Proof. First take a periodic complex on the whole cover. Conjugation by \(e^{-\rho h}\) changes its differential to \[d_\rho=d+\rho\,\bar\partial h\wedge.\] Fourier transformation of the deck variable decomposes this complex as a direct integral of the compact complexes on \(X\) with multipliers \(e^{\rho+i\theta}\), \(0\le\theta\le2\pi\). One can see both the identification and the norm assertion on a compact fundamental region: the sequence of its translates is an \(\ell^2\) sequence with values in local Sobolev spaces, and multiplication by \(e^{-\rho h}\) changes the norm only by bounded factors on that region. Plancherel’s theorem gives the direct integral. The endpoint gauges at \(\theta=0,2\pi\) agree by the unitary gauge \(e^{2\pi i h}\); thus a global logarithm of the multiplier is unnecessary.

By (10), every compact complex on this circle is acyclic. Its Hodge operator \(d_\rho+d_\rho^*\) is consequently invertible. The compact elliptic estimate, the bounded inverse theorem, and a finite covering of the parameter circle give a uniform inverse from \(L^2\) to \(W^1\). For completeness, locally in the parameter an already invertible operator remains invertible by a Neumann series; the difference of two nearby operators is a small bounded map \(W^1\to L^2\). Applying the compact elliptic estimate to the inverse gives the uniform higher-order bounds. Fourier inversion therefore gives an inverse for the conjugated periodic Hodge operator, and the compact homotopy identities integrate to an acyclic complex with a homotopy gaining one derivative. Duality gives the same bounds at negative Sobolev orders.

The adjoint used for a general weight is specifically the adjoint in that weighted norm: \[ d_{E,\phi}^*=e^{2\phi}d_E^*e^{-2\phi}. \tag{14}\] Thus \(d_E+d_{E,\phi}^*\), conjugated by \(e^{-\phi}\), is \(\widetilde d_E+\widetilde d_E^*\), where \(\widetilde d_E=e^{-\phi}d_Ee^\phi\). At the positive end its periodic limit is the operator just considered with \(\rho=a\); at the negative end it is the one with \(\rho=-b\). The difference has order zero and all its coefficients tend to zero, by (12).

Here is the parametrix argument explicitly. On each tail use the inverse of the corresponding whole-cover periodic limit, with nested cutoffs equal to one sufficiently far along that tail. On a relatively compact middle region use an interior elliptic parametrix. Compose their sum with the conjugated Hodge operator on either side. Commutators with the cutoffs are order-zero operators supported in compact collars. Following or preceding the order-\(-1\) parametrices they are compact, by Rellich’s theorem. Multiplication by a coefficient tending to zero is compact from \(W^1\) to \(L^2\): truncate it to a compact set, apply Rellich there, and bound the remaining operator norm by its supremum on the tails. This treats the end errors as well. The resulting left and right parametrices are inverses modulo compact operators. In particular the Hodge operator is Fredholm from \(W^1\) to \(L^2\).

This construction also proves the useful estimate \[ \|u\|_{1;a,b}\le C\bigl( \|(d_E+d_{E,\phi}^*)u\|_{0;a,b} +\|u\|_{L^2(C)}\bigr) \tag{15}\] for a fixed compact set \(C\), and gives the analogous estimate for the adjoint. The global first-order elliptic estimate shows that the maximal domain of the conjugated Hodge operator is \(W^1\). To justify the assertion for an initially distributional element of that domain, apply the local estimate on the uniform charts and sum; cutoffs in \(h/R\) then exhaust \(Y\), with bounded commutators tending to zero. The same cutoffs show that compactly supported smooth sections are a core. The operator is therefore self-adjoint on \(W^1\).

Because \(d_E^2=0\), its square is the degree-preserving Hodge Laplacian. Fredholmness and self-adjointness give a spectral gap away from its finite-dimensional kernel. Let \(P\) be the orthogonal projection onto that kernel and let \(G\) be the inverse of the Laplacian on its orthogonal complement, zero on the kernel. Uniform elliptic estimates give \[G:W^t_{a,b}\longrightarrow W^{t+2}_{a,b} \quad(t\in\mathbb Z),\qquad Q=d_{E,\phi}^*G:W^t_{a,b}\longrightarrow W^{t+1}_{a,b}.\] For negative \(t\), these follow also by transposing the positive-order estimates after conjugation. Elements of \(\operatorname{im}P\) have all weighted Sobolev orders, so \(P\) extends to all these spaces. Commutation with the differential, first on smooth vectors and then by density, gives \[ I=P+d_EQ+Qd_E. \tag{16}\] On the stepped complex \(Q\) maps degree \(q\) into degree \(q-1\) with exactly the required gain of one derivative. A closed element has zero class precisely when its projection under \(P\) is zero. This proves closed ranges, finite-dimensionality, and independence of the Sobolev realization. It proves the same statements for the maximal \(L^2\) complex, completing the proof. ◻

Proof. Choose \(\phi_{-a,-b}=-\phi_{a,b}\), so their exponential weight factors are reciprocal. On weighted \(L^2\) representatives the integrand is absolutely integrable by Cauchy–Schwarz, since the two weights cancel. Integration by parts is valid by inserting the cutoffs in \(h/R\): their derivatives are bounded and the \(L^2\) tails tend to zero. Thus exact classes pair trivially with closed classes. The Hermitian Hodge star and the bundle metric give an antilinear map \(J\) from \((p,q)\) forms with values in \(E\) to complementary forms with values in \(E^{-1}\), characterized by the pointwise Hermitian inner product under wedge integration. The map \(u\mapsto e^{-2\phi}Ju\) takes the weighted harmonic equations to the opposite-weight harmonic equations; this follows immediately by transposing \(d_E\) and using (14). It is an isomorphism, and the pairing of \(u\) with its image is its weighted squared norm, up to the harmless fixed degree sign. Harmonic representatives therefore give a nondegenerate pairing. Both spaces are finite-dimensional by Proposition 10. At the cochain level, the dual of the order \(s-q\) term has order \(q-s\). Writing the complementary degree as \(2-q\) shows that the dual stepped complex has base order \(2-s\), since \((2-s)-(2-q)=q-s\). Thus opposite weights also have this complementary Sobolev shift; independence of the shift, already proved above, is what allows the notation in (17) to suppress it. ◻

Proof. The logarithm of the norm of a holomorphic section satisfies \(L\log|s|\ge-M\) in distributions, including at its zeros, for a fixed \(M\) depending on the curvature bound. In a holomorphic frame this is the fact that the logarithm of the modulus of a holomorphic function is plurisubharmonic, plus the bounded curvature contribution of the metric. Consequently \(u=\log|s|+(M/c)h\) is subharmonic for \(L\). The chain rule and the bound on \(dh\) give, for a sufficiently small \(\eta>0\), \[L(e^{-\eta h}) =e^{-\eta h}\bigl(-\eta c+\eta^2\,\sigma_L(dh,dh)\bigr)\le0,\] where \(\sigma_L(dh,dh)\) is bounded and nonnegative.

Choose a regular value \(h_0>H_0\) and a finite upper bound \(C\) for \(u\) on its compact level. Given \(A>\max(1,-C)\), choose a regular \(H>h_0\) so large that \(u\le-A\) on \(\{h=H\}\). Such an \(H\) exists by the assumed decay. The superharmonic function \[v_A=-A+(A+C)e^{-\eta(h-h_0)}\] has value \(C\) on the lower boundary and is at least \(-A\) on the upper boundary. The maximum principle on the compact slab gives \(u\le v_A\). It applies to the upper-semicontinuous subharmonic logarithm with its value \(-\infty\) at zeros, either directly or by local regularization. For a fixed point of height greater than \(h_0\), take \(H\) beyond that point and let \(A\to\infty\). Since \(1-e^{-\eta(h-h_0)}>0\), we obtain \(u=-\infty\) there. No connectedness assumption on the level sets was used. ◻

Proof. We transform a cut-off section to the compact base, continue the resulting solutions through the exceptional multiplier, and recover the improved rates by contour inversion.

The transformed equation. Consider first the positive tail and choose a cutoff \(\chi\) equal to one far along it and zero off a slightly longer tail. In the periodic end frame, extend \(u=\chi s\) by zero to \(Y\). Then \[ du=f=(d\chi)s-\chi Bs. \tag{18}\] If \(s\) belongs to rate \(r\), multiplication by \(B\) puts the second term at rate \(r-\gamma\); the first term is compactly supported. Estimates with any strictly intermediate rate avoid boundary-convergence issues.

For a positive-tail-supported form \(v\), its Fourier–Laurent transform is, initially in a right half-plane, \[ \widehat v(z)(x)= \sum_{k\in\mathbb Z}e^{-z(h(x)+k)}(T^{k})^*v(x). \tag{19}\] The expression is invariant under \(T\), hence a form on \(X\). If \(v\) has rate \(r\), Cauchy–Schwarz on the translates gives normal convergence in compact-manifold Sobolev norms when \(\operatorname{Re}z>r\). The differentiated series converges on every smaller half-plane, including any polynomial factors in \(k\), so it is holomorphic there. The transform of (18) is \[ d(z)\widehat u(z)=\widehat f(z),\qquad d(z)=d+z\,\bar\partial h\wedge. \tag{20}\] Its multiplier is \(\lambda=e^z\); this also fixes our sign convention. The source is holomorphic on \(\operatorname{Re}z>r-\gamma\). Moreover \(d(z)\widehat f(z)=0\) there: the identity holds in the initial half-plane because \(d^2u=0\), and extends by the identity theorem.

Continuation on the compact base. For \(e^z\ne1\), compact acyclicity gives a unique degree-zero solution of (20). It depends holomorphically on \(z\). To verify the latter assertion without choosing a nonholomorphic family of Hodge inverses, fix \(z_0\) where degree-zero cohomology vanishes. Compact ellipticity makes \[B_0=(d(z_0)^*d(z_0))^{-1}d(z_0)^*\] a bounded left inverse, from \(W^t\) to \(W^{t+1}\). Writing \(d(z)=d(z_0)+(z-z_0)A\), any existing solution obeys \[ \widehat u(z)= [I+(z-z_0)B_0A]^{-1}B_0\widehat f(z). \tag{21}\] The right side is holomorphic near \(z_0\). In the canonical case degree-zero cohomology also vanishes at \(e^{z_0}=1\), so the same formula continues the solution across that point. The full equation, not only its image under \(B_0\), holds by analytic continuation. Higher canonical cohomology at this parameter creates no pole in this degree-zero calculation.

In the scalar case the constants prevent such a left inverse at the exceptional point. A local gauge reduces that point to \(z_0=0\). We first solve for the component orthogonal to the constants, then determine the constant coefficient. Let \(P_0\) be projection onto the constants and use the compact left inverse \(B_0\) with \(B_0d=I-P_0\). Write a prospective solution as \(c+v\), with \(v\) orthogonal to the constants, and put \[R(z)=(I+zB_0A)^{-1},\qquad v=R(z)B_0\widehat f-zcR(z)B_0A1.\] The inverses here act on the complement of the constants. Set \(w(z)=1-zR(z)B_0A1\); the candidate solution is \(R(z)B_0\widehat f+c w(z)\). The class of \(A1=\bar\partial h\) in \(H^1(X,\mathcal O_X)\) is nonzero. Indeed, if \(\bar\partial h=\bar\partial v\) for a function on \(X\), then \(h-v\) would be a holomorphic function on \(Y\) changing by \(1\) under \(T\), contrary to Lemma 6. Choose a continuous scalar functional \(\ell\) on degree-one forms which annihilates \(\operatorname{im}d\) and for which \(\ell(A1)\ne0\); pairing with the harmonic representative of this nonzero class gives such a functional. Applying \(\ell\) to the transformed equation determines the candidate coefficient \[ c(z)= \frac{\ell\bigl(\widehat f-d(z)R(z)B_0\widehat f\bigr)} {\ell(d(z)w(z))}, \qquad \ell(d(z)w(z))=z\ell(A1)+O(z^2). \tag{22}\] This coefficient is meromorphic with at most a simple pole. For every sufficiently small nonzero \(z\), compact acyclicity already supplies a unique solution of the full transformed equation. That solution must satisfy both the complement equation and the displayed scalar equation, so it equals our candidate. Thus the candidate solves the full equation on the punctured neighborhood and continues it meromorphically across zero. The residue is a constant, since the pole part in the differential equation is annihilated by \(d\). This calculation supplies the required finite-order meromorphic continuation without applying a Fredholm theorem to the overdetermined map \(d:W^{t+1}(\Lambda^{0,0})\to W^t(\Lambda^{0,1})\).

Recovering the end expansion. For a desired positive-end rate \(r_1\) with \(r-\gamma<r_1<r\), choose a contour real part \(\sigma\ne0\) such that \[r-\gamma<\sigma<r_1<r.\] Start from any vertical line with real part greater than \(r\). Both lines lie in the source’s domain of holomorphy, and the continued solution has bounded Sobolev norm on each multiplier circle. Moving the inversion contour to real part \(\sigma\) gives, on a fixed fundamental region, \[ (T^k)^*u(x)=\frac1{2\pi i} \int_{\sigma-i\pi}^{\sigma+i\pi} e^{z(h(x)+k)}\widehat u(z)(x)\,dz, \tag{23}\] plus the residues crossed when moving from the initial line to \(\sigma\). Horizontal boundary terms cancel in the contour rectangle: the transform changes by the gauge \(e^{-2\pi i h}\) under \(z\mapsto z+2\pi i\), and the factor in (23) changes by its inverse. This justifies the contour deformation. Cauchy’s estimate on the new line bounds its integral contribution on the \(k\)th translate by \(Ce^{\sigma k}\) in every fixed Sobolev norm. After applying the target weight \(e^{-r_1 k}\), its squared norms on the positive translates are summable because \(r_1-\sigma>0\). The constants in (22) are the only residues; there are none in the canonical case.

At the negative end use a cutoff supported in \(\{h<H\}\). Its transform converges first in a left half-plane. If the allowed negative-end rate is \(b\), the initial domain is \(\operatorname{Re}z<-b\) and the source extends to \(\operatorname{Re}z<-b+\gamma\). Given \(b_1\) with \(b-\gamma<b_1<b\), choose a new contour with \(-b_1<\sigma<-b+\gamma\) and \(\sigma\ne0\). On the \((-l)\)th translate its integral is \(O(e^{-\sigma l})\), whose squared norms become summable after multiplication by \(e^{-b_1l}\) because \(\sigma+b_1>0\). Thus moving the contour to the right improves the negative-end rate; the residue discussion is unchanged.

It remains to check the iteration, since \(\gamma\) is fixed. For a homogeneous perturbed equation, each improvement in \(s\) improves the source \(-Bs\) by the same amount. Choose, for example, improvements smaller than \(\gamma/3\) at each step, perturbing contour positions if necessary to avoid real part zero. Canonical sections have no residue, so iteration reaches every finite decay rate. Scalar sections can cross zero once, giving \(s=c_\pm+r\) with \(r\) at a strictly negative allowed rate. If \(c_\pm=0\), the same homogeneous iteration continues indefinitely. If \(c_\pm\ne0\), then \[(d+B\wedge)r=-B c_\pm;\] only a finite decay rate for this remainder is claimed. Uniform local elliptic estimates give the stated decay of all its derivatives. ◻

Proof. Apply Lemma 13. The Chern curvatures of the metrics used here are bounded: those of the periodic bundles are bounded, and the connection coefficients and their first derivatives in the end frames are bounded by (12). Every canonical section, and a scalar-type section with zero positive coefficient, decays faster than every exponential on the positive tail. Lemma 12 makes it zero there. Unique continuation for a holomorphic section on the connected \(Y\) then gives the global injectivity assertion. The negative-end expansion uses no maximum principle at that end and makes no vanishing assertion there. ◻

Proof. By Lemma 11 the dual of the group in question is \(H^{2,0}_{-a,-b}(Y,\mathcal O)\). Its elements are canonical sections with finite exponential growth on the positive end; they vanish by Proposition 15, applied to the periodic trivial datum. Closed range and finite-dimensionality turn the dual vanishing into the asserted vanishing. For \(f\) use the homotopy \(Q\) from (16) with weights \((-R,-R)\). The top-degree harmonic projection is zero, so \(\alpha=Qf\) solves the equation. Elliptic regularity and the higher-order bounds for \(Q\) give all the asserted weighted Sobolev orders. ◻

Proof. Choose a smooth \((0,1)\)-connection \(d_0\) on \(E\) which is standard in the given end trivializations. A partition of unity constructs one, because the space of such connections is affine. Its curvature \(d_0^2\) is a compactly supported scalar \((0,2)\)-form \(\kappa\): the endomorphism bundle of a line bundle is canonically trivial. In complex dimension two it is automatically \(\bar\partial\)-closed. Apply Lemma 16, already proved solely for the trivial periodic bundle, to solve \[\bar\partial\alpha=-\kappa\] with weights \((-R,-R)\), where \(R>\gamma\). Define \(d_E=d_0+\alpha\wedge\). In a local frame its square is \[d_E^2=\kappa+\bar\partial\alpha+\alpha\wedge\alpha=0.\] The last term is zero because \(\alpha\) is scalar-valued and has degree one. The local integrability theorem for a \((0,1)\)-connection therefore makes \(d_E\) a holomorphic line-bundle structure. In rank one this theorem is just the local Dolbeault lemma applied to its closed connection form, followed by multiplication of the frame by the exponential of a local primitive. Weighted Sobolev embedding on the uniform charts gives \(|\nabla^j\alpha|\le C_j e^{-R|h|}\) on the tails, hence the prescribed bound with \(\gamma\). The finitely many middle charts impose no end condition.

Now weighted duality identifies the dual of \(H^{0,2}_{a,b}(Y,E)\) with \(H^{2,0}_{-a,-b}(Y,E^{-1})\). The latter vanishes by Proposition 15. This proves (24). The section assertions follow from the same proposition, whose finite-growth hypothesis permits every positive finite rate. The scalar top-degree vanishing used to construct \(d_E\) preceded, and did not assume, the existence of this holomorphic structure. ◻

Proof. A global weighted holomorphic function has a positive-end constant coefficient. Subtract that constant; the resulting holomorphic function has zero positive coefficient and vanishes by Proposition 15. Constants belong to the space because both end rates are positive. The top-degree assertion is Lemma 16.

Here are the details in degree one. Represent a class by a smooth weighted harmonic \((0,1)\)-form \(u\), so all its weighted Sobolev norms are finite. Choose \(\chi_++\chi_-=1\) with \(\chi_+\) zero far down the negative end and one far up the positive end. The compactly supported \((0,2)\)-form \(g=\bar\partial(\chi_+u)=-\bar\partial(\chi_-u)\) admits a primitive \(v\) decaying at both ends at any prescribed finite rate, by Lemma 16. Choose that rate larger than \(a+b+1\). Then \[u_+=\chi_+u-v,\qquad u_-=\chi_-u+v\] are closed, with \(u=u_++u_-\). The first belongs to the single-linear space \((a,-a)\), because its negative tail is \(-v\); the second belongs to the single-linear space \((-b,b)\), because its positive tail is \(v\). Both complexes are acyclic by Proposition 10. Choose \(\bar\partial f_+=u_+\) and \(\bar\partial f_-=u_-\) in those respective spaces. Each of these spaces embeds into the two-growth-end space \((a,b)\): one end has the same weight and the other has a stronger decay condition. Thus \(u=\bar\partial(f_++f_-)\) in the original complex. ◻

Proof. The even-to-odd operator \[D_E^+=d_E+d_{E,\phi}^*: W^1_{a,b}(\Lambda^{0,0}E\oplus\Lambda^{0,2}E) \longrightarrow L^2_{a,b}(\Lambda^{0,1}E)\] has index equal to the Euler characteristic in the statement, by the Hodge decomposition proved above. Replace \(d_E\) by any smooth \((0,1)\)-connection \(d_0\) standard at the ends. The difference, after conjugation, is multiplication by exponentially decaying coefficients, and is compact from \(W^1\) to \(L^2\). The index is unchanged. We do not require \(d_0^2=0\) for this replacement: \(D_0^+\) is still a first-order elliptic operator with the same principal symbol and the same invertible end limits.

Transfer the datum and this connection to \(X_n\), making them trivial and standard across the cut collars. Denote the resulting smooth bundle by \(E_n\) and its elliptic operator by \(D_{E_n}^+\). Modify \(\phi_{a,b}\) on a compact set so that it is zero on a neighborhood of the entire transferred piece and its collars; this does not change the index. The following excision uses the same compact cut-and-paste principle as [24]; we give the operator comparison in the present setting: \[ \mathop{\mathrm{ind}}D_{E,Y}^++\mathop{\mathrm{ind}}D_{\mathcal O,X_n}^+ =\mathop{\mathrm{ind}}D_{\mathcal O,Y}^++\mathop{\mathrm{ind}}D_{E_n,X_n}^+. \tag{26}\] To see it directly, compare these two direct sums. On the common compact interior interchange the two copies, carrying the \(E\) copy to the transferred \(E_n\) copy and the trivial copy to the trivial copy. Outside the identified region use the identity. In each collar both bundles are trivialized; interpolate between interchange and identity using the real rotation matrix \[\begin{pmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{pmatrix}, \qquad 0\le\theta\le\pi/2.\] Use the same rotation on the even and odd bundles. This defines bounded invertible maps on the direct-sum domains and ranges. On the interior the operators are interchanged exactly. On the collars their principal symbols are the identical Dolbeault symbols, which commute with the rotation; differentiating the rotation only adds order-zero terms. Outside the compact comparison region they agree. The conjugated operators therefore differ by compactly supported order-zero terms, which are compact from \(W^1\) to \(L^2\). Their indices agree, proving (26).

The transferred connection on \(X_n\) need not be integrable. To compute its index nevertheless by ordinary Hirzebruch–Riemann–Roch, recall from Lemma 8 that \(H^2(X_n,\mathcal O_{X_n})=0\). The exponential sequence then shows that every integral class in \(H^2(X_n;\mathbb Z)\) is the first Chern class of a holomorphic line bundle. Smooth complex line bundles are classified by their first Chern class, so \(E_n\) admits some holomorphic structure. Replacing the transferred connection by its Dolbeault operator changes the closed elliptic operator only in order zero and hence preserves its index. Compact-complex Riemann–Roch [2] gives \[\mathop{\mathrm{ind}}D_{E_n,X_n}^+-\mathop{\mathrm{ind}}D_{\mathcal O,X_n}^+ =\chi(X_n,E_n)-\chi(X_n,\mathcal O_{X_n}) =\frac12\bigl(c_1(E_n)^2-K_{X_n}\cdot c_1(E_n)\bigr).\] These products equal \(e^2\) and \(K_Y\cdot e\), since the class was transferred in its fixed trivialized collars. Combining (26) with Lemma 18 proves (25).

When the correction term is zero, Proposition 17 gives \(H^{0,2}=0\) and Proposition 15 gives \(\dim H^{0,0}\le1\). The index equality then reads \(\dim H^{0,0}-\dim H^{0,1}=1\). Both dimensions are nonnegative, so they are respectively \(1\) and \(0\), with the asserted nonzero positive coefficient. ◻

Proof. Proposition 17 supplies the stated holomorphic structure with the prescribed finite rate \(\gamma\). Choose any \(\varepsilon>0\) and use the pair \((\varepsilon,\varepsilon)\). Proposition 19 gives \[\mathop{\mathrm{ind}}\bar\partial_E =1+\frac12(e^2-K_Y\cdot e)=1.\] By the weighted degree-two vanishing and the positive-end uniqueness in Propositions 17 and 15, \(H^{0,2}_{\varepsilon,\varepsilon}(Y,E)=0\) and the positive leading coefficient is an injective map \[H^{0,0}_{\varepsilon,\varepsilon}(Y,E)\longrightarrow\mathbb C.\] The index identity therefore implies that this degree-zero space is one-dimensional (and the degree-one space is zero). Take its nonzero section and normalize the positive leading coefficient to one. Proposition 15 gives a constant negative leading coefficient \(c_-\) and exponentially small errors at both ends.

Suppose \(c_-\neq0\). Uniform convergence to nonzero constants would make \(s\) nowhere zero outside a compact height slab. Its zero divisor \(D\) would then be compact. Moreover its compact divisor class would equal the particular relative class \(e\), not just its ordinary image. To see the role of the framings, write \(s=f_\pm\tau_\pm\) on the ends in the fixed frames. Far enough out, \(f_+\) lies in a disk about \(1\) that avoids zero and \(f_-\) lies in a disk about \(c_-\) that avoids zero. Straight-line homotopies in these disks deform \(f_\pm\) through nowhere-zero maps to their limiting constants. Each nonzero constant is in turn homotopic to \(1\) in \(\mathbb C^*\). Thus the trivializations provided by \(s\) on the ends are homotopic to the prescribed ones. The relative zero-divisor formula for \(c_1\) consequently identifies the compact divisor class with \(e\).

The restriction of \(\pi\) to the compact divisor is proper. Its pushforward \(\pi_*D\) is an effective divisor on \(X\), with positive integral multiplicities equal to the corresponding generic mapping degrees. Since \(K_Y=\pi^*K_X\), the projection formula gives \[K_X\cdot\pi_*D=K_Y\cdot D=K_Y\cdot e=-m<0.\] This contradicts Lemma 5; if the divisor were empty, the left side would instead be zero and give the same contradiction. Therefore \(c_-=0\), proving (27).

The positive limit excludes zeros for all sufficiently large \(h\). The zero set of a nonzero holomorphic section is a locally finite effective divisor. The local divisor construction identifies \(E\simeq\mathcal O_Y(D)\), so \([D]=c_1(E)=j(e)\) in ordinary cohomology; the corresponding Chern-current identity is the local form of [7]. Under Poincaré duality for the oriented noncompact four-manifold \(Y\), this says that its locally finite fundamental cycle is the dual of \(j(e)\). In particular, if \(\alpha\) is any compactly supported closed two-form, then \[ \int_Y\alpha\wedge j(e)=\int_D\alpha. \tag{28}\] The right side is well-defined by local finiteness. No assertion that \(D\) is a compact divisor is being made here. ◻

Proof. Here are details of the quantitative propagation used in the proof. Choose a compact set \(Q\subset Y\) whose deck translates cover \(Y\). Choose finitely many small coordinate balls covering \(Q\), and paths joining their centers to a point \(p\), together with a path from \(p\) to \(T^{-1}p\). The union of these paths is compact. Subdividing the paths and slightly enlarging the balls produces a finite collection of ball chains. Repeating its translates connects a starting ball to every ball covering \(T^{-l}Q\) in at most \(N(l+1)\) transitions, for one fixed \(N\). All balls in such a chain lie in a fixed-width enlargement of the height interval traversed by the chain. These statements use only connectedness of \(Y\), compactness of the chosen paths, and the identity \(h\circ T=h+1\); connectedness of individual height levels is unnecessary.

We recall the elementary analytic estimate for a transition. If \(F\) is holomorphic on a Euclidean ball of radius \(r_3\), and \(0<r_1<r_2<r_3\), the three-circle inequality is \[ \log\sup_{B_{r_2}}|F| \le \vartheta\log\sup_{B_{r_1}}|F| +(1-\vartheta)\log\sup_{B_{r_3}}|F|, \qquad \vartheta=\frac{\log(r_3/r_2)}{\log(r_3/r_1)}. \tag{30}\] Indeed, restricting to each complex line through the center gives the one-variable three-circle inequality, and then taking the supremum over the lines gives the displayed inequality. Choose the ball chains so that a small ball at one stage is contained in a middle ball at the next stage, while that stage’s large ball remains inside its coordinate chart. This is possible by subdividing the paths further. Only finitely many radius ratios and coordinate changes are involved. Consequently the corresponding numbers \(\vartheta\) have a positive lower bound \(\vartheta_0\). The same construction, with a bounded number of additional transitions, reaches the small concentric balls needed for all the local integrations below.

On a sufficiently negative tail write the operator as \(\bar\partial+\alpha\) in the prescribed smooth frame. Since \(\alpha\to0\) smoothly and \(\bar\partial\alpha=0\), on each of our large coordinate balls there is a function \(u\) with \(\bar\partial u=\alpha\) and a uniform bound \(|u|\le C_0\) on a slightly smaller ball. This is the local Dolbeault lemma with its interior estimate. One can obtain the estimate directly by the Cauchy–Green operator in the first coordinate and then in the second: closedness removes the first antiholomorphic derivative from the remaining coefficient; the integrable kernels give a bound in terms of finitely many coefficient norms on the larger polydisc. The finite atlas makes these bounds uniform. Moving the start of the tail makes the coefficient norms at most one, independently of the original decay rate. The frames \(e^{-u}\) are holomorphic and their norms, and the absolute values of transition functions between these frames, are bounded above and below by geometric constants.

Let \(F\) denote the coefficient of \(s\) in such a holomorphic frame. The boundedness of \(s\) gives a common upper bound \(\log|F|\le U\) on the tail, where \(U\ge1\) may depend on \(s\). At every transition, (30), the inclusion of the preceding small ball in the next middle ball, and the bounded frame changes show the following. If the logarithmic supremum on the preceding small ball is at least \(-M\), then on the next small ball it is at least \[-cM-C_1(U+1),\qquad c=\vartheta_0^{-1}>1.\] Increasing \(c\) if needed accommodates any transition with equal radii. Since \(s\) is nonzero, unique continuation supplies a finite starting value \(M\). Start sufficiently far down the tail that all repeated chains stay in the region where the frame bounds hold. Iteration over at most \(N(l+1)\) transitions gives \[ \sup_V\log|F|\ge -C_2 c^{Nl} \tag{31}\] on every small ball \(V\) needed to cover the \(l\)th negative cell. Here and below a change of the starting cell changes \(C_2\), not \(c\) or \(N\). A slab \(Y_l\) meets only a fixed number of these cells, because \(h\) is bounded on \(Q\).

For completeness the conversion from a logarithmic supremum to an \(L^1\) bound is elementary. Arrange nested coordinate balls of radii \(r,2r,4r\) and choose \(x\in B_r\) with \(\log|F(x)|\ge-M-1\), using (31). The nonnegative function \(v=U-\log|F|\) is superharmonic. The ball \(B(x,3r)\) contains \(B_{2r}\) and lies in \(B_{4r}\), so its mean inequality gives \[\int_{B_{2r}}v\le \int_{B(x,3r)}v \le |B_{3r}|\,v(x)\le |B_{3r}|(U+M+1).\] The usual limiting interpretation applies at zeros. Thus \(\int_{B_{2r}}|\log|F||\le C_3(U+M+1)\). Choose a smooth cutoff \(\chi\) equal to one on \(B_r\) and supported in \(B_{2r}\). Poincaré–Lelong [7] and distributional integration by parts give, up to the fixed normalization of \(d^c\), \[\int_D\chi\,\omega =c_0\int\log|F|\,dd^c(\chi\omega) \le C_4\int_{B_{2r}}|\log|F||.\] The coefficients of \(dd^c(\chi\omega)\) are uniformly bounded. No closedness of the Hermitian form \(\omega\) is needed for this local identity. Summing over the fixed number of balls covering a slab and using (31) proves (29), for example with any fixed \(A>N\log c\). Finitely many slabs outside the chosen tail are compact and are absorbed in \(C_{E,s}\).

A reduced component has no more area than the effective divisor. Since \(T\) is an isometry and raises height by one, replacing a component by \(T^kB\) shifts the slab index by \(k\). After absorbing the finitely many remaining slabs in the constant, its exponent is still \(A\). A fixed-width enlarged slab is contained in a bounded number of unit slabs whose indices differ from \(l\) by a bounded amount. This proves the final assertion as well. ◻

Proof. Finite-dimensionality is Proposition 10 with \(s=-3\). We first give the direct exactness argument needed for bounded-above support. If \(J\) is scalar and zero on the positive tail, it belongs to the single-linear-rate complex \((-b,b)\): its negative weight is unchanged and its positive part has compact support. That complex is acyclic. Its primitive has rates \((-b,b)\) and therefore also belongs to \((-\varepsilon,b)\), since \(b>\varepsilon\).

The full scalar degree-one vanishing can also be checked without a contour argument in higher degree. Keep \(b\) fixed and continuously vary the positive rate from \(-b\) to \(-\varepsilon\). After conjugation the even-to-odd Hodge operators have the same domain \(W^1\) and range \(L^2\), and vary continuously in operator norm: only bounded order-zero coefficients depending on the derivative of the weight change. They remain Fredholm by Proposition 10, since the positive rate stays negative. Their index is consequently constant, and is zero at the acyclic single-linear pair \((-b,b)\). At every point of this path, \(H^{2,0}=0\) by positive-tail uniqueness. By weighted duality \(H^{2,2}\) is dual to the scalar holomorphic functions with a positive rate at the positive end and the negative rate \(-b\) at the negative end. Every such global scalar function is constant: subtract its positive-end coefficient and use Proposition 15. The negative-end decay requirement excludes a nonzero constant. Thus \(H^{2,2}=0\) as well. Index zero now implies \(H^{2,1}=0\).

For the primitive assertion, on \(\{h>C\}\) the distributional section \(\beta\) solves the degree-zero equation for \(K_Y\otimes F\). Local elliptic regularity makes it smooth there. Uniform interior estimates on shorter tails turn its weighted \(W^{-3}\) bound into finite exponential growth in every positive Sobolev order. Proposition 15 therefore makes it zero on a positive tail. This holds for every primitive in the stated space.

To verify the last membership assertion, localize to the uniformly bounded coordinate charts on one enlarged slab. In real dimension four, \(W^4\) embeds continuously into \(C^0\), with a uniform constant on these charts. An order-zero current of mass \(M\) thus has \(W^{-4}\) norm at most \(CM\). A uniformly locally finite partition of unity and the bounded derivatives of the weight give an upper bound for the squared weighted norm by a constant multiple of \[\sum_{l\ge0} e^{-2bl}(C_0e^{\mu l})^2 =C_0^2\sum_{l\ge0}e^{-2(b-\mu)l}<\infty.\] There are only finitely many additional slabs between the zero level and the upper support bound, and none beyond it. This proves membership in \(W^{-4}_{-\varepsilon,b}\), which is degree one of (35). The constants of individual currents enter their norms, not the choice of spaces or the dimension of their common cohomology group. ◻

Proof. The mass criterion in Theorem 22 puts the currents in the indicated complex. Suppose a finite linear combination is exact: \[\sum_{i=1}^r a_i C_{\eta_i}=\bar\partial_F\beta.\] The sum is zero on a positive tail. The primitive assertion in Theorem 22 says that \(\beta\) is zero there as well. On \(Y\setminus\bigcup_iB_i\), its distributional Dolbeault derivative vanishes. Local elliptic regularity makes it a holomorphic section of \(K_Y\otimes F\) on that complement. The complement is connected: a path in the connected real four-manifold can be perturbed off the smooth real-codimension-two strata of a finite union of analytic curves, and off its locally discrete singular strata. Only finitely many charts are needed along the compact path. Analytic continuation from the positive tail therefore gives \[\mathop{\mathrm{supp}}\beta\subset\bigcup_{i=1}^rB_i.\]

Suppose \(a_i\ne0\). Choose a smooth point of \(B_i\), outside the other curves, where \(\eta_i\ne0\). Such a point exists since the singular and intersection sets are locally discrete and a nonzero holomorphic differential has isolated zeros. In holomorphic coordinates \((z,w)\) and a holomorphic frame of \(F\), the curve is \(w=0\), the differential is \(a(z)\,dz\) with \(a\not\equiv0\), and \(\beta=u\,dz\wedge dw\) for a distribution supported on \(w=0\). Its local finite normal-jet expansion has the form \[ u=\sum_{p+q\le N}u_{pq}(z)\, \partial_w^p\partial_{\bar w}^q\delta_0(w), \tag{36}\] with distributional tangential coefficients. This is the support theorem for distributions [19]; locally it also follows by taking a finite-order bound for \(u\), Taylor-expanding a test function in its normal variables, and cutting off in \(|w|<\delta\). A test function whose normal derivatives through order \(N\) vanish on \(w=0\) has \(C^N\) norm \(O(\delta)\) after that cutoff, and hence pairs to zero as \(\delta\to0\). Thus only the displayed finite Taylor jet can matter. The normal jets in (36) are independent, as is seen by prescribing those jets of test functions.

The coefficient of the normal exterior-form direction in \(\bar\partial\beta\) is \(\partial_{\bar w}u\); its tangential direction cannot contribute to that coefficient. The corresponding coefficient of the right-hand side is a nonzero constant multiple of \(a_i a(z)\delta_0(w)\). But normal differentiation in (36) increases \(q\) by one. Comparing the largest \(q+1\), then descending, forces every \(u_{pq}\) to vanish and leaves zero in place of this nonzero order-zero delta mass. This contradiction proves independence. Notice that the familiar principal-value meromorphic primitive of a residue is not supported on the divisor; the support conclusion above is essential.

For \(F=\mathcal O\), Theorem 22 makes every closed current of the kind under consideration exact: its support is bounded above, so a single linear weight \((-b,b)\) supplies the scalar primitive. The independence assertion then rules out even one nonzero current. ◻

Proof. Let \(\eta\) be a square-integrable holomorphic differential on \(R\). Cauchy–Schwarz in (34) gives \[\mathop{\mathrm{Mass}}(C_\eta;Y_l) \le c\|\eta\|_{L^2(R)} \left(\int_{f^{-1}(Y_l)}|df|^2dA_R\right)^{1/2} \le C\|\eta\|_{L^2(R)}e^{Al/2}.\] Here the energy of the holomorphic normalization is a fixed multiple of the reduced curve area; ramification and singular points are measure-zero sets. Lemma 21 gives the last bound. The scalar case of Proposition 23 forces \(\eta=0\).

We give the topological consequence also for open \(R\). A handle in an oriented surface contains two embedded circles with algebraic intersection one. Smooth closed forms \(\alpha,\beta\), supported in small annular neighborhoods of these circles and dual to them, have \[\int_R\alpha\wedge\beta=1.\] For example a bump function of the transverse coordinate, times its differential and with integral one, gives each such dual form. Choose any smooth conformal metric and let \(V\) be the closure of \(dC_c^\infty(R,\mathbb R)\) in the real Hilbert space of \(L^2\) one-forms. Set \(u=\alpha-\operatorname{proj}_V\alpha\). Forms in \(V\) are weakly closed, so \(du=0\); orthogonality to \(dC_c^\infty\) gives \(d^*u=0\). Integration by parts against the compactly supported closed form \(\beta\), followed by \(L^2\) continuity, gives \[\int_Ru\wedge\beta=\int_R\alpha\wedge\beta=1.\] Thus \(u\ne0\). Local elliptic regularity makes \(u\) smooth. With the complex orientation, \(u+i*u\) is a nonzero \((1,0)\)-form; the two equations for \(u\) make it closed and hence holomorphic. It is \(L^2\), a contradiction. No closed-range assertion for the de Rham differential on a noncompact surface has been used.

There is consequently no handle in any compact subsurface of \(R\). This is genus zero. Equivalently, every Jordan curve separates: a nonseparating Jordan curve can be joined from one side to the other in its complement, producing a second circle with intersection one and hence a handle in a compact neighborhood. A compact genus-zero Riemann surface is \(\mathbb P^1\). For noncompact \(R\), apply Koebe’s general planar uniformization theorem in the form [18]: every planar Riemann surface admits a conformal embedding into the Riemann sphere. The criterion that every Jordan curve separates is its planarity hypothesis. The theorem allows arbitrary connectivity, so no finite-type assumption on \(R\) is being added. Its image is open, and thus is the asserted domain. ◻

Proof. Choose \(\delta\) strictly larger than twice the maximum height oscillation of the balls \(U_p(4r)\), with a fixed positive margin; a uniform multiple of \(8Kr\|dh\|_\infty+1\) suffices after increasing the chart comparison constant.

Choose \(P\geq2\) large enough that \[ q:=\frac{2K^2}{e^P-1}\leq\frac12, \tag{40}\] and set \(Q=3P+4\). Cauchy’s estimate on source disks of radius one, inside \(C_j^{2P+2}\), gives a uniform bound \(D_0\) for \(u\) on \(\overline{C_j^{2P}}\) whenever \(S_j\) is flat. The norm of a coordinate map with image in \(U_p(r)\) is at most \(r\), so this bound depends only on \(K,r\), and the fixed disk radius. In particular it bounds \(u\) on \(S_j\).

It remains to prove a uniform positive energy cost. The issue is that the source domain may omit part of the thickened cylinder. We show that a sequence with energy tending to zero has a horizontal graph limit; proper projection near that limit then forces the missing source points to be present. If no \(\varepsilon_0\) works, there are maps \(f_\nu\), nonflat strips, and marked points as in the statement for which the integral in (39) tends to zero. Translate source indices to \(j=0\) and translate the targets and their height intervals by the same powers of \(T\) so that the marked images lie in a fixed compact set. After a subsequence, the marked source points tend to \(v_\infty\in S_0\) and their images tend to \(p\in Y\). Denote the corresponding translated intervals by \(I_\nu\); their lengths play no role in what follows. Work in the fixed coordinate ball \(U_p(4r)\). For large \(\nu\), the marked image is arbitrarily close to \(p\), so the strict margin in \(\delta\) makes every point of this ball have height in the corresponding \(I_\nu+[-\delta,\delta]\), even when the marked height lies at an endpoint of \(I_\nu\). The energy of the graph portions in \(C_0^Q\times U_p(4r)\) therefore tends to zero in the target direction.

These portions \[\Gamma_\nu=\{(v,f_\nu(v)):v\in\Omega_\nu\cap C_0^Q, \ f_\nu(v)\in U_p(4r)\}\] are closed analytic curves in that product. To check closedness, suppose \((v_k,f_\nu(v_k))\) converges in a compact subset of the product. Properness of \(f_\nu\) puts the \(v_k\) in a compact subset of \(\Omega_\nu\); their limit is consequently in its domain and belongs to the graph. On a half-cylinder obtained by cutting a proper full normalization, the same proof works on compact source disks away from its artificial finite boundary; properness is applied to the ambient map. This is the place where properness is essential.

The product Hermitian area of a graph is the sum of its horizontal and vertical areas. Its horizontal area is at most \(\mathop{\mathrm{Area}}(C_0^Q)\), because the source projection has at most one point in each fiber. Its vertical area tends to zero. Bishop’s compactness theorem [4], in the analytic-cycle and support formulation of [17], therefore gives a subsequence converging locally as positive analytic cycles, and with convergence of their supports, to a nonempty pure one-dimensional analytic cycle \(\Gamma_\infty\) through \((v_\infty,p)\). These theorems apply in local product coordinates (equivalently, identify the finite cylinder with an annulus). The locally bounded graph areas give their mass hypothesis. Convergence of positive currents makes the limit’s vertical area zero. On the regular part of each component this says that the target coordinate functions have zero derivative, so each component is horizontal. Analytic continuation in the connected cylinder then gives \[C_0^Q\times\{p\}\subset\mathop{\mathrm{supp}}\Gamma_\infty.\] For completeness, horizontal components cannot disappear at an interior source point: the defining holomorphic functions of the limit, restricted to the horizontal slice, vanish on an open set and hence on the whole connected slice.

We spell out why convergence to this slice fills a smaller cylinder. Fix a source disk \(V\Subset C_0^Q\). Choose a small target ball \(B(p,\eta)\Subset U_p(r)\) whose closure meets no other horizontal component of the limit. Such a ball exists by local finiteness of the analytic cycle. On a slightly larger source disk the approximating graphs eventually avoid the target boundary \(\partial B(p,\eta)\): otherwise a convergent sequence of graph points would give a limit point on that boundary. Thus the projection of the graph portion inside \(V\times B(p,\eta)\) is proper. It is finite, since its fibers have at most one point, and it has nonempty image by convergence to the horizontal slice. A nonconstant finite holomorphic map of analytic curves is open, and a proper map is closed; its image is therefore all of the connected disk \(V\). Its degree is exactly one, by the same single-point fiber bound. In particular every point of \(V\) belongs to \(\Omega_\nu\), and its image is in \(B(p,\eta)\). If there were a second horizontal component, the same argument in a disjoint target ball would give two graph points over one source point. Thus there is no second component.

Cover \(\overline{C_0^{2P+2}}\) by finitely many such disks. It follows that for all sufficiently large \(\nu\) this entire compact cylinder belongs to \(\Omega_\nu\) and its image is in \(U_p(r)\). Interior Cauchy estimates also give convergence of its coordinate derivatives to zero on smaller cylinders. In particular \(S_0\) is flat, contradicting its choice. This proves (39) and all the asserted uniformities. ◻

Proof. Take a flat strip \(S_j\), and express \(f\) in its coordinate chart as a vector-valued periodic holomorphic function \(F(v)\). Write \(H=\partial_vF\). Since \(F\) is single valued around the cylinder, \[\int_0^{2\pi}H(t+i\theta)\,d\theta =\frac1i\int_0^{2\pi}\partial_\theta F(t+i\theta)\,d\theta=0.\] Consequently the Laurent–Fourier expansion is \(H(v)=\sum_{n\ne0}c_ne^{nv}\). Bound the positive coefficients on the circle \(t=j+1+P\) and the negative coefficients on \(t=j-P\). For \(j\leq t\leq j+1\) their geometric sums give \[|H(t+i\theta)|\leq \frac{2}{e^P-1} \max\left\{\sup_{t=j-P}|H|,\sup_{t=j+1+P}|H|\right\}.\] The statement is valid for vector values by taking norms in the Fourier coefficient integrals. Comparing the coordinate and target norms yields \[ m_j\leq q\max_{|k-j|\leq P}m_k \tag{42}\] whenever all the indices on the right belong to the chain. Each application uses the chart for its own central strip; the numbers \(m_k\) are intrinsic, so changing charts causes no further loss.

All \(m_j\) in the chain are at most \(D_0\). Iterating (42) inside \(J\) therefore bounds \(m_j\) by \(D_0q^{\lfloor d(j)/P\rfloor}\), where \(d(j)\) is the integer distance to the nearer finite end of \(J\). On a half-infinite interval use its single finite end. At most \(2P\) indices have any prescribed value of \(\lfloor d(j)/P\rfloor\), so one may take \[L_0=\frac{2PD_0}{1-q}.\] If \(J=\mathbb Z\), the same iteration is possible arbitrarily often at every index and gives \(df=0\); analytic continuation contradicts nonconstancy.

Starting at a marked point of the chain, move in the \(t\) direction at its fixed angular coordinate toward either finite endpoint. The change of \(h\circ f\) is at most \(\|dh\|_\infty\sum_{j\in J}m_j\leq\Lambda\). The shared boundary with the neighboring nonflat strip is in the domain because the endpoint flat strip has a full thickening. That boundary point belongs to the neighboring closed strip and has height in \(I+[-\Lambda,\Lambda]\). This proves the charging assertion. Figure 1 illustrates the derivative decay along the chain and the charge to a neighboring nonflat strip. ◻

Proof. Take \(R_0=\Lambda+\delta\), with constants from the two preceding lemmas. Let \(\mathcal N_I\) be the set of nonflat strips containing a point of height in \(I+[-\Lambda,\Lambda]\). The source cylinders \(C_j^Q\) have overlap at most \(2Q+3\). Summing (39) over any finite subset of \(\mathcal N_I\) gives \[ \#\mathcal N_I\leq(2Q+3)\varepsilon_0^{-1}E_I. \tag{44}\] This also proves that the set being counted is finite. In particular, the set \(\mathcal N'_I\) of nonflat strips hitting \(I\) has this bound. Cauchy–Schwarz, first on each strip restricted to the target slab and then on the finite sum, gives \[\begin{split} &\sum_{j\in\mathcal N'_I} \int_{\Omega\cap S_j\cap(h\circ f)^{-1}(I)}u\,w\,dA_v\\ &\qquad\leq M_w(\#\mathcal N'_I)^{1/2} \left(\sum_{j\in\mathcal N'_I} \int_{\Omega\cap S_j\cap(h\circ f)^{-1}(I)}u^2dA_v\right)^{1/2} \leq C M_w E_I. \end{split}\] Only the intervals appearing explicitly in these integrals are used; no estimate of the energy outside the enlarged target interval is being assumed.

On one flat chain, (41) and \(\|w\|_{L^1(S_j)}\leq\sqrt{2\pi}M_w\) give \[\sum_{j\in J}\int_{S_j}u\,w\,dA_v \leq\sqrt{2\pi}M_wL_0.\] Every chain meeting \(I\) has a finite endpoint adjacent to a member of \(\mathcal N_I\), by Lemma 26. Each nonflat strip has at most two adjacent chains. Thus at most \(2\#\mathcal N_I\) chains contribute, and their integral over the target slab is bounded by their full chain integrals. Combining this bound with (44) proves (43) on the full cylinder.

For a half-cylinder, omit the fixed finite block of at most \(2Q+4\) strips on which the small-energy neighborhoods meet its initial boundary. On this block Cauchy–Schwarz gives \(CM_wE_I^{1/2}\leq CM_w(1+E_I)\). Apply the preceding argument to the remaining indices. There is at most one chain reaching the new finite endpoint, and its integral is at most \(\sqrt{2\pi}M_wL_0\). All other chains are charged as before. Increasing \(C_0\) proves the half-cylinder assertion. ◻

Proof. By Proposition 24, the normalization \(R\) is a domain in \(\mathbb P^1\). If it omits two distinct points, send them by a Möbius transformation to \(0\) and \(\infty\). Then \(R\subset\mathbb C^*\) is a domain in the logarithmic cylinder and \(\alpha=dz/z=dv\) is a globally defined nonzero holomorphic differential on \(R\). Formula (45) with \(w=1\) bounds the mass of \(f_*\alpha\) on every unit height interval by \(C_Be^{A\ell}\): pulling back a test one-form of norm at most one costs at most a fixed multiple of \(u\), while \(|dv|\) is constant. The scalar case of Proposition 23 forbids this nonzero differential. Thus \(\mathbb P^1\setminus R\) has at most one point. It has at least one point because \(R\) is noncompact. Removing that single point gives \(R\simeq\mathbb C\). ◻

Proof. Let \(a_0(t)=(2\pi)^{-1}\int_0^{2\pi}a(t,\theta)d\theta\). Its primitive \[g_0(t)=2\int_{-\infty}^t a_0(s)ds\] is bounded by \(\pi^{-1}\|a\|_{L^1}\), and \(g_0'=2a_0\in L^2(\mathbb R)\). This bounds its full \(W^{1,2}\) norm on each unit strip, including its mean. For \(a_\perp=a-a_0\), take Fourier series in \(\theta\) and Fourier transform in \(t\). For every integer \(n\ne0\) define \[\widehat{g_\perp}(\xi,n) =\frac{2\widehat{a_\perp}(\xi,n)}{i\xi-n}.\] Since \(|n|\geq1\), each of the multipliers \(1/(i\xi-n)\), \(\xi/(i\xi-n)\), and \(n/(i\xi-n)\) is bounded in absolute value by one. Plancherel therefore gives \(\|g_\perp\|_{W^{1,2}(\mathcal C)}\leq C\|a\|_{L^2}\). The sum \(g=g_0+g_\perp\) has the asserted periodicity, equation, and stripwise bound. Local elliptic regularity for \(\partial_t+i\partial_\theta\) gives the last assertion.

Let \(B\) be a uniform bound for these stripwise norms. The two-dimensional Trudinger inequality on a fixed bounded strip [31], including the \(L^2\) term in the norm, gives constants \(c,C>0\) independent of \(j\) such that \[\int_{S_j}\exp\left(\frac{c|\operatorname{Re}g|^2}{B^2}\right)dA_v \leq C\] when \(B>0\). One can cut the cylinder at a fixed angle and apply the bounded-rectangle form of that inequality; no boundary value of \(g\) is prescribed. The zero-norm case is immediate. For each finite \(p\), the inequality \(p|x|\leq cx^2/B^2+p^2B^2/(4c)\) proves (46). In particular both \(e^{\operatorname{Re}g}\) and \(e^{-\operatorname{Re}g}\) have bounded \(L^p\) norms on unit strips, for every fixed finite \(p\). ◻

Proof. It remains, by Proposition 24 and Corollary 28, to exclude a component with proper normalization \(f:\mathbb C\to B\subset Y\). Let \(M_B=\sup_Bh<\infty\) and \(M_D=\sup_Dh<\infty\). The curves \(B_k=T^kB\) are distinct, since their height suprema are \(M_B+k\). For every sufficiently large positive integer \(k\) this supremum exceeds \(M_D\), so \(B_k\) is not contained in \(D\). Write \(f_k=T^k\circ f\). We will construct on each of these curves the nonzero \(f_k^*E\)-valued holomorphic differential \[ \alpha_k=(f_k^*s)\,dz \tag{47}\] and prove that its pushforward has growth exponent \(A\).

The key point is the cancellation of the growth of \(dz\). Properness sends the exterior of the plane to the negative end of \(Y\). After a periodic change of frame there, the section coefficient becomes an ordinary holomorphic function tending to zero at infinity, hence of size \(O(1/|z|)\). Since \(dz=z\,dv\), this leaves only an exponential gauge factor, whose stripwise \(L^2\) bound is exactly the multiplier hypothesis in Proposition 27.

Fix one such \(k\) temporarily. Properness, together with the upper height bound, implies \[ h(f_k(z))\longrightarrow-\infty \quad\hbox{uniformly as }|z|\longrightarrow\infty. \tag{48}\] Indeed, otherwise a sequence with \(|z|\to\infty\) would have image in some compact height slab \(\{c\leq h\leq M_B+k\}\), contradicting properness. On an exterior disk \(|z|>R_k\) we may therefore use the prescribed smooth negative-end frame of \(E\). Choose its Hermitian norm to be one on that end. In this frame write \(\bar\partial_E=\bar\partial+\eta\), where \(|\eta|\leq C e^{-\gamma|h|}\); this is the choice (32). In the logarithmic coordinate \(v=\log z\), write \(f_k^*\eta=a(v)d\bar v\). Then \[ |a(v)|\leq C e^{-\gamma|h(f_k(v))|}\,u(v). \tag{49}\]

After increasing \(R_k\), the entire image of this exterior disk has negative height. Summing (49) over its target height intervals, using (45) with \(w=1\) for the first estimate and (38) for the second, gives \[\begin{align*} \int |a|\,dA_v &\leq C_k\sum_{\ell\geq0}e^{-(\gamma-A)\ell}<\infty,\\ \int |a|^2\,dA_v &\leq C_k\sum_{\ell\geq0}e^{-(2\gamma-A)\ell}<\infty. \end{align*}\] An initial compact annulus only adds a finite quantity. Smoothly cut \(a\) off toward the finite side and extend by zero to the full cylinder. This retains both bounds and agrees with \(a\) on a slightly smaller exterior disk. Lemma 29 gives a single-valued periodic smooth gauge \(g\) there, with \(\bar\partial g=a\,d\bar v\) and uniformly bounded stripwise \(W^{1,2}\) norms.

Let \(\sigma(v)\) be the coefficient of \(f_k^*s\) in the same smooth frame. Holomorphicity says \(\bar\partial\sigma+a\sigma\,d\bar v=0\). Hence \[H(v)=e^{g(v)}\sigma(v)\] is ordinary holomorphic and periodic on the exterior cylinder. By Proposition 20 and (48), \(\sup_{S_j}|\sigma|\to0\) as \(j\to\infty\). The exponential estimate (46) therefore yields \[\int_{S_j}|H|^2dA_v \leq\left(\sup_{S_j}|\sigma|^2\right) \int_{S_j}e^{2\operatorname{Re}g}dA_v \longrightarrow0.\] The mean-value inequality on fixed small disks in the cylinder, each contained in the union of three adjacent strips, implies \(H(v)\to0\) uniformly as \(t\to\infty\). Viewed as a holomorphic function of \(z\), it consequently extends holomorphically across \(z=\infty\) with value zero. Its Taylor expansion in \(1/z\) gives \[ |H(z)|\leq C_k|z|^{-1} \tag{50}\] on a further exterior disk.

Since \(dz=e^v\,dv\), Equations (47) and (50) give the coefficient estimate \[|\alpha_k|_v=|e^v\sigma(v)| \leq C_k e^{-\operatorname{Re}g(v)}.\] Here \(|\cdot|_v\) is the norm of its coefficient relative to \(dv\) and the unit smooth frame. The right-hand factor has uniformly bounded \(L^2\) norms on unit strips by (46). Apply (45) with \(w=e^{-\operatorname{Re}g}\). Pullback of a test one-form costs at most \(Cu\), so this proves \[ \mathop{\mathrm{Mass}}\left((f_k)_*\alpha_k\bigm|_{h^{-1}([-\ell-1,-\ell])}\right) \leq C_k'e^{A\ell}\qquad(\ell\geq0). \tag{51}\] The omitted compact disk in the source has compact image and contributes only on finitely many height intervals, which are absorbed in \(C_k'\). The section \(f_k^*s\) is nonzero because \(B_k\not\subset D\), and \(dz\) is nowhere zero; thus \(\alpha_k\) is a nonzero holomorphic bundle-valued differential.

Finally fix, once and for all, an admissible \(\varepsilon>0\) and a number \(b>\max\{A,\varepsilon\}\). The currents in (51) all belong to the same degree-one space \(W^{-4}_{-\varepsilon,b}\) of the shifted complex from Theorem 22; their coefficients take values in the same bundle \(E\). Their positive supports are bounded separately, and any finite collection has a common upper support bound. Proposition 23 therefore makes the classes of every finite collection linearly independent in the single space \(H^{2,1}_{-\varepsilon,b}(Y,E)\). That space is finite dimensional by Theorem 22. Infinitely many distinct \(B_k\) contradict this finite dimension. The constants \(C_k'\) may vary; no bound on their supremum has been used. The exponent \(A\), the weights, and the Sobolev orders were fixed throughout. This excludes the plane case and completes the proof. ◻

Proof of Corollary 2. Theorem 1 supplies a global spherical shell. Kato’s deformation theorem [20] then gives a one-parameter deformation with each nonzero fiber a modification of a primary Hopf surface; his Corollary 1 gives \(\pi_1(X)\cong\mathbb Z\). In complex dimension two, a modification between smooth surfaces factors as a finite succession of point blowups, with possibly infinitely near centers [3].

After shrinking \(\Delta\), the family is a proper smooth submersion. Ehresmann’s theorem identifies its fibers by diffeomorphisms preserving their continuously varying complex orientations. Every primary Hopf surface is diffeomorphic to \(S^1\times S^3\) [16], and hence has second Betti number zero. Each complex point blowup increases the second Betti number by one and replaces the oriented smooth manifold by its connected sum with \(\overline{\mathbb{CP}}^{\,2}\) [3]. Since the fibers have second Betti number \(b\), there are exactly \(b\) blowups, and the asserted oriented smooth type follows. The product orientation can be chosen compatibly because \(S^1\times S^3\) admits an orientation-reversing diffeomorphism. ◻

Proof of Corollary 3. Albanese–Di Cerbo–Lombardi [1] prove these conclusions except for the possible case of a class \(\mathrm{VII}_0^+\) surface violating the Global Spherical Shell conjecture. Theorem 1 excludes that case. Corollary 2 also explains the non-asphericity: such a surface has \(\pi_1\cong\mathbb Z\) and \(b_2>0\), but an aspherical manifold with fundamental group \(\mathbb Z\) is homotopy equivalent to \(S^1\). ◻