Proof. As before, let \( [ R , \infty ) \times \ensuremath { \mathbb { T } } ^ { 3 } = \{ ( r , \theta ) | r \geq R , \theta = ( \theta _ { 1 } , \theta _ { 2 } , \theta _ { 3 } ) \in \ensuremath { \mathbb { R } } ^ { 3 } / \Lambda \} \). Let \( \Lambda ^ { * } \) be the dual lattice. Choose \( \lambda \in \Lambda ^ { * } - \{ 0 \} \) with minimal length. Choose \( ( a _ { 1 } , a _ { 2 } , a _ { 3 } ) \in \mathbb { S } ^ { 2 } \) such that

\[ ( a _ { 1 } I ^ { * } + a _ { 2 } J ^ { * } + a _ { 3 } K ^ { * } ) \mathrm { d } r = - \frac { \lambda _ { 1 } \mathrm { d } \theta _ { 1 } + \lambda _ { 2 } \mathrm { d } \theta _ { 2 } + \lambda _ { 3 } \mathrm { d } \theta _ { 3 } } { | \lambda | } . \]

Use \( a _ { 1 } I \! + \! a _ { 2 } J \! + \! a _ { 3 } K \) as the complex structure. Then the function \( e ^ { 2 \pi | \lambda | r } e ^ { 2 \pi i < \lambda , \theta > } \) is holomorphic. The growth rate of this function is exactly \( O ( e ^ { 2 \pi | \lambda | r } ) \).

Now we pull back this function from \( [ R , \infty ) \times \mathbb { T } ^ { 3 } \) to M, cut it off and fill in with 0 inside K, we obtain a function f satisfying

\[ \bar { \partial } _ { g } f = \phi = { \cal O } ( e ^ { ( 2 \pi | \lambda | - \mu ) r } ) , \]

where \( \mu \) is the constant in Theorem 4.17. So \( \phi \in \underline { { L } } _ { - 2 \delta } ^ { 2 } \) for any non-critical positive number \( \delta \in ( 2 \pi | \lambda | - \mu , 2 \pi | \lambda | ) \). By Theorem 4.20, there exists \( \psi \in \underline { { H ^ { 2 } } } _ { 2 \delta } \) satisfying

\[ \phi = \Delta \psi = - ( { \bar { \partial } } ^ { * } { \bar { \partial } } + { \bar { \partial } } { \bar { \partial } } ^ { * } ) \psi \]

in the distribution sense. Elliptic regularity implies that \( \psi \) is a smooth (0, 1)-form. As before, \( \bar { \partial } \psi = \xi \omega ^ { + } \) is a harmonic (0,2)-form. So \( \xi \) is a harmonic function of order \( O ( e ^ { \delta r } ) \).

Now we use a cut-off function and the diffeomorphism to average g and the pull back of h. We get a smooth metric \( g ^ { \prime } \) on M which is identically the pull back of h outside a very big ball. Now let \( \nu \) be the inferior of positive \( \nu ^ { \prime } \) such that \( \xi \) is bounded by \( O ( e ^ { \nu ^ { \prime } r } ) \). If \( \nu > \) 0, then \( \Delta _ { g ^ { \prime } } \xi \, \in \, \underline { { L } } _ { - 2 \delta } ^ { 2 } \) 0 for any positive \( \nu > \delta ^ { \prime } > \nu - \mu \). It follows that there exists a function in \( \underline { { L } } _ { - 2 \delta ^ { \prime } } ^ { 2 } \) whose Laplacian \( \Delta _ { g ^ { \prime } } \) is \( \Delta _ { g ^ { \prime } } \xi \). The difference of those two functions is a \( g ^ { \prime } . \)-harmonic function. By Lemma 4.21, it must have at most linear growth rate since the growth rate is below the first nonlinear harmonic function. It follows that \( \xi \) must lie in \( O ( e ^ { \delta ^ { \prime } r } ) \), a contradiction. So \( \nu \) = 0. Therefore, \( \xi \) is bounded by any exponential growth function.

So \( \Delta _ { g ^ { \prime } } \xi \) decay exponentially. In particular, it’s in \( L _ { 1 - \epsilon } ^ { 2 } \). By Theorem 4.12, we can find out a function in \( H _ { - 3 - \epsilon } ^ { 2 } \) whose \( \Delta _ { g ^ { \prime } } \) is \( \Delta _ { g ^ { \prime } } \xi \). Therefore, we know that \( \xi \) is actually in \( O ( r ^ { 1 + \epsilon } ) \). Of course, \( \bar { \partial } \psi = \xi \omega ^ { + } \) has the same estimate.

By Lemma 4.4, the harmonic (0,1)-form \( \bar { \partial } ^ { \ast } \bar { \partial } \psi = \bar { \partial } ( f + \bar { \partial } ^ { \ast } \psi ) \) is in \( O ( r ^ { \epsilon } ) \)) and its covariant derivative is in \( O ( r ^ { - 1 + \epsilon } ) \). The Weitzenb¨ock formula implies that \( \nabla ^ { * } \nabla ( \bar { \partial } ^ { * } \bar { \partial } \psi ) = 0 \). Therefore

\[ \int _ { M } | \nabla ( \bar { \partial } ^ { * } \bar { \partial } \psi ) | ^ { 2 } \chi \leq \int _ { M } | \bar { \partial } ^ { * } \bar { \partial } \psi | | \nabla ( \bar { \partial } ^ { * } \bar { \partial } \psi ) | | \nabla \chi | \]

for any smooth compactly supported \( \chi \). Let \( \chi = \chi ( r \! - \! R ) \), the right hand side converges to 0. Therefore \( \partial ^ { * } \bar { \partial } \psi \) is a covariant constant (0,1)-form. If this form is non-zero, it would be invariant under the holonomy of any loop. However, elements in \( \mathrm { S U } \)\( 2 \)) have no fixed point except the identity matrix. So \( ( M , g ) \) must have trivial holonomy. Therefore, it’s \( \mathbb { R } ^ { 4 - k } \times \mathbb { T } ^ { k } \) with flat metric. It’s a

contradiction with our non-splitting assumption. So actually \( \bar { \partial } ^ { \ast } \bar { \partial } \psi \) is identically 0. \( f + \bar { \partial } ^ { * } \psi \) is a global holomorphic function on M.

# 4.7 Compactification of ALG and ALH-non-splitting in-stantons

In Theorem 4.14 and 4.22, we proved the existence of global holomorphic func-tion u in ALG and ALH-non-splitting cases such that any far enough fiber is biholomorphic to a complex torus. Notice that du is never zero on far enough fiber. Define a holomorphic vector field X by \( \omega ^ { + } ( Y , X ) = \mathrm { d } u ( Y ) \). Then since \( X ( u ) \, = \, \mathrm { d } u ( X ) \, = \, \omega ^ { + } ( X , X ) \, = \, 0 . \)\( _ { \rightmoon } , \) is well defined when it’s restricted to each far enough fiber. On each fixed far enough fiber, there exists a unique holomorphic form \( \phi \) such that \( \phi ( X ) \) = 1. Locally

\[ \omega ^ { + } = f ( u , v ) \mathrm { d } u \wedge \mathrm { d } v , X = f ^ { - 1 } ( u , v ) \frac \partial { \partial v } , \phi = f ( u , v ) \mathrm { d } v . \]

Notice that each far enough fiber is topologically a torus. So actually, we can integrate the from \( \phi \) to get a holomorphic function \( v \in \mathbb { C } / \mathbb { Z } \tau _ { 1 } ( u ) \oplus \mathbb { Z } \tau _ { 2 } ( u ) \) up to a constant. We can fix this constant locally by choosing a holomorphic section of u as the base point. Therefore M is biholomorphic to \( U \times \mathbb { C } / ( u , v ) \sim \)\( ( u , v + m \tau _ { 1 } ( u ) + n \tau _ { 2 } ( u ) ) \), where \( \tau _ { 1 } ( u ) \) and \( \tau _ { 2 } ( u ) \) are locally defined holomorphic functions. Actually, they are the integral of \( \phi \) in the basis of \( H _ { 1 } \) of each fiber. This gives a holomorphic torus fiberation locally.

Recall that there is a diffeomorphism from M minus a large compact set to the standard fiberation. Denote the inverse image of the zero section by s. s is again a section outside large compact set because du differ with the standard one by a decaying error. Write \( \bar { \partial } s \) as \( e ( u ) \mathrm { d } \bar { u } \otimes X \), then e is a function defined on the inverse of the punctured disc with polynomial growth rate. So there is an at most polynomial growth function E on the inverse of punctured disc such that \( \bar { \partial } E ( u ) = e ( u ) \mathrm { d } \bar { u } \). Now we apply the flow −E(u)X to the section s to get a holomorphic section \( s _ { 0 } \) on the neighborhood of infinity. View \( s _ { 0 } \) as the zero section, we know that M minus a large compact set is biholomorphic to \( ( \mathbb { C } - B _ { R } ) \times \mathbb { C } / ( u , v ) \sim ( u , v + m \tau _ { 1 } ( u ) + n \tau _ { 2 } ( u ) \)) globally, where \( \tau _ { 1 } ( u ) \) and \( \tau _ { 2 } ( u ) \) are multi-valued holomorphic functions.

As proved in Kodaira’s paper [1], there exists an (unique) elliptic fiberation B over \( B _ { R ^ { - 1 } } \) with a section such that B minus the central fiber D is biholo-morphic to \( ( B _ { R ^ { - 1 } } - \{ 0 \} ) \times \mathbb { C } / ( \tilde { u } , v ) \sim ( \tilde { u } , v + m \tau _ { 1 } ( \tilde { u } ^ { - 1 } ) + n \tau _ { 2 } ( \tilde { u } ^ { - 1 } ) ) \)). We can naturally identity points and get a compactification \( \bar { M } \) of M. So \( \bar { M } \) is a compact complex surface with a meromorphic function \( u = \tilde { u } ^ { - 1 } \). Now since the subvari-ety of critical points \( \{ \mathrm { d } u = 0 \} \) is a finite union of irreducible curves (On those irreducible curves, u is of course constant) and points, we know that except finite critical values in \( \mathbb { C P } ^ { 1 } \), any fiber of u has no intersection with \( \{ \mathrm { d } u = 0 \} \). Therefore, a generic fiber has genus 1 and must be an elliptic curve. In other words, \( \bar { M } \) is a compact elliptic surface. In conclusion, we’ve proved the second main theorem.