The product has annular frequency of size \(2^j\). After multiplying by \(2^{js}\), its pointwise sequence estimate contains \[ 2^{-(r-s)(j-k)}\,2^{ks}|v_{k\ell}|. \] Convolution with this positive summable sequence is bounded on \(\ell^2\) when \(sN+d\). Summation over \(\ell\) now converges in operator norm and proves F2 in order zero. For general order \(m\), apply this result to the symbol \(a(x,\xi)\langle\xi\rangle^{-m}\) and the input \(\langle D\rangle^m v\). This gives \[ C^rS^m_{1,1}:H^{s+m,p}\longrightarrow H^{s,p} \quad(0