Abundance conjecture

TL;DR

This paper sketches a proof of the abundance conjecture, establishing the equality of Kodaira and numerical Kodaira dimensions for compact complex algebraic manifolds using advanced techniques from complex geometry and number theory.

Contribution

It provides a new proof of the abundance conjecture by combining Simpson's method, semi-positivity, and Nevanlinna theory techniques, addressing key cases and assumptions.

Findings

Proof of the abundance conjecture under certain conditions.

Verification of the coincidence of foliations and fibrations for the canonical bundle.

Application of Gelfond-Schneider and Nevanlinna theory techniques.

Abstract

We sketch a proof of the abundance conjecture that the Kodaira dimension of a compact complex algebraic manifold equals its numerical Kodaira dimension. The proof consists of the following three parts: (i) the case of numerical Kodaira dimension zero, (ii) the general case under the assumption of the coincidence of the numerically trivial foliation and fibration for the canonical bundle, and (iii) the verification of the coincidence of the numerically trivial foliation and fibration for the canonical bundle. Besides the use of standard techniques such as the L2 estimates of d-bar, the first part uses Simpson's method of replacing the flat line bundle in a nontrivial flatly twisted canonical section by a torsion flat line bundle. Simpson's method relies on the technique of Gelfond-Schneider for the solution of the seventh problem of Hilbert. The second part uses the semi-positivity of the direct image of a relative pluricanonical bundle. The third part uses the technique of the First Main Theorem of Nevanlinna theory and its use is related to the technique of Gelfond-Schneider in the first part.

Equations

$${\kappa }_{\mathrm{kod}}(X)=\underset{m\to \infty }{\mathrm{limsup}}\frac{\log {\dim }_{\mathbb{C}}\Gamma \left(X,m{K}_X\right)}{\log m}$$

$${\kappa }_{\mathrm{num}}(X)=\underset{k\ge 1}{\sup }\left[\underset{m\to \infty }{\mathrm{limsup}}\frac{\log {\dim }_{\mathbb{C}}\Gamma \left(X,m{K}_X+kA\right)}{\log m}\right],$$

$$s_1^{(m,L)},\cdots ,s_{q_{m,L}}^{(m,L)}\in \Gamma \left(X,mL\right)$$

$${\Phi }_L=\sum _{m=1}^{\infty }{\epsilon }_{m,L}\sum _{j=1}^{q_{m,L}}{\left|s_j^{(m,L)}\right|}^{\frac{2}{m}},$$

$${\Theta }_L=\frac{\sqrt{-1}}{2\pi }\partial \bar{\partial }\log {\Phi }_L.$$

$$\frac{1}{{\Phi }_{m_{\nu }{K}_X+A}}$$

$$\frac{1}{m_{{\nu }_k}}{\Theta }_{m_{{\nu }_k}{K}_X+A}$$

$$\frac{1}{m_{\nu }}{\Theta }_{m_{\nu }{K}_X+A}$$

$$\frac{1}{m_{\nu }}\left(m_{\nu }{K}_X+A\right)\cdot A^{n-1}$$

$${\Xi }_{K_X}=\sum _{j=1}^J{\gamma }_j\left[V_j\right]+R,$$

$$\underset{m\in \mathbb{N}}{\sup }{\dim }_{\mathbb{C}}\Gamma \left(X,m{K}_X+kA\right)\le C_k<\infty \mathrm{for}\mathrm{all}k\in \mathbb{N}$$

$$\underset{m\in \mathbb{N}}{\sup }{\dim }_{\mathbb{C}}\Gamma \left(X,m{K}_X+\ell \left(n+2\right)A\right)=\infty$$

$${\Xi }_{K_X}=\sum _{j=1}^J{\gamma }_j\left[V_j\right]$$

$${\Xi }_{K_X}=\sum _{j=1}^J{\hat{\gamma }}_j{V}_j$$

$$V_k=\sum _{j=1}^{J_0}{\beta }_{k,j}{V}_j\mathrm{for}{J}_0+1\le k\le J$$

$${\Xi }_{K_X}=\sum _{j=1}^J{\gamma }_j{V}_j=\sum _{j=1}^{J_0}{\gamma }_j^{\ast }{V}_j+\sum _{k=J_0+1}^J{\gamma }_k\left(V_k-\sum _{j=1}^{J_0}{\beta }_{kj}{V}_j\right),$$

$${\gamma }_j^{\ast }={\gamma }_j+\sum _{k=J_0+1}^J{\gamma }_k{\beta }_{kj}.$$

$${\Xi }_{K_X}=\sum _{j=1}^{J_0}{\gamma }_j^{\ast }{V}_j.$$

$$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr 0.0pt{\hfil$\displaystyle\Xi_{K_{X}}=\sum_{j=1}^{J_{0}}\gamma^{*}_{j}V_{j}+\sum_{k=J_{0}+1}^{J}\gamma_{k}\left(V_{k}-\sum_{j=1}^{J_{0}}\beta_{kj}V_{j}\right)\cr 0.0pt{\hfil$\displaystyle=\sum_{j=1}^{J_{0}}\gamma^{*}_{j}V_{j}+\sum_{k=J_{0}+1}^{J}\left(\gamma_{k}-\gamma_{k}^{\prime}\right)\left(V_{k}-\sum_{j=1}^{J_{0}}\beta_{kj}V_{j}\right)+\sum_{k=J_{0}+1}^{J}\gamma_{k}^{\prime}\left(V_{k}-\sum_{j=1}^{J_{0}}\beta_{kj}V_{j}\right)\cr 0.0pt{\hfil$\displaystyle=\sum_{j=1}^{J}\hat{\gamma}_{j}V_{j}+\sum_{k=J_{0}+1}^{J}\left(\gamma_{k}-\gamma^{\prime}_{k}\right)\left(V_{k}-\sum_{j=1}^{J_{0}}\beta_{kj}V_{j}\right),\crcr}}}}$$

$${\hat{\gamma }}_j={\gamma }_j^{\ast }-\sum _{k=J_0+1}^J{\gamma }_k^{\prime }{\beta }_{kj}={\gamma }_j+\sum _{k=J_0+1}^J\left({\gamma }_k-{\gamma }_k^{\prime }\right){\beta }_{kj}$$

$${\Xi }_{K_X}=\sum _{j=1}^J{\hat{\gamma }}_j{V}_j$$

$$\prod _{j=1}^J{\left(s_{V_j}\right)}^{\hat{m}{\hat{\gamma }}_j}$$

$$\frac{\partial f_j}{\partial z_k}=R_{jk}\left(f_1,\cdots ,f_{d+1}\right)$$

$$\Vert \Xi \Vert =\frac{1}{(d-1)!}\Xi \wedge {\left(\frac{\sqrt{-1}}{2}\sum _{j=1}^{d}d{z}_j\wedge d\overline{z_j}\right)}^{d-1}.$$

$$\Theta =\frac{1}{s}\frac{\sqrt{-1}}{2\pi }\partial \bar{\partial }\log {\left|P\left(f_1,\cdots ,f_{n+1}\right)\right|}^2$$

$$\Vert \Theta \Vert \left(B_r\right)\le \frac{1}{{\mathrm{Vol}}_{d-1}}\left(\left(d+1\right)\rho +o(1)\right)\left[K:\mathbb{Q}\right]\mathrm{for}\mathrm{fixed}r.$$

$$\log |f(x)|=\frac{1}{2\pi }{\int }_{\partial G}\log |f(\xi )|\frac{\partial g(\xi ,x)}{\partial n}ds+\sum _{G}g\left(x,b_{\nu }\right)-\sum _{G}g\left(x,a_{\mu }\right),$$

$$\log \left|f(0)\right|=\frac{1}{2\pi }{\int }_0^{2\pi }\log \left|f\left(r{e}^{i\theta }\right)\right|d\theta -\sum _{|a_{\mu }|<r}\log \frac{r}{\left|a_{\mu }\right|}+\sum _{|b_{\nu }|<r}\log \frac{r}{\left|b_{\mu }\right|},$$

$$\sum _{|a_{\mu }|<r}\log \frac{r}{\left|a_{\mu }\right|}\ge \left(\log \frac{r}{r_0}\right)n\left(r_0\right)$$

$$\log \left|F(w_0)\right|\le \underset{w\in \partial B_R}{\max }\log \left|F(w)\right|-(1-\tau )\Vert T\Vert \left(B_{|w_0|}\right)\log \frac{\frac{\tau R}{6d}}{\left|w_0\right|},$$

Full text

ABUNDANCE CONJECTURE

Dedicated to Shing-Tung Yau

Yum-Tong Siu 111Partially supported by Grant 0500964 of the National Science Foundation. Written for the Festschrift of the sixtieth birthday of Shing-Tung Yau

§0. Introduction. For a compact complex algebraic manifold $X$ we define its Kodaira dimension

[TABLE]

and its numerical Kodaira dimension

[TABLE]

where $A$ is any ample line bundle on $X$. The definition of $\kappa_{\rm num}(X)$ is independent of the choice of the ample line bundle $A$ on $X$. We use the analytic viewpoint and formulate the abundance conjecture as $\kappa_{\rm kod}(X)=\kappa_{\rm num}(X)$. For terminology and formulations in the context of algebraic geometry see [Kawamata1985, p.569] and [Nakayama2004] and the remarks at the end of Chapter 7 of [Kollár-Mori1998]. In this note we present a sketch of the proof of the following theorem which affirms the abundance conjecture. Complete details are being written up and will be available later elsewhere.

(0.1) Theorem. For a compact complex algebraic manifold $X$ the Kodaira dimension $\kappa_{\rm kod}(X)$ of $X$ equals the numerical Kodaira dimension $\kappa_{\rm num}(X)$ of $X$.

In this sketch some parts are given in details while for some others only the key techniques and steps are explained. Even though in the parts where only a sketch is given, enough of the essential arguments is presented for the reader to understand how the proof works. The emphasis of our presentation is to highlight what the important arguments are and why they work and to minimize the amount of prerequisite background material.

(0.2) Three Parts of the Proof. The proof is divided into the following three parts. The first part is the special case of zero numerical Kodaira dimension. The second part is the general case of positive numerical Kodaira dimension under the additional assumption that the numerically trivial foliation for the canonical line bundle coincides with the numerically trivial fibration for the canonical line bundle. In other words, the assumption is that the leaves of the numerically trivial foliation for the canonical line bundle are knonwn to be compact. The third part verifies that the numerically trivial foliation for the canonical line bundle always coincides with the numerically trivial fibration for the canonical line bundle.

In all three parts the arguments work only for the canonical line bundle and do not work for general holomorphic line bundles for the reasons which can be very briefly described as follows. For the first part the Kodaira-Serre duality is used to identify the space of all flatly twisted canonical sections with the dual of a cohomology group with coefficients in a flat bundle. In the second part the variation of Hodge structure is used. In the third part the adjunction for foliation is used.

Besides the usual analytic techniques of $L^{2}$ estimates of $\bar{\partial}$ and the standard techniques of algebraic geometry, for the first part we use the technique developed first by Gelfond and Schneider [Gelfond1934, Schneider1934] for their solution of Hilbert’s seventh problem. The technique of Gelfond and Schneider comes in as an ingredient of the method of Simpson [Simpson1993] to replace the flat line bundle in a nontrivial flatly twisted canonical section by one which is a torsion flat line bundle. For the second part we use the technique of the semipositivity, and the condition for positivity, of the zeroth direct image of the relative pluricanonical line bundle. For the third part we use the technique of the First Main Theorem of Nevanlinna theory which is related to the technique of Gelfond and Schneider in the first part.

(0.3) Algebraic Properties of Solutions of Algebraically Defined Differential Equations. It is interesting to point out that the special techniques used for the first and second parts can be considered as originating from the same theme of algebraic properties of solutions of algebraically defined differential equations. Moreover, the technique of the third part is related to that of the first part. By an algebraically defined differential equation is meant a differential equation whose coefficients are algebraic functions (which may also be required to be defined over an algebraic number field). The technique of Gelfond-Schneider [Gelfond1934, Schneider1934] concerns the problem of algebraic values of solutions of algebraically defined differential equations.

The technique of semipositivity of the zeroth direct image of the relative canonical line bundle can be traced to the 1873 paper of Schwarz [Schwarz1873] where he studied the problem of Gauss about the condition for the existence of algebraic solutions for the Gauss hypergeometric differential equation [Gauss1812]. After the work of Newton on laws of motion and gravitation the question of algebraic solutions of singular differential equations such as the Gauss hypergeometric differential equation is significant because the motions of heavenly bodies satisfy second-order differential equations with singularity coming from mass and algebraic solutions are considered in order to study periodic solutions.

Schwarz [Schwarz1873] studied the problem by using Euler’s integral representation of the hypergeometric function by beta functions [Euler1778] and interpreting it as the period of an elliptic curve so that the Gauss hypergeometric differential equation is the differential equation from the variation of the period (which in modern terminology means the variation of Hodge structure) with the differentiation used in the differential equation defined by the Gauss-Manin connection. Actually in the situation considered by Schwarz it is the equivariant variation of Hodge structure with cyclic group action rather than the usual variation of Hodge structure.

The zeroth direct image of the relative canonical line bundle is a holomorphic subbundle of the flat bundle whose fibers are the Betti groups endowed with the natural metric defined by the integration of the exterior product of the representative differential form with its complex conjugate. The curvature of a holomorphic subbundle of a flat vector bundle is seminegative if the metric of the ambient flat vector bundle is positive definite, otherwise the curvature may be semipositive if the second fundamental form has value in the part of the ambient flat vector where the metric is negative definite.

In the 1973 paper of Schwarz he did not consider the curvature and the second fundamental form. Instead he considered the monodromy which he showed in his case to be in the biholomorphism group of an open $1$-disk (i.e., elements of $U(1,1)$), which means that in his case the second fundamental form has value in the part of the ambient flat vector bundle where the metric is negative. In Picard’s presentation [Picard1885] of the method of Schwarz and its generalization to the case of complex dimension two the rôle of the invariant circle [Picard1885, p.365, Equation(C)] and the rôle of $U(2,1)$ [Picard1885, p.378] are more clearly spelled out.

With the benefit of hindsight of more than a century later one can say that the seed is already in the 1973 paper of Schwarz for the statement that the curvature of the zeroth direct image of the relative canonical line bundle is semipositive and is given by the second fundamental form in the flat vector bundle whose fibers are the Betti groups. The papers of Deligne-Mostow [Deligne-Mostow1986] and Mostow [Mostow1987] gave a historic account of the developments from the work of Euler, Schwarz, Picard, Levavasseur [LeVavasseur1893] and their own work from the viewpoint of the monodromy of hypergeometric functions and nonlattice integral monodromy.

(0.4) Notations. For complex manifold $Y$ we denote by $T_{Y}$ its tangent bundle (of type (1,0)), denote by $T_{Y}^{*}$ the dual of $T_{Y}$, denote by $K_{Y}$ its canonical line bundle, and denote by $\Omega^{p}_{Y}$ the bundle of $(p,0)$-forms on $Y$. For a holomorphic family $\pi:Y\to S$ of complex manifolds over a complex manifold, we denote by $\Omega^{p}_{Y|S}$ the relative bundle of $(p,0)$-forms which is the vector bundle on $Y$ whose restriction to a fiber $Y_{s}$ is equal to the bundle $\Omega_{Y_{s}}^{p}$ of $(p,0)$-forms on the fiber $Y_{s}$ and we denote by $K_{Y|S}$ the relative canonical line bundle which is the line bundle on $Y$ whose restriction to a fiber $Y_{s}$ is equal to the canonical line bundle $K_{Y_{s}}$ of the fiber $Y_{s}$. The topological boundary of a set $E$ is denoted by $\partial E$ and its topological closure is denoted by $\bar{E}$. If there is no confusion, for a divisor $D$ in a complex manifold we also use the same notation $D$ to denote the holomorphic line bundle on $Y$ associated to the divisor $D$ and we use $s_{D}$ to denote the holomorphic section of the holomorphic line bundle $D$ whose divisor is the divisor $D$. The notations ${\mathbb{N}}$, ${\mathbb{Z}}$, ${\mathbb{Q}}$, ${\mathbb{R}}$, and ${\mathbb{C}}$ denote respectively the natural numbers, the integers, the rational numbers, the real numbers, and the complex numbers. The complex projective space of complex dimension $n$ is denoted by ${\mathbb{P}}_{n}$. The open disk in ${\mathbb{C}}$ of radius $r$ centered at the origin is denoted by $\Delta_{r}$. When $r=1$, $\Delta_{r}$ is also denoted by $\Delta$. We also use the notation $\Delta$ for the Laplacian where its meaning will be explicitly spelled out or clear from the context.

(0.5) Remark on the Application of Arithmetic Technique Involving Heights to Analytic Objects Not Defined Over the Field of Algebraic Numbers. An arithmetic technique involving heights which was introduced by Gelfond and Schneider for the solution of Hilbert’s seventh problem is used in Part I. The key point in the use of heights is a conclusion about a positive lower bound to guarantee the nonvanishing of a limit. This lower bound is obtained from the self-evident statement that the norm of an integer over a number field which is nonzero is bounded from below by $1$. In analysis it is difficult to obtain a positive lower bound to guarantee the nonvanishing of a limit, for which, for example, one of the available techniques is Harnack’s inequality. Here the arithmetic technique involving heights is assuming this rôle of producing a positive lower bound to guarantee the nonvanishing of a limit. Since the application of this arithmetic technique involving heights requires the analytic objects under consideration to be defined over the field $\overline{\mathbb{Q}}$ of all algebraic numbers, we would like to remark how the general case is handled which does not satisfy this condition of definition over $\overline{\mathbb{Q}}$.

For the situations which we consider we have a complex algebraic manifold $X$ defined over ${\mathbb{C}}$ by a finite number of homogeneous polynomials and we also have holomorphic vector bundles or other analytic objects ${\mathfrak{F}}$ which are defined algebraically over ${\mathbb{C}}$. We can choose a finite number of complex numbers $c_{1},\cdots,c_{k}$ which are algebraically independent over $\mathbb{Q}$ such that all these analytic objects ${\mathfrak{F}}$ under consideration are defined algebraically over a field $K$ which is a finite extension of the purely transcendental extension field ${\mathbb{Q}}\left(c_{1},\cdots,c_{k}\right)$ of the rational number field ${\mathbb{Q}}$. In this note our problem where the arithmetic technique involving heights is used deals with the existence or nonexistence of certain analytic objects. We choose generic elements $\gamma_{1},\cdots,\gamma_{k}$ in $\overline{\mathbb{Q}}$ and introduce another set of analytic objects $\tilde{X}$ and $\tilde{\mathfrak{F}}$ which are obtained by replacing $c_{\ell}$ by $\gamma_{\ell}$ in the coefficients of equations defining them. When our conclusions hold for the set of analytic objects $\tilde{X}$ and $\tilde{\mathfrak{F}}$ for a generic choice of elements $\gamma_{1},\cdots,\gamma_{k}$ in $\overline{\mathbb{Q}}$, the same conclusions hold for our original set of analytic objects $X$ and ${\mathfrak{F}}$. In the sketch of our proof, when we have to use the arithmetic technique involving heights this way of handling the general case of analytic objects which may not be defined over $\overline{\mathbb{Q}}$ is implicitly assumed without any further explicit mention.

Table of Contents

PART I. The Case of Zero Numerical Kodaira Dimension

• §1.

Curvature Current and Dichotomy

• §2.

Gelfond-Schneider’s Technique of Algebraic Values of Solutions of Algebraically Defined Differential Equations

• §3.

Final Step of the Case of Zero Numerical Kodaira Dimension

PART II. The Case of General Numerical Kodaira Dimension Under the Assumption of Coincidence of Numerical Trivial Foliation and Fibration

• §4.

Numerically Trivial Foliations and Fibrations for Canonical Line Bundle

• §5.

Curvature of Zeroth Direct Image of Relative Canonical and Pluricanonical Bundle

• §6.

Strict Positivity of Direct Image of Relative Pluricanonical Bundle Along Numerically Trivial Fibers in the Base of Numerically Trivial Fibration

PART III. Coincidence of Numerically Trivial Foliation and Fibration for Canonical Bundle

• §7.

Technique of Nevanlinna’s First Main Theorem for Proof of Compactness of Leaves of Foliation

PART I. The Case of Zero Numerical Kodaira Dimension.

§1. Curvature Current and Dichotomy

A general method of using the dichotomy in the decomposition of the curvature current to produce holomorphic sections of holomorphic line bundles was introduced in [Siu2006, §6] for the analytic proof of the finite generation of the canonical ring [Siu2006, Siu2008, Siu2009]. The advantage of using curvature currents is that in the context of curvature currents one can use the roots of line bundles and the limits of flat twistings of multiples of line bundles with the desired amount of effective control.

(1.1) Curvature Current of the Canonical Line Bundle. Let $L$ be a holomorphic line bundle over a compact complex algebraic manifold $X$ of complex dimension $n$ such that $\Gamma\left(X,m_{0}L\right)$ is nonzero for some positive integer $m_{0}$. For any positive integer $m$ let

[TABLE]

be a ${\mathbb{C}}$-basis of $\Gamma\left(X,mL\right)$. Let

[TABLE]

where $\varepsilon_{m,L}$ is a sequence of positive numbers going decreasingly to [math] fast enough to guarantee the convergence of the infinite series. Let

[TABLE]

The reciprocal $\frac{1}{\Phi_{L}}$ of $\Phi_{L}$ defines a (possibly singular) metric along the fibers of $L$ whose curvature current $\Theta_{L}$ is a closed positive $(1,1)$-current on $X$. The metric $\frac{1}{\Phi_{L}}$ as defined depends on the choice of $s_{1}^{(m,L)},\cdots,s_{q_{m,L}}^{(m,L)}$.

Suppose the numerical Kodaira dimension $\kappa_{\rm num}(X)$ of $X$ is $\geq 0$. We have $\dim_{\mathbb{C}}\Gamma\left(X,m_{\nu}K_{X}+A\right)\geq 1$ for some sequence of positive integers $m_{\nu}\to\infty$ and we can define the metric

[TABLE]

of $m_{\nu}K_{X}+A$ and its curvature current $\Theta_{m_{\nu}K_{X}+A}$ for $\nu\in{\mathbb{N}}$. We can now define a curvature current $\Xi_{K_{X}}$ of the canonical line bundle $K_{X}$ of $X$ as the weak limit, in the space of closed positive $(1,1)$-currents on $X$, of

[TABLE]

as $k\to\infty$ for some subsequence $m_{\nu_{k}}$ of the sequence $m_{\nu}$. It is possible to get such a weak limit because the total mass of

[TABLE]

with respect to a strictly positive smooth curvature form $\omega_{A}$ of the ample line bundle $A$ is equal to

[TABLE]

which is bounded uniformly in $\nu$. The curvature current $\Xi_{K_{X}}$ of the canonical line bundle $K_{X}$ of $X$ depends on many choices.

(1.2) Second Case of Dichotomy. The curvature current $\Xi_{K_{X}}$ defined in (1.1) admits the following decomposition as a closed positive $(1,1)$-current

[TABLE]

where $V_{j}$ is an irreducible nonsingular hypersurface of $X$ and $\gamma_{j}>0$ and the Lelong number of $R$ is [math] outside a countable union of subvarieties of codimension $\geq 2$ in $X$ [Siu1974]. We have the dichotomy into the following two cases.

• (i)

Either $J=\infty$ or $R\not=0$.

• (ii)

$J<\infty$ and $R=0$.

We claim that under the assumption

[TABLE]

we must have the second case of the dichotomy with $J<\infty$ and $R=0$. Choose a positive integer $\ell$ such that $\ell A$ is very ample. If we are in the first case of the dichotomy with either $J=\infty$ or $R\not=0$, by the argument given in [Siu2006, §6] (and also [Siu2009, Proposition (5.4)]) of using the curve which is the intersection of $n-1$ divisors of generic sections of $\ell A$, we conclude that

[TABLE]

which contradicts the assumption (1.2.1) and thus finishes the verification of the claim. We now discuss how the rationality of the coefficients $\gamma_{j}$ (in possibly some other representation of $\Xi_{K_{X}}$) follows from a rather trivial linear algebra argument of linear independence of cohomology classes.

(1.3) Rationality of Coefficients in Some Representation of Curvature Current as Element of the (1,1)-Cohomology Group. We consider $\Xi_{K_{X}}$ as an element of $H^{1,1}(X)$, which means that we use the same notation to denote the element of $H^{1,1}(X)$ defined by the closed positive $(1,1)$-current $\Xi_{K_{X}}$. Here $H^{1,1}(X)$ is the Hodge group which is isomorphic to the set of harmonic $(1,1)$-forms on $X$ with respect to some chosen Kähler metric of $X$. We can regard the equation

[TABLE]

also as an equation for elements of $H^{1,1}(X)$ with $J<\infty$. We claim that there is another representation

[TABLE]

as elements of $H^{1,1}(X)$ such that each $\hat{\gamma}_{j}$ is a positive rational number for $1\leq j\leq J$.

The claim is verified as follows by using a rather trivial linear algebra argument of linear independence of cohomology classes. By relabeling $V_{1},\cdots,V_{J}$ we can assume without loss of generality that $V_{1},\cdots,V_{J_{0}}$ are ${\mathbb{R}}$-linearly independent as elements of $H^{1,1}(X)$ and we can write

[TABLE]

as elements in $H^{1,1}(X)$ for some $\beta_{k,j}\in{\mathbb{Q}}$. In the space of closed positive $(1,1)$-currents we have

[TABLE]

where

[TABLE]

As elements of $H^{1,1}(X)$ we have

[TABLE]

Since the elements of $H^{1,1}(X)$ induced by $\Xi_{K_{X}}$ and $V_{1},\cdots,V_{J_{0}}$ all come from $H^{2}\left(X,{\mathbb{Q}}\right)$ and since the elements of $H^{1,1}(X)$ induced by $V_{1},\cdots,V_{J_{0}}$ are linearly independent, it follows that the numbers $\gamma_{k}^{*}$ are all rational for $J_{0}+1\leq k\leq J$. For any rational numbers $\gamma_{k}^{\prime}$ for $J_{0}+1\leq l\leq J$, in the space of closed positive $(1,1)$-currents we have

[TABLE]

where $\hat{\gamma}_{j}=\gamma^{*}_{j}-\sum_{k=J_{0}+1}^{J}\gamma^{\prime}_{k}\beta_{kj}$ for $1\leq j\leq J_{0}$ and $\hat{\gamma}_{j}=\gamma^{\prime}_{j}$ for $J_{0}+1\leq j\leq J$ are all rational. For $\gamma_{k}^{\prime}$ sufficiently close to $\gamma_{k}$ for $J_{0}+1\leq k\leq J$, the numbers

[TABLE]

for $1\leq j\leq J_{0}$ and $\hat{\gamma}_{j}=\gamma^{\prime}_{j}$ for $J_{0}+1\leq j\leq J$ are all positive. Thus as elements of $H^{1,1}(X)$ we have

[TABLE]

with $\hat{\gamma}_{j}$ all positive and rational for $1\leq j\leq J$. This finishes the verification of the claim.

(1.4) Nontrivial Section of Flatly Twisted Pluricanonical Line Bundle. Choose a sufficiently divisible positive integer $\hat{m}$ such that $\hat{m}\hat{\gamma}_{j}$ is a positive integer for $1\leq j\leq J$. Then $\hat{K}_{X}$ is equal to the line bundle $\sum_{j=1}^{J}\hat{m}\hat{\gamma}_{j}V_{j}$ plus a flat line bundle. Let $s_{V_{j}}$ be the canonical section of the line bundle $V_{j}$ on $X$. Then

[TABLE]

is a non identically zero holomorphic section of $\hat{m}K_{X}+F$ over $X$ for some flat line bundle $F$ over $X$.

§2. Gelfond-Schneider’s Technique of Algebraic Values of Solutions of Algebraically Defined Differential Equations

(2.1) Bombieri’s Higher Dimensional Formulation. Gelfond [Gelfond1934] and Schneider [Schneider1934] independently introduced the technique of algebraic values of solutions of algebraically defined differential equations to solve Hilbert’s seventh problem [Hilbert1900], which later was put into the setting of group-homomorphisms and generalized to the higher-dimensional case by Lang and Bombieri [Lang1962, Lang1965, Lang1966, Bombieri1970, Bombieri-Lang1970]. The technique is actually more transparent when formulated for the higher-dimensional case. We give here Bombieri’s higher-dimensional formulation and indicate why the arguments work, in order to prepare for its comparison with the use in Part III later of the related argument of using the First Main Theorem in Nevanlinna theory (see (7.6) below).

(2.1.1)Theorem [Bombieri1970]. Let $f_{1},\cdots,f_{d+1}$ be meromorphic functions on ${\mathbb{C}}^{d}$ of finite order $\leq\rho$ in the sense that each is the quotient of two entire functions of growth order $O\left(e^{|z|^{\rho+\varepsilon}}\right)$ for any $\varepsilon>0$ where $z=\left(z_{1},\cdots,z_{d}\right)$ are the coordinates of ${\mathbb{C}}^{d}$ and $\left|z\right|^{2}=\sum_{j=1}^{d}\left|z_{j}\right|^{2}$. Let $K$ be a finite extension field of ${\mathbb{Q}}$. Assume that $f_{1},\cdots,f_{d+1}$ do not satisfy any polynomial equation with coefficient in $K$ and that the following differential equation

[TABLE]

holds for $1\leq j\leq d+1$ and $1\leq k\leq d$, where $R_{jk}$ is a rational function in $d+1$ variables with coefficients in $K$. Let $S$ be the set of all $z\in{\mathbb{C}}^{n}$ such that $f_{j}\left(z\right)\in K$ for $1\leq j\leq d+1$. Then $S$ is contained in the zero-set of a polynomial of degree $\leq d(d+1)\rho[K:{\mathbb{Q}}]+2d$ in $z_{1},\cdots,z_{d}$.

(2.1.2) Sketch of the Main Arguments for its Proof. We give a sketch of its proof here for the purpose of highlighting some key points which will be used for comparison with the related argument in Part III of using the First Main Theorem of Nevanlinna theory to prove the coincidence of the numerically trivial foliation and fibration for the canonical line bundle (see (7.6) below). We will do this sketch only for the special case where $f_{1},\cdots,f_{d+1}$ are all entire, because the general case of meromorphic functions on ${\mathbb{C}}^{d}$ can be reduced to the case of entire functions on ${\mathbb{C}}^{d}$ by simply multiplying the $d+1$ meromorphic functions by an appropriate entire function on ${\mathbb{C}}^{d}$. Let $m\in{\mathbb{N}}$ and $S_{m}$ be a subset of $m$ distinct points of $S$. Choose a non-identically-zero $P\left(\xi_{1},\cdots,\xi_{n+1}\right)$ polynomial of degree $J$ in the indeterminates $\xi_{1},\cdots,\xi_{d+1}$ with coefficients in $K$ such that $P\left(f_{1},\cdots,f_{n+1}\right)$ vanishes to order at least $L$ at every point of $S_{m}$. The algebraic independence of $f_{1},\cdots,f_{d+1}$ over $K$ is to guarantee that $P\left(f_{1},\cdots,f_{d+1}\right)$ cannot be identically zero on ${\mathbb{C}}^{d}$. The polynomial $P\left(\xi_{1},\cdots,\xi_{n+1}\right)$ in the indeterminates $\xi_{1},\cdots,\xi_{n+1}$ is chosen by solving a system of linear equations with its coefficients $a_{j_{1},\cdots,j_{d+1}}$ in $K$ for $0\leq j_{\nu}\leq J$ as unknowns where the linear equations are given by the vanishing of $P\left(f_{1},\cdots,f_{d+1}\right)$ to order at least $J$ at every point of $S_{m}$.

We also want appropriate height estimates for the coefficients of the polynomial $P\left(\xi_{1},\cdots,\xi_{n+1}\right)$. The height of $\frac{p}{q}$ with $p,q\in{\mathbb{Z}}$ relatively prime is the maximum of $|p|$ and $|q|$ and its definition can be naturally generalized to elements of $K$. Instead of using the terminology of heights we follow Bombieri’s terminology in [Bombieri1970, p.278]. For $\alpha\in K$ let the denominator ${\rm den}(\alpha)$ be the smallest positive natural integer $d$ such that $d\alpha$ is an integer in $K$. Let $\left\|\alpha\right\|=\max_{\sigma}\left|\sigma\alpha\right|$ and ${\rm size}(\alpha)=\max_{\sigma}\left(\log{\rm den}(\alpha),\log\left|\sigma\alpha\right|\right)$, where $\sigma$ ranges over all embeddings of $K$ into ${\mathbb{C}}$. The solution of the linear equations to get $a_{j_{1},\cdots,j_{d+1}}$ in $K$ with estimates of $\left\|a_{j_{1},\cdots,j_{d+1}}\right\|$ and ${\rm den}\left(a_{j_{1},\cdots,j_{d+1}}\right)$ is obtained by Siegel’s lemma of the pigeon-hole principle [Bombieri1970, p.279, Lemma 2]. The differential equation $(2.1.1.1)$ is to guarantee that the coefficients of the linear equations are in $K$ and also to guarantee that we have the height estimate ${\rm size}\left(a_{j_{1},\cdots,j_{d+1}}\right)=O\left(L\right)$ when $J^{d+1}$ is the integral part $\left\lfloor mL^{d}\log L\right\rfloor$ of $mL^{d}\log L$.

For a closed positive $(1,1)$-current $\Xi$ on an open subset $\Omega$ of ${\mathbb{C}}^{d}$ let $\left\|\Xi\right\|$ be the mass measure of $\Xi$ which is the $(d,d)$-current defined by

[TABLE]

For a subset $E$ of $\Omega$ we denote by $\left\|\Xi\right\|(E)$ the value of the measure $\left\|\Xi\right\|$ at the set $E$. Let $s$ be the minimum vanishing order of $P\left(f_{1},\cdots,f_{n+1}\right)$ at points of $S_{m}$ which is achieved at some point $Q$ of $S_{m}$. The key point of Bombieri’s higher-dimensional version of the technique of Gelfond-Schneider is that, as $J\to\infty$ and $S_{m}$ exhausts $S$, the sequence of closed positive $(1,1)$-current

[TABLE]

satisfies the following uniform bound for its mass.

(2.1.2.1) The mass measure $\left\|\Theta\right\|$ of $\Theta$ satisfies

[TABLE]

for fixed $r>0$, where $B_{r}$ is the ball $B_{r}$ of ${\mathbb{C}}^{d}$ of radius $r$ centered at the origin and ${\rm Vol\,}_{d-1}$ is the the volume of the unit ball in ${\mathbb{C}}^{d-1}$.

This estimate is obtained from the higher-dimensional version of Nevanlinna’s technique of integration by parts twice. In Nevanlinna’s original paper [Nevanlinna1925], in order to derive his First Main Theorem [Nevanlinna1925, p.18, Erster Hauptsatz] he used the formula [Nevanlinna1925, p.5, $({\rm A})^{\prime\prime}$]

[TABLE]

where $f(x)$ is meromorphic on G=\left\{\,x\in{\mathbb{C}}\,\big{|}\,r_{1}<|x|<r_{2}\,\right\} with zeroes $\left\{a_{\mu}\right\}$ and poles $\left\{b_{\nu}\right\}$ and $g$ is the Green’s function of $G$. This formula specializes to the special case of the Poisson-Jensen formula [Nevanlinna1925, p.5, (B)] that

[TABLE]

where $f$ is a meromorphic function on ${\mathbb{C}}$ with zero-set $\left\{a_{\mu}\right\}$ and pole-set $\left\{b_{\nu}\right\}$ and has no zeroes and no poles at [math] and on \left\{\,x\in{\mathbb{C}}\,\big{|}\,\left|x\right|=r\,\right\}. Note that, when $f$ is entire, though the mass of the $(1,1)$-current $\frac{\sqrt{-1}}{2\pi}\partial\bar{\partial}\log\left|f\right|$ on the disk $\Delta_{r}$ of radius $r$ in ${\mathbb{C}}$ centered at [math] is the number $n(r)$ of points $a_{\mu}$ in $\Delta_{r}$, the expression $\sum_{\left|a_{\mu}\right|<r}\log\frac{r}{\left|a_{\mu}\right|}$ is used in the above formula instead, because Green’s function $g$ of $G$ involves the logarithmic function. What is used in the technique of Gelfond-Schneider is that

[TABLE]

for $0<r_{0}<r$ so that it is possible to extract from it the factor $\log\frac{r}{r_{0}}$ which is of the same order as $\log r$ as $r\to\infty$ when we focus on the the lower bound of the number $n\left(r_{0}\right)$ of points $a_{\mu}$ in $\Delta_{r_{0}}$ for fixed $r_{0}$. The following is the higher-dimensional version of (2.1.2.1) [Bombieri1970, p.273, Proposition 4] whose term for the mass measure of the closed positive $(1,1)$-current for the zero divisor contains also a logarithmic factor.

(2.1.2.3) If $F$ is a holomorphic function on some open neighborhood of the topological closure of the open ball $B_{R}$ in ${\mathbb{C}}^{d}$ of radius $R$ centered at [math], then for $0<\tau<1$ and $w_{0}\in B_{\frac{\tau R}{6d}}$

[TABLE]

where $T$ is the closed positive $(1,1)$-current $\frac{\sqrt{-1}}{2\pi}\partial\bar{\partial}\log\left|F\right|^{2}$ for the zero divisor of $F$ whose mass measure is denoted by $\left\|T\right\|$.

From (2.1.2.3) an upper bound for $\left\|T\right\|\left(B_{\left|w_{0}\right|}\right)$ needs to come from a lower bound of $\log\left|F(w_{0})\right|$ and an upper bound of $\max_{w\in\,\partial B_{R}}\log\left|F(w)\right|$. By applying (2.1.2.3) to $F=P\left(f_{1},\cdots,f_{d+1}\right)$ with $w_{0}$ averaged over $\partial B_{r}$ and with $R=s^{\kappa}$ for any $0<\kappa<\frac{1}{(d+1)\rho}$, we get

[TABLE]

for any fixed $r>0$ with $Q\in B_{\frac{r}{2}}$ and for any $0<\tau<1$ as $L\to\infty$ and $S_{m}$ exhausting $S$ for the following reason.

The factor $\kappa\log s$ on the left-hand side of (2.1.2.4) comes from $\log\frac{\frac{\,\tau R\,}{6d}}{\,\left|w_{0}\right|\,}$ with $R=s^{\kappa}$. The factor $s$ on the left-hand side of (2.1.2.4) comes from the fact that $\Theta$ is obtained by dividing by $s$ the closed positive $(1,1)$-current for the zero divisor of $F$. In the application of (2.1.2.3) the contribution from $\max_{w\in\,\partial B_{R}}\log\left|F(w)\right|$ is no more than $o\left(s\right)$, because the upper bound of $\log\left|F\right|$ on $B_{R}$ is no more than

[TABLE]

when $\varepsilon>0$ is chosen with $\kappa<\frac{1}{(d+1)\left(\rho+\varepsilon\right)}$.

By applying the Cauchy integral formula for some nonzero partial derivative $\left(D^{s}F\right)(Q)$ of $F$ of order $s$ at $Q$ in terms of the value of $F$ on $\partial B_{r}$, one concludes that the average of $\log\left|F(w_{0})\right|$ over $w\in\partial B_{r}$ is bounded from below by the sum of $-\left(1+o(1)\right)s\log s$ and $\log\left|\left(D^{s}F\right)(Q)\right|$, because the Cauchy integral formula for the $s$-th derivatives contains the factor $\frac{s!}{2\pi i}$ whose logarithm can be estimated by $s\log s\left(1+o(1)\right)$. The lower bound $-\left(\left[K:{\mathbb{Q}}\right]-1\right)\left(1+o(1)\right)s\log s$ for $\log\left|\left(D^{s}F\right)(Q)\right|$ comes from differentiating (2.1.1.1) $s-1$ times and from the use of the chain rule, because we have ${\rm size}\left(a_{j_{1},\cdots,j_{d+1}}\right)=O\left(L\right)$ and because there is the factor $\left[K:{\mathbb{Q}}\right]-1$ from the relation among ${\rm size}(\alpha)$, ${\rm den}(\alpha)$ and $\left|\sigma\alpha\right|$ for $\alpha\in K$ and an embedding $\sigma$ of $K$ into ${\mathbb{C}}$ [Bombieri1970, p.278].

Since the Lelong number of the weak limit of $\Theta$ is at least $1$ at every point of $S$, the conclusion of Bombieri’s higher-dimensional formulation now follows from the usual $L^{2}$ estimates of $\bar{\partial}$.

(2.1.2.5) Gelfond-Schneider’s Solution of Hilbert’s 7th Problem. The solution of Hilbert’s seventh problem by Gelfond and Schneider [Gelfond1934, Schneider1934] is a special case of (2.1.1) with $f_{1}(z)=e^{z}$ and $f_{2}(z)=e^{\beta z}$ and $K={\mathbb{Q}}\left[\alpha,\alpha^{\beta}\right]$ so that if $\alpha^{\beta}$ is not transcendental for some algebraic $\alpha,\beta$ with $\alpha\not=0,1$ and $\beta$ irrational, then $S$ contains ${\mathbb{Z}}\log\alpha$, contradicting (2.1.1).

(2.1.3) Formulation for Group-Homomorphism and Torsion Points. In the technique of Gelfond-Schneider the system of differential equations can be replaced by the differential of the group law when we have a holomorphic group-homomorphism from ${\mathbb{C}}^{n}$ to an algebraic group instead of $n+1$ meromorphic functions of finite order on ${\mathbb{C}}^{n}$ which are algebraically independent over the number field $K$. This kind of formulation was introduced by Lang [Lang1962, Lang1965, Lang1966]. We need the following statement from this kind of formulation.

(2.1.3.1) Let $\varphi:{\mathbb{C}}^{n}\to G$ be a holomorphic map into a group $G$ which has an algebraic group structure over a number field $K$. Assume that $\varphi$ is a group homomorphism. Let $\vec{e_{1}},\cdots,\vec{e_{n}}$ be a ${\mathbb{C}}$-linear basis of ${\mathbb{C}}^{n}$ such that $\varphi\left(2\pi\sqrt{-1}\,\vec{e_{j}}\right)$ is a $K$-point of $G$ for $1\leq j\leq n$. If the differential $d\varphi$ with respect to the ${\mathbb{C}}$-linear basis $\vec{e_{1}},\cdots,\vec{e_{n}}$ of ${\mathbb{C}}^{n}$ is algebraic and defined over $K$, then there is a finite subgroup $E$ of $G$ such that $\varphi\left(2\pi\sqrt{-1}\,\vec{e_{j}}\right)$ is in $E$ for $1\leq j\leq n$.

(2.1.3.2) Here is its proof. Without loss of generality we can assume that $\vec{e_{1}},\cdots,\vec{e_{n}}$ are the standard basis elements of ${\mathbb{C}}^{n}$. The group $G$ can be embedded into some ${\mathbb{P}}_{N}$ with the embedding defined over $K$. Let $f_{1},\cdots,f_{N}$ be the $N$ meromorphic functions on ${\mathbb{C}}^{n}$ which are the pullbacks via $\varphi$ of the inhomogeneous coordinates of ${\mathbb{P}}_{N}$. Let $f_{N+j}=e^{z_{j}}$ for $1\leq j\leq n$, where $z_{1},\cdots,z_{n}$ are the $n$ coordinate functions of ${\mathbb{C}}^{n}$. Let $\psi:{\mathbb{C}}^{n}\to G\times{\mathbb{C}}^{n}$ be defined by $\varphi$ and the identity map of ${\mathbb{C}}^{n}$. From the differential $d\psi$ and the group law of $G$ over $K$ and from the fact that the differential $d\varphi$ at the origin of ${\mathbb{C}}^{n}$ is defined over $K$ we get a system of differential equations on ${\mathbb{C}}^{n}$ whose unknowns are $N+n$ meromorphic functions $f_{1},\cdots,f_{N+n}$ and which expresses any first-order partial derivative of each of $f_{1},\cdots,f_{N+n}$ as a rational function of $f_{1},\cdots,f_{N+n}$ with coefficients in $K$. Note that since the group law of $G$ is defined over $K$, by the chain rule in order for the differential equations to be defined over $K$ it suffices to have $d\varphi$ defined over $K$ at one point of ${\mathbb{C}}^{n}$.

The $N$ meromorphic functions $f_{1},\cdots,f_{N}$ are of finite type, because $\varphi$ is a group homomorphism from which we can estimate the growth of meromorphic functions $f_{1},\cdots,f_{N}$ on the ball of radius $r$ in ${\mathbb{C}}^{n}$ as $r\to\infty$ by using the group law of $G$. Let $\Gamma$ be the discrete subgroup $\sum_{j=1}^{n}{\mathbb{Z}}\left(2\pi\sqrt{-1}\,\vec{e_{j}}\right)$ of ${\mathbb{C}}^{d}$. Since $\Gamma$ is not contained in any algebraic hypersurface of ${\mathbb{C}}^{n}$ of finite degree and since $\psi\left(\Gamma\right)$ is contained in the algebraic points of $G\times{\mathbb{C}}^{n}$, it follows from $(2.1.1)$ that the $N+n$ meromorphic functions $f_{1},\cdots,f_{N+n}$ form a field of transcendence degree $\leq n$ over $K$.

Let $\pi:{\mathbb{C}}^{n}\to{\mathbb{C}}^{n}\left/\Gamma\right.$ be the natural projection and let $\psi:{\mathbb{C}}^{n}\to G\times\left({\mathbb{C}}^{n}\left/\Gamma\right.\right)$ be defined by $\varphi$ and $\pi$. Since ${\mathbb{C}}^{n}\left/\Gamma\right.$ is biholomorphic to $\left({\mathbb{C}}-\left\{0\right\}\right)^{n}$ under the map $\left(z_{1},\cdots,z_{n}\right)\mapsto\left(e^{z_{1}},\cdots,e^{z_{n}}\right)$ and since the meromorphic functions $f_{1},\cdots,f_{N},e^{z_{1}},\cdots,e^{z_{n}}$ form a field of transcendence degree $\leq n$ over $K$, it follows that the image $\psi\left({\mathbb{C}}^{n}\right)$ of ${\mathbb{C}}^{n}$ in $G\times\left({\mathbb{C}}^{n}\left/\Gamma\right.\right)$ is a subvariety of complex dimension $n$ in $G\times\left({\mathbb{C}}^{n}\left/\Gamma\right.\right)$. Moreover, the restriction, to $\psi\left({\mathbb{C}}^{n}\right)$, of the projection $G\times\left({\mathbb{C}}^{n}\left/\Gamma\right.\right)\to{\mathbb{C}}^{n}\left/\Gamma\right.$ to its second factor makes $\psi\left({\mathbb{C}}^{n}\right)$ a finite-sheeted branch cover of ${\mathbb{C}}^{n}\left/\Gamma\right.$ whose fiber over the origin is projected by the natural projection $G\times{\mathbb{C}}^{n}\to G$ to a finite subset $E$ of $G$. This means that $\varphi\left(\Gamma\right)$ is contained in the finite subset $E$ of $G$ and the subgroup $\varphi\left(\Gamma\right)$ of $G$ is a finite subgroup of $G$. The proof is finished.

(2.1.3.3) Remark on the Normalizing Factor. The normalizing factor $2\pi\sqrt{-1}$ used in (2.1.3.1) is the same one from the Cauchy integral formula or from the curvature form of a holomorphic line bundle with a Hermitian metric along its fibers.

(2.1.4) Formulation for Group-Homomorphism and Coset of Subgroup. In our application of the technique of Gelfond-Schneider in the case of zero numerical Kodaira dimension the points and their images under a group-homomorphism are not known to be defined over $K$ as in the assumption of $(2.1.3)$, but only a coset and its image under a group-homomorphism is known to be defined over $K$ as subvarieties. A modification is needed. This modification was introduced by Simpson to enable him to draw conclusions about torsion translates of triple tori [Simpson1993, p.370, Corollary 3.5], which is used for the case of zero numerical Kodaira dimension. We will not go into Simpson’s setting to define what he means by torsion translates of triple tori. Instead we will just extract from his work the simple argument which we need for our situation. Here is the modification of (2.1.3.1) for the coset formulation.

(2.1.4.1) Let $\varphi:{\mathbb{C}}^{n}\to G$ be a holomorphic map into a group $G$ which has an algebraic group structure over a number field $K$ such that $\varphi$ is a group homomorphism. Let $N$ be a ${\mathbb{C}}$-linear subspace of ${\mathbb{C}}^{n}$ defined by a ${\mathbb{C}}$-basis over $K$ and let $v$ be a point of ${\mathbb{C}}^{n}$ such that the coset $N+v$ is defined by the vanishing of polynomials on ${\mathbb{C}}^{n}$ with coefficients in $K$. Assume that $\varphi\left(N+v\right)$ is a subvariety of $G$ defined over $K$. If the normalized differential $\frac{\sqrt{-1}}{2\pi}d\varphi$ at the origin of ${\mathbb{C}}^{n}$ is algebraic and defined over $K$, then there is an element $w$ of $N+v$ such that $\varphi(w)$ is a torsion point of $G$.

(2.1.4.2) The proof is as follows. In the application of (2.1.3.1) we replace $G$ by the quotient $G/\varphi(N)$ and replace ${\mathbb{C}}^{n}$ by the ${\mathbb{C}}$-line in ${\mathbb{C}}^{n}/N$ containing the image $\hat{v}$ of $v$ in ${\mathbb{C}}^{n}/N$ and replace $\vec{e_{1}}$ by $\hat{v}$. Then from the application of (2.1.3.1) with $n=1$ we conclude that $\hat{v}$ is a torsion element of $G/\varphi(N)$ and as a consequence there is an element $w$ of $N+v$ such that $\varphi(w)$ is a torsion point of $G$.

(2.2) Two Algebraic Descriptions of Flat Line Bundles. In order to apply (2.1.4.1) we need two different algebraic descriptions of flat line bundles on a compact complex manifold $X$ so that one which plays the rôle of the algebraic group $G$ is the integration of the other which plays the rôle of ${\mathbb{C}}^{n}$ and both are related by the holomorphic group-homomorphism $\varphi$. For the algebraic description which plays the the rôle of ${\mathbb{C}}^{n}$ we use the set of all holomorphic $1$-forms with their algebraic structure for the description. For the algebraic description which plays the the rôle of the algebraic $G$ we use transition functions with their algebraic structure for the description.

(2.2.1) Let $L$ be a flat line bundle over $X$ which is described by a finite open cover $\left\{U_{j}\right\}_{j\in J}$ of $X$ and transition functions $g_{jk}$ on $U_{j}\cap U_{k}$ such that each $g_{jk}$ is constant with $\left|g_{jk}\right|\equiv 1$ on $U_{j}\cap U_{k}$ and there are local holomorphic frames $F_{j}$ of $L$ on $U_{j}$ (i.e., each $F_{j}$ is a holomorphic section of $L$ on $U_{j}$ generating ${\mathcal{O}}(L)$ on $U_{j}$) with $F_{j}=g_{jk}F_{k}$ on $U_{j}\cap U_{k}$. For any section $s=\left\{s_{j}\right\}$ of $L$ on an open subset $G$ of $X$ with $s_{j}$ being a function on $G\cap U_{j}$ we can use $\left\|s\right\|^{2}=\left|s_{j}\right|^{2}$ which is independent of $j$ as a Hermitian metric along the fibers of $L$ whose curvature form is identically zero on $X$.

For any holomorphic $1$-form $\omega$ on $X$ and for any complex number $\lambda\in{\mathbb{C}}$, we define a new complex structure $L_{\lambda,\omega}$ on the smooth line bundle $L$ as follows. A local smooth section $s$ of $L_{\lambda}$ is defined to be holomorphic if and only if $\bar{\partial}s=\lambda s\,\bar{\omega}$. In other words, the new differentiation in the $(0,1)$-direction for local smooth sections of $L_{\lambda}$ is $\bar{\partial}_{\lambda}=\bar{\partial}-\lambda\,\bar{\omega}$.

(2.2.2) We can determine the transition functions for $L_{\lambda,\omega}$ as follows. By replacing the finite open cover $\left\{U_{j}\right\}_{j\in J}$ of $X$ by a refinement if necessary, we can assume without loss of generality that there exists a holomorphic function $f_{j}$ on $U_{j}$ such that $df_{j}=\omega$ on $U_{j}$ and $f_{j}-f_{k}=c_{kj}$ for some constant $c_{jk}$ on $U_{j}\cap U_{k}$. Let $s_{j,\lambda}=e^{\lambda\overline{f_{j}}}F_{j}$ on $U_{j}$. Then $\bar{\partial}s_{j,\lambda}=\lambda\,s_{j,\lambda}\bar{\omega}$ on $U_{j}$. The collection

[TABLE]

can be used as local holomorphic frames for the holomorphic line bundle $L_{\lambda}$ on $X$. The transition functions $\left\{g_{jk,\lambda,\omega}\right\}_{j,k\in J}$ for $L_{\lambda,\omega}$ with respect to the cover $\left\{U_{j}\right\}_{j\in J}$ are given by

[TABLE]

on $U_{j}\cap U_{k}$ which is a constant. One conclusion we can immediately get from this explicit expression for the transition function is that the line bundle $L_{\lambda,\omega}$ is holomorphic in the variable $\lambda\in{\mathbb{C}}$.

(2.2.3) For reasons explained in Remark(0.4) we can assume without loss of generality that $X$ as well as all relevant analytic objects over $X$ which come into consideration in our discussion is assumed to be defined over $\overline{\mathbb{Q}}$. We can describe the two algebraic structures for the set of flat line bundles over $X$.

The first description is to give the flat line bundle $L$ with constant transition functions $g_{jk}$ (not necessarily of absolute-value $1$) the algebraic structure for the complex variables $g_{jk}$. This is the same as giving the set of flat line bundles the complex structure of ${\rm Hom}\left(\pi_{1}(X),{\mathbb{C}}^{*}\right)={\rm Hom}\left(H_{1}\left(X,{\mathbb{Z}}\right),{\mathbb{C}}^{*}\right)$ defined by the generators of $\pi_{1}(X)$ (or $H_{1}\left(X,{\mathbb{Z}}\right)$) and their relations. We call this description the algebraic structure from transition functions.

The second description is to give $L_{\lambda,\omega}$ the algebraic structure for the holomorphic line bundle $\omega$ and for the complex variable $\lambda$ when the line bundle $L$ which we start out with is the trivial line bundle. We call this description the algebraic structure from holomorphic $1$-forms.

As we have mentioned in (2.2) the algebraic structure from transition functions is obtained from the algebraic structure from holomorphic $1$-forms by using integration or the process of finding local primitives of $1$-forms.

(2.3) Coset Defined by Lower Bound of Flatly Twisted Hodge Number. In order to apply the formulation of the technique of Gelfond-Schneider for group-homomorphisms and cosets of groups as given in (2.1.4.1), we would like to produce a coset in the second description of algebraic structures for the space of flat bundles. For this purpose we have the following statement which is a special case of the general result of Simpson on torsion translates of triple tori [Simpson1993].

(2.3.1) Let $X$ be a compact complex algebraic manifold of complex dimension $n$ and $L$ is a flat line bundle over $X$. Let $m$ be the complex dimension of the space $\Gamma\left(X,T_{X}^{*}\right)$ of all holomorphic $1$-forms on $X$. Let $\omega$ be a holomorphic $1$-form on $X$. Then for any nonzero complex number $\lambda\in{\mathbb{C}}^{*}$ the dimension of $\Gamma\left(X,L_{\lambda,\omega}+K_{X}\right)$ is independent of $\lambda\in{\mathbb{C}}^{*}$ (but may depend on $\omega$). As a consequence, for any $k\in{\mathbb{N}}$ the set of all flat line bundles $F$ (as points of ${\mathbb{C}}^{m}$ and given the algebraic structure from holomorphic $1$-forms) which satisfy $\dim_{\mathbb{C}}\Gamma\left(X,F+K_{X}\right)\geq k$ is of the form $N+v$ for some ${\mathbb{C}}$-linear subspace $N$ of ${\mathbb{C}}^{m}$ and some $v\in{\mathbb{C}}^{m}$.

(2.3.2) For the following reason the above statement is a consequence of the Hodge decomposition with flat twisting. Let $H^{n}_{\rm top}\left(X,F\right)$ be the topologically defined $n$-th cohomology group of $X$ with coefficients in the local system given by the flat line bundle $F$. By standard arguments of topology and homological algebra the complex dimension of the topologically defined $H^{n}_{\rm top}\left(X,L_{\lambda,\omega}\right)$ is independent of $\lambda\in{\mathbb{C}}^{*}$ for any prescribed holomorphic $1$-form $\omega$ on $X$, because the transition function $g_{jk,\lambda,\omega}$ of $L_{\lambda,\omega}$ on $U_{j}\cap U_{k}$ is given by $e^{-\lambda\overline{c_{jk}}}g_{jk}$, where $g_{jk}$ is the transition function for $L$ on $U_{j}\cap U_{k}$ and $c_{kj}=f_{j}-f_{k}$ with $df_{j}=\omega$ on $U_{j}$. Note that these arguments of topology and homological algebra work only for $\lambda\in{\mathbb{C}}^{*}$ and not for $\lambda\in{\mathbb{C}}$ when $\lambda\to 0$.

By Hodge theory with flat twisting $L_{\lambda,\omega}$, we know that $\Gamma\left(X,L_{\lambda,\omega}+K_{X}\right)$ is a summand in a direct sum decomposition of $H^{n}_{\rm top}\left(X,L_{\lambda,\omega}\right)$ where the complex dimension of each summand is an upper semi-continuous function of $\lambda\in{\mathbb{C}}^{*}$. Hence the dimension of $\Gamma\left(X,L_{\lambda,\omega}+K_{X}\right)$ is independent of $\lambda\in{\mathbb{C}}^{*}$ for any prescribed holomorphic $1$-form $\omega$ on $X$. By linear algebra arguments, for any $k\in{\mathbb{N}}$ the set of all flat line bundles $F$ (as points of ${\mathbb{C}}^{m}$ and given the algebraic structure from holomorphic $1$-forms) which satisfy $\dim_{\mathbb{C}}\Gamma\left(X,F+K_{X}\right)\geq k$ is of the form $N+v$ for some ${\mathbb{C}}$-linear subspace $N$ of ${\mathbb{C}}^{m}$ and some $v\in{\mathbb{C}}^{m}$.

(2.3.3) Remark on Dimension Jump of Cohomology for Flat Line Bundles Parametrized by Nonzero Complex Numbers at Zero Value of Parameter. Note that in (2.3.1) there is in general a jump of $\dim_{\mathbb{C}}\Gamma\left(X,L_{\lambda,\omega}+K_{X}\right)$ when $\lambda\in{\mathbb{C}}^{*}$ approaches [math], otherwise $\dim_{\mathbb{C}}\Gamma\left(X,F+K_{X}\right)$ would be independent of the flat line bundle $F$, which is not true. This phenomenon is related to the fact that the natural metric of line bundle ${\mathcal{L}}_{\omega}$ over $X\times{\mathbb{C}}$ whose restriction to $X\times\left\{\lambda\right\}$ is $L_{\lambda,\omega}$ does not have a semipositive curvature form. The reason is as follows. There is a natural Hermitian metric $e^{-\varphi_{\lambda,\omega}}$ along the fibers of $L_{\lambda,\omega}$ which on $U_{j}$ is given by $e^{-\varphi_{j,\lambda,\omega}}=e^{-2{\rm Re}\left(\lambda\overline{f_{j}}\right)}$ so that

[TABLE]

on $U_{j}\cap U_{k}$.

A natural Hermitian metric $e^{-\varphi_{{\mathcal{L}}_{\omega}}}$ for the line bundle ${\mathcal{L}}_{\omega}$ on $X\times{\mathbb{C}}$ is given by

[TABLE]

on $U_{j}\times{\mathbb{C}}$. Its curvature current in general may not be semipositive, because that the function $2{\rm Re}\left(\lambda\overline{f_{j}}\right)$, though pluriharmonic in the variables along $X$ so that the curvature of $e^{-\varphi_{\lambda}}$ is zero on $X$ for a fixed $\lambda$, may not be pluriharmonic or plurisubharmonic jointly in both the variable $\lambda$ and the variables along $X$, because the function $\overline{f_{j}}$ is locally conjugate-holomorphic while the function $\lambda$ is holomorphic.

If the curvature form of ${\mathcal{L}}_{\omega}$ were semipositive, by the extension theorem of Ohsawa-Takegoshi [Ohsawa-Takegoshi1987] (or the pluricanonical case in [Siu1998, Siu2002, Paun2007, Varolin2008]), $\dim_{\mathbb{C}}\Gamma\left(X,L_{\lambda,\omega}+K_{X}\right)$ (and even $\dim_{\mathbb{C}}\Gamma\left(X,L_{\lambda,\omega}+qK_{X}\right)$ for any fixed $q\in{\mathbb{N}}$) would have been independent of $\lambda\in{\mathbb{C}}$, which is not true when $\lambda\to 0$. If we could replace $\lambda$ by $\overline{\lambda}$ or $\frac{1}{\ \overline{\lambda}\ }$ (which holds only when $|\lambda|=1$) or some other holomorphic function in $\overline{\lambda}$, the curvature form of ${\mathcal{L}}_{\omega}$ would be semipositive, which of course is impossible because we need to make the family of holomorphic line bundles $L_{\lambda,\omega}$ holomorphic in the variable $\lambda$.

(2.4) Simpson’s Result on Torsion Flat Bundle for Flatly Twisted Canonical Section. Let $X$ be a compact complex algebraic manifold of complex dimension $n$ and $G$ be the moduli space of all flat line bundles on $X$ given the algebraic structure from the transition functions. For $\gamma\in G$ let $F_{\gamma}$ be the flat line bundle on $X$ associated to $\gamma$. For $k\in{\mathbb{N}}$ let $V_{k}$ be the set of all $\gamma\in G$ such that $\dim_{\mathbb{C}}\Gamma\left(X,F_{\gamma}+K_{X}\right)\geq k$. Then $V_{k}$ is an algebraic subvariety of $G$ from the general theory in algebraic geometry concerning the behavior of the dimension of the cohomology group of a varying fiber.

By letting $L^{(0)}$ be the trivial line bundle on $X$ and setting $\lambda=1$, to a holomorphic $1$-form $\omega$ we can assign the flat line bundle $L^{(0)}_{1,\omega}$ to $\omega$. Let $m$ be the complex dimension of the space of all holomorphic $1$-forms on $X$ so that the space of all holomorphic $1$-forms on $X$ can be identified with points of ${\mathbb{C}}^{m}$. We now consider the map $\varphi:{\mathbb{C}}^{m}\to G$ which assigns to each holomorphic $1$-form $\omega$ on $X$ the element $\gamma$ of $G$ such that the flat line bundle $L^{(0)}_{1,\omega}$ is the same as $F_{\gamma}$. Now the following special case of a result of Simpson [Simpson1993] is a consequence of (2.3.1) and (2.1.4.1).

(2.4.1) Let $X$ be a compact complex algebraic manifold and $k\in{\mathbb{N}}$. If there exists a flat line bundle $F$ over $X$ such that $\dim_{\mathbb{C}}\Gamma\left(X,F+K_{X}\right)\geq k$, then there exists a flat line bundle $E$ on $X$ such that $pE$ is equal to the trivial line bundle on $X$ for some positive integer $p$ and $\dim_{\mathbb{C}}\Gamma\left(X,E+K_{X}\right)\geq k$.

(2.4.2) Brieskorn’s Use of Algebraic Values of Solutions of Algebraically Defined Differential Equations. We would like to remark that as mentioned by Simpson in [Simpson1993, p.362], Brieskron in [Brieskorn1070, p.114, (4), (5), (6) in Proof of Satz 4] already used Gelfond-Schneider’s method of algebraic values of solutions of algebraically defined differential equations to draw conclusions on the monodromy of isolated singularities of hypersurfaces in arguments similar to Simpson’s. What is new in Simpson’s technique is the use of the quotient of abelian variety with respect to an algebraically defined abelian subvariety in order to get an algebraic point in the quotient [Simpson1993 p.370, Proof of Theorem 3.3] as explained in (2.1.4.1) and (2.1.4.2).

§3. Final Step of the Case of Zero Numerical Kodaira Dimension

(3.1) Use of Branched Cover to Construct Pluricanonical Section From Flatly Twisted Pluricanonical Section. For the case of a compact complex algebraic manifold $X$ with zero numerical Kodaira dimension, we have already got to the point of the existence of a nontrivial flatly twisted pluricanonical section on $X$. What is left is to get rid of the flat twisting which we will do by lifting the flatly twisted pluricanonical section to a flatly twisted canonical section on the manifold $\hat{X}$ constructed from a branched cover over $X$ and then pushing forward to get a nontrivial pluricanonical section on $X$ after we invoke (2.4.1) to get a nontrivial pluricanonical section on $\hat{X}$.

(3.1.1) Construction of Manifold from Branched Cover and Lifting of Flatly Twisted Pluricanonical Section to Flatly Twisted Canonical Section. Suppose $L$ is a flat line bundle over a compact complex algebraic manifold $X$ of zero numerical Kodaira dimension and suppose $m$ is a positive integer $\geq 2$. Assume we have a nonzero element $s$ of $\Gamma\left(X,L+mK_{X}\right)$. We would like to construct from it a nonzero element of $\Gamma\left(X,m^{\prime}K_{X}\right)$ for some positive integer $m^{\prime}$. Let $N$ be the order of the torsion subgroup of $H^{2}\left(X,{\mathbb{Z}}\right)$ which is the same as the order of the torsion subgroup of $H_{1}\left(X,{\mathbb{Z}}\right)$ by the universal coefficient theorem. The flat line bundle $NL$ belongs to the identity component of the real-analytic group of all flat line bundles over $X$. As a result, we can write $NL$ as $mNL^{\prime}$ for some flat line bundle $L^{\prime}$ over $X$. By replacing $s$ by $s^{N}$ and $L$ by $NL=mNL^{\prime}$ and then replacing $mN$ by $m$, we can assume without loss of generality that we have a nonzero element $s$ of $\Gamma\left(X,L+mK_{X}\right)$ such that $L=mL^{\prime}$ for some flat line bundle $L^{\prime}$ over $X$.

Let $\tilde{X}$ be the graph of the multi-valued section $s^{\frac{1}{m}}$ of $L^{\prime}+K_{X}$ over $X$ and let $\pi:\tilde{X}\to X$ be the natural projection. Then from $s^{\frac{1}{m}}$ we get a holomorphic section $s^{\prime}$ of $\pi^{*}\left(L^{\prime}+K_{X}\right)$ over $\tilde{X}$. At regular point of the divisor of $s$ defined by $\zeta=0$ locally, an $m$-th root $w$ of $\zeta$ is given by $\zeta=w^{m}$ which yields $d\zeta=mw^{m-1}dw$ so that the Jacobian determinant of $\pi:\tilde{X}\to X$ is equal to $m-1$ times the divisor of $s^{\prime}$ at a regular point of the divisor of $s$. Let $\hat{X}$ be the desingularization of (the normalization of) $\tilde{X}$ with projection $\hat{\pi}:\hat{X}\to X$ which is the composite of the desingularization projection $\hat{X}\to\tilde{X}$ and $\pi:\tilde{X}\to X$. Then the canonical line bundle $K_{\hat{X}}$ is equal to $\hat{\pi}^{*}\left(K_{X}+(m-1)\left(K_{X}+L^{\prime}\right)\right)+D$ for some nonnegative divisor $D$ on $\hat{X}$ whose $\hat{\pi}$ image is contained in the zero-set of $s$. Let $s_{D}$ be the canonical section of the line bundle over $\hat{X}$ associated to $D$. Since $\hat{\pi}^{*}s$ is an element of $\Gamma\left(\hat{X},\hat{\pi}^{*}\left(mK_{X}+mL^{\prime}\right)\right)$, it follows that $s_{D}\,\hat{\pi}^{*}s$ is a nonzero element of

[TABLE]

(3.1.2) Removal of Flat Twisting and Pushing Forward to Get Pluricanonical Section. The complex dimension of $\Gamma\left(\hat{X},K_{\hat{X}}+\hat{\pi}^{*}L^{\prime}\right)$ is $\geq 1$, because it contains the nonzero element $s_{D}\,\hat{\pi}^{*}s$. By (2.4.1) there is a flat line bundle $L^{\#}$ on $\hat{X}$ such that the complex dimension of $\Gamma\left(\hat{X},K_{\hat{X}}+L^{\#}\right)$ is $\geq 1$ and $\hat{m}L^{\#}$ is the trivial line bundle on $\hat{X}$ for some positive integer $\hat{m}$. Hence there exists some nonzero element $\hat{s}$ in $\Gamma\left(\hat{X},\hat{m}K_{\hat{X}}\right)$ for some positive integer $\hat{m}$.

By taking the product of the values of $\hat{s}$ on the $m$ sheets of $\hat{X}$ over $X$ outside the divisor of $s$, we get a non identically zero meromorphic section $\sigma$ of $m\hat{m}K_{X}$ over $X$ whose pole-set is contained in the divisor of $s$. We claim that $\sigma$ must be holomorphic. Assume the contrary and we are going to derive a contradiction. Since the pole-set of $\sigma$ is contained in the divisor of $s$, it follows that for some sufficiently large $N$ the product $\sigma s^{N}$ is a holomorphic section of the line bundle $m\left(\hat{m}+N\right)K_{X}+NL$ over $X$ whose divisor is different from that of $s^{\hat{m}+N}$. Since $L$ is flat, we can choose its transition functions to be constant of absolute value $1$ with respect to some local holomorphic frames for some finite cover of $X$