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.