数理解析研究所講究録1182 巻 2001 年 73-79

73$\mathrm{F}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{h}\mathrm{e}1_{\mathrm{S}}^{1}$

problem and some examples of algebraic varieties
名古屋商科大学商学部

上野亮-(Ryoichi Ueno)

This talk is concerned with a topic in the theory of brached coverings, called
problem in Namba [9] : the problem on the existence of finite Galois coverings of complex
manifolds with given branch divisors. This problem originates inconjecture oncompact Riemann surfaces, proved by Bundgaard-Nielsen [2], Fox [3] early in. Ishall give some higher dimensional analogues of it, where compact Riemann surfaces are
replaced by the complex projective plane, products of compact Riemann surfaces, or complex
tori. I shall then mention some examples of projective manifolds with ample canonical bundles
obtained as an application of our results.
$\dagger|\mathrm{F}\mathrm{e}\mathrm{n}\mathrm{C}\mathrm{e}\mathrm{l}\dagger \mathrm{S}$$\mathrm{F}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{h}\mathrm{e}1_{\mathrm{S}}|$$1950^{\mathfrak{l}}\mathrm{s}$1 First we recall some basic definitions and facts about finite Galois coverings in the categoryof complex-analytic spaces. Let $M$ be a connected complex manifold. A 遺漏 e $Ga]_{ois}$ cove 吻 gof $M$ is a finite, s 両 ective, proper holomorphic mapping $f:Xarrow M$ from an irreducible, normal,
complex space $X$ onto 1 砿 such that the coveIitJg
gtoup of $f$ acts transitively oneach fibre of . Here, the covering transforation group means the group of biholomorphic
mappmgs : $Xarrow X$ such that
. When $f:Xarrow M$ is afinite Galois covering, we call it
frequently the $Ga]_{ois}$ gtoup of . Afinite Galois covering is called an 3be 肱 (resp. cyc 面。,$solv\theta b]e)$
covering if its Galois group is abelian (resp cyclic, solvable).
Let $D$ be an lrreducible hypersurface of a connected complex manifold $M$ and $f:Xarrow M$
afinite Galois covering. The ta l 茄 c 訪 oIl 加 dex of along 1) is defined as follows. Take anonsingular point of $D$. Then every point
is nonsingular point of both $X$ and $f^{-1}(D)$ .
Moreover, if $W$ is a sufficiently small corected open neighbourhood ofwith a coordinate
systemsuch thatand $D\cap W=\{(w_{\mathrm{f}}, \ldots, w_{\iota},)\in W|w_{\nu\iota}=0\}$, thenthere is a coordinate systemin the connected component
with $q\in U$
and a positive integer such thatand$tt\mathit{8}ns\tau or\mathrm{m}\mathrm{a}tion$$f$$f\mathrm{o}\mu=f$

$\mu$

$f$

$f$

$q\in \mathrm{f}^{arrow\iota}(p)$

$p$

$p$

$p=(\mathrm{O}, \ldots, \mathrm{o})$$(w_{1}, \ldots, w_{r\mathrm{t}})$$U\subset\tau^{-l}(w)$$(Z_{\{}, \ldots,Z_{\iota},)$$e$$q=(\mathrm{o},\ldots,\mathrm{o})$$f:(z_{\mathrm{t}}, \ldots,z_{\mathrm{n}})\in U\vdasharrow(_{W_{l}}, \ldots, w_{\mathrm{t}},)=(z,$$,$$\ldots,$.

$zh^{-}|’ z_{n}^{e_{)\in W}}$the.We
be distinct irreducible hypersurfaces on . For a positive divisor$D,$$\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{i}\mathrm{w}\mathrm{e}.\mathrm{n}_{\mathrm{f}\mathrm{i}_{\mathrm{C}}}\mathrm{t}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{i}\mathrm{h}\mathrm{a}\mathrm{t}_{G\mathrm{i}}\mathrm{n}\mathrm{d}\mathrm{s}\mathrm{e}\mathrm{x}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{t}\mathrm{f}$$\mathrm{r}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{d}f\mathrm{a}\mathrm{l}\mathrm{o}\mathrm{n}\mathrm{g}\mathrm{o}D\mathrm{n}\mathrm{l}.\mathrm{y}\mathrm{b}\mathrm{I}\mathrm{f}$$\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}e=\mathrm{l},f\mathrm{p}$$\mathrm{i}\mathrm{S}\mathrm{S}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{n}_{\mathrm{i}\mathrm{d}\mathrm{t}\mathrm{o}}\mathrm{t}1\mathrm{y}\mathrm{o}\mathrm{f}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{i}\mathrm{c}_{\mathrm{i}}\mathrm{b}\mathrm{e}\mathrm{u}\mathrm{n}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{f}\mathrm{i}\mathrm{e}\mathrm{f}\mathrm{d}$$\mathrm{a}1_{0}\mathrm{n}\mathrm{g}D\mathrm{a}\mathrm{n}\mathrm{d}U.$Now let $D,$

$D_{\mathrm{n}}$$,$$\ldots,$

call

$e$

$M$

$\mathrm{k}$to$D=_{CD\prime}bf\mathit{3}Rch_{\mathit{8}t}D,\mathrm{i}|+\cdots \mathrm{f}^{+e_{n^{D_{n^{\mathrm{W}}}\mathrm{i}}}}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{C}\mathrm{h}j(\uparrow \mathrm{h}Cj21^{\frac{\geq}{\leqq}\underline{4}\mathrm{C}}j1\leqq j_{\frac{\leq}{\mathrm{h}\mathrm{e}}),\cdotarrow M ,\mathrm{t}\mathrm{a}}k\mathrm{a}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\iota_{\mathrm{o}\mathrm{n}\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{f}f1}\mathrm{e}\mathrm{G}\mathrm{a}1_{0}\mathrm{i}\mathrm{S}\mathrm{C}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}k),\mathrm{r}\mathrm{m}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{e}\mathrm{x}\mathrm{o}\mathrm{a}\mathrm{o}\mathrm{n}\mathrm{g}D\mathrm{i}\mathrm{S}ej\mathrm{a}\mathrm{n}\mathrm{d}f\mathrm{i}\mathrm{n}\mathrm{g}f.x_{i}\mathrm{i}_{\mathrm{S}\mathrm{s}}\mathrm{a}\mathrm{i}\mathrm{d}$unramified along any irreducible hypersurface other than 1}, ...,.
Take areference pointin. We denote byalassoround: the homotopy class of aclosed path inwhich starts from ,moves to apoint near the nonsingular locus of $Di$ , tums once round 勿血 the positive direction,and return to(Figure1).
$D_{n}$$*$$M\backslash (D_{l}\cup\cdots\cup D_{n})$$\mathit{7}i(1\leqq j\leqq k)$$M\backslash (D_{l}\cup\cdots\cup D_{n})$$D_{j}$$*$$\overline{l^{\neg}}\mathrm{I}^{\backslash }.\#^{\iota_{\mathrm{t}}\mathrm{r}_{\mathrm{C}}}L$

$*$

74
The following is fundamental.
There is an oIle-fo-otle correspondence between the set of finte
Theorem (cf. Namba [9]).

and the set of normal subgroups
Galois $cove\dot{p}\mathit{1}gsf:xarrow M$ branching at$*)$ with ffiite indices such that the orderof the fundamental groupin the quotient group $\pi(M\backslash (D_{1^{\cup\cdots\cup D}k}), *)/N$ is $e_{i}(l\leqq j=^{k})<$ .
of the image of$D=e_{(}D_{f}+\cdots+e_{\mathrm{A}}D_{h}$$N$$\pi_{\mathrm{f}}(M\backslash (D_{k},^{U}\cdots UD),$$r_{i}$The quotient group $\pi_{l}(M\backslash (D_{i}\cup\cdots\cup D_{k}), *)/N$ is nothing but the Galois group of .

$f$problem. Let $M$ be a compact Riemann surface
We now start a discussion on$M$distinct points on . For integers $e_{i}\geqq 2(1\leqq j\leqq k)$ , we consider theof genus andconjecture, proved by Bundgaard-Nielsen [2],.
positive divisor
Fox [3], asserts that $tBere$ is a fmite Galois covering $f:Xarrow Mbra\Pi ChiI\mathit{1}g$ at $Dif$ an only if oneof the following conditions is satisfied.

2.

$\mathrm{F}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{h}\mathrm{e}1_{\mathrm{S}}^{\dagger}$$g$$p_{l},\ldots,$$p_{\mathrm{A}}$$\mathrm{F}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{h}\mathrm{e}1^{\dagger}\mathrm{S}$$D=e_{lp_{\mathrm{f}}}+\cdots+e_{\mathrm{A}}p_{\mathrm{A}}$$d$$\cdot$,(i)(ii) $g=0$ and(iii) $g=0,$$g\underline{\geq}\mathit{1}$(The case where$g\geqq 1$,$k_{=}>_{\mathit{3}}$$k=\mathit{2}$.
is due to Bundgaar-Nielsen; The case where $g=0$ is due to Fox.)
and

$e,$

$=e_{L}$As a generalization to higher dimension, we consider the following problem raised in the
problem: Given a connected complex manifold $M$ andbook of Namba [9], called
on $M$, we consider the positive divisor $D=e’ D,$
distin irreducible hypersurfaces
for the existence of a finite. Then, give a condition on
Galois covermg $f:Xarrow Mb_{Ia}I\mathrm{J}chiIJg$ at $D$ :In group-theoretic terms, as we explained in section 1, the existence of such a covering issuch that each orderequivalent to that of a finite quotient group of.): lassos round. ( : a reference point,of the image ofis$\pi_{(}(M\backslash \{pl ,..., p_{k}\}, *)$ is isomorphic
When $M$ is a compact Riemann surface of genus
to
, if
,
i)
, if $g=0$ ,ii)$[ , ]$ being the commutator.
Bunggaard-Nielsen, or Fox proved the assertion by finding out adroitly finite quotients of
is of higher dimension,$M\backslash (D_{|h}M\backslash \{p"\ldots.’.p\mathfrak{b}^{\}}D^{*}’)^{)\mathrm{v}}*$$\mathrm{F}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{h}\mathrm{e}1^{\dagger}\mathrm{s}$$ct$$D_{f_{f}}\ldots,$$+$$D_{h}$$...+e_{k}D_{\mathrm{A}}(e_{i}\geqq \mathit{2}, l_{=^{j_{=}}}<<_{k})$$e,,\ldots,e_{k}$$\pi_{\{}(M\backslash (D_{\int}\cup\cdots\cup D_{h}), *)$$\gamma_{i}$$*$$e\dotplus\geqq 2(1\leqq j\leqq k)$$D_{\dot{d}}$$\tau i$$g,$$\langle\alpha, , \beta \mathrm{f} ,..., \alpha_{3}, l*, 7_{1},\ldots, \gamma’k|[\alpha_{t}, \beta, ]\cdots[\alpha_{j}, \beta q]_{\mathcal{T}_{l}\mathit{7}_{k}}\ldots=1\rangle$$g\geqq 1$$\langle\gamma_{\{},\ldots, \tau_{b}|\gamma_{\mathrm{f}}’\cdots\gamma_{k}’=1\rangle$$\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\{$.

)$\mathrm{i}_{\mathrm{S}}\mathrm{i}\mathrm{n}\mathrm{n}\cup\cdot,\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}1\mathrm{t}\mathrm{o}\mathrm{o}\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}1\mathrm{i}\mathrm{C}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}\mathrm{t}\mathrm{o}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{S}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{f}\mathrm{y}\mathrm{i}\mathrm{g}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{r}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{S}.\mathrm{H}_{0}\mathrm{W}\mathrm{e}\mathrm{e}\mathrm{r},\mathrm{W}\mathrm{h}\mathrm{e}\mathrm{n}M1\mathrm{d}\mathrm{e}$$\pi_{l}\pi_{\mathrm{f}\{}$

can be so chosen as to(Remark 1)
proof implies that, in the casebe solvable.
(Remark 2) Their proof does not make it clear which finite groups occur as the quotients, or the
Galois groups. Recently Matsuno [7], improving their proof, gave a method to compute the Galois
groups effectively.
(Remark 3) In some special cases, evert if $M$ is of higer dimension, the fundamental groups canreport in this volume.
be dealt with. See$\mathrm{g}\geqq 1,$$\mathrm{B}\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{g}\mathrm{a}\mathrm{a}\mathrm{r}\mathrm{d}-\mathrm{N}\mathrm{i}\mathrm{l}\mathrm{e}\mathrm{l}\mathrm{s}\mathrm{e}\mathrm{n}^{1}\mathrm{s}$$f$$\mathrm{M}\mathrm{a}\mathrm{t}\mathrm{s}\mathrm{u}\mathrm{n}\mathrm{o}^{\uparrow}\mathrm{s}$result [3] and a technique on linear pencils,Kato [6], Namba [9],By a combination ofevading the difficulty of the fundamental groups in higher dimension, give some solutions forproblem, for example, in the following cases:$M$ : the complex projective plane,$D,$on each line (Figure 2),: lines such that there is at least one point of multiplicity

3.

$\mathrm{F}\mathrm{o}\mathrm{X}^{\mathrm{t}}\mathrm{S}$$\mathrm{F}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{h}\mathrm{e}1_{\mathrm{S}}^{\dagger}$$P^{2}$.

$,\ldots,$,$M:P^{1}$$D,$ $,\ldots,$.

$\geqq 3$$D_{k}$$D:\mathrm{A}$conics such that for each
(Figure 3),,$D_{i}$

there is another$\mathrm{M}:P^{\lambda}$$D_{l},\ldots,$$D_{k}:3$

lines circumscribing a conic (Figure 4).

$D_{k}$

touching at 2 distinct points

75

Theorem ([9], Theorem 1.5.8). Let $D,,\ldots,$be distin $cti\Pi educible$ conics on.
Suppose that, for each $Di$ , there issuch thatandare tangent at two dintin
poits. Then, for any irtegets, there is ffiite Gdois covering $f:Xarrow P^{z}$branching at $D=e_{t^{D};}+\cdots+e_{k}D_{t}$ .
$P^{\mathrm{z}}$$D_{k}$$\mathit{8}not\mathrm{A}erD_{k}$$D_{i}$$e_{\dot{d}}\underline{\geq}\mathit{2}(l<=^{j}=<k)$$D_{k}$

Theorem ([9], Proposition 1.5.9). Let $D,,$
be 3 distin$ci_{I}cvmSc\Gamma ibirg$ an irreducible conic C.
Then, for any integers
covering $f:Xarrow P^{2}bI\beta nchiRg$ at a $(D, +D_{X}+D_{\mathit{3}})+bC$ .
$D_{2,\mathit{3}}D$$Ct$lines on, there is a$a,b\underline{\geq}\mathit{2}$$P^{\mathrm{Z}}$$B\dot{l}Iite$

Galois(For further discussions in this line, refer Namba [9].)4.

.

Let us now give some other results on.
.
.

$\mathrm{F}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{h}\mathrm{e}1_{\mathrm{S}}^{1}$

problem. We treat the following cases:$\mathrm{M}:P^{2},\cdot$$D,$: lines in a near-pencil arrangement, that is, lines passing through one point andanother line not passing the point (Figure 5),$,\ldots,$$D_{k}$,$\mathrm{M}:P^{1}$$D_{2},\ldots,$$\mathrm{M}$lines circumscribing a conic (Figure 6),: a product of compact Riemann surfaces of genus: any hypersurfaces,$D_{f},\ldots,$$\mathrm{M}$$D_{k}:II(\geqq 3)$

$ct$

$a$

$\geqq 1$,$D_{k}$: a complex torus,$D:k$ any hypersurfaces.

$D_{l},\ldots,$$\mathrm{F}_{1^{\backslash }}f\iota_{\mathrm{t}}r4\sigma$$F_{\rho\}^{\backslash },1t\mathrm{c}\mathit{6}$76
In fact, by means of a topological method, we obtain the following:a point on . Suppose that distinct
Theorcm 1 (cf. [10], Prositiotion 7.2, 7.3).
be,...,does not. Let$D,,\ldots,$
line
that
anotherandthrough
pass
lines
at
covefing
Then,
a
is
there
integers–2.
ifand only ifone of the following conditions is satiesfied:
,
(i)
,
and
(ii)
(
is one ofthe following.
and
(iii)
$T\mathit{8}ke$

$p$

,

$n$

$e$

$D_{\infty}$$p$$D_{\mathrm{h}}$$P^{2}$$e_{\mathrm{L}},,$$d$$f:Xarrow p^{x}bt\mathit{3}\mathit{1}IChi_{Il}g$$I\mathrm{i}\dot{n}iteG\mathit{3}\mathit{1}ois$$D=e_{1}D_{\mathrm{t}}+\cdots+e_{t\iota\kappa}D+dD_{\infty}$$I\mathit{1}\underline{\geq}\mathit{4}$$=$$e^{-t},+e_{2}^{-_{l}}+e^{-\int},\leqq \mathit{1}$$n=\mathit{3}$$n=\mathit{3},$$(e,, e_{\mathrm{a}}, e_{S}, d)$$e_{\iota^{1}}^{-}+e_{\mathrm{a}}^{-}/+e_{S}>\mathit{1}$($e,,$e,$e_{z\prime}$$d$)(.Remark 4) When I gave my talk, on the necessary condition, I could prove only the first case.

Theorem 2 ([11]). Let$D_{1},$, be,$,\ldots,$

distinct lines on $P^{l}citc\iota rSctibiJlg$ an. Then, there is a finite
and an in teger$l\mathfrak{l}(_{=}>\mathit{3})$$D_{\iota}$, be integersirreducible conic C.
Galois covering $f:Xarrow P$ branching at
is one of the following:if$\ldots$$\geq \mathit{2}$$\mathrm{I}x\mathrm{t}e,$

all cases, stimulated$e_{\mathrm{t}}$$d$$\geqq \mathit{1}$$D=e_{1}D_{1}+\cdots+e_{\mathrm{t}},D_{\mathrm{n}}+\mathit{2}dC$$(e_{\mathrm{t}},\ldots, e_{\hslash}, d)$

(i)
(ii)
(iii)

,

$n_{=}>\mathit{4}$$n=\mathit{3}\mathit{3}I\mathrm{z}de^{-1},+e_{\mathrm{a}}^{-}+|<e_{\mathrm{p}}^{-_{t}}=^{\mathit{1}}$$n=\mathit{3},$I do not know about the necessary conditions.

(Remark 5)

andhypersurfaces on $M=M;\cross\cdots\cross M_{n}$ . Then, for any integers, there is a solvable covering $f:Xarrow M$ branching $atD=e,Dl+\cdots+e\mathrm{g}D_{\mathrm{A}}$ .

Theorem 3 ([11]). Let$D,$ $,\ldots,$$D_{k}$’$d=l$ and $e^{-f},+e^{-l}+G^{-}>\mathit{1}zj\iota$ .

distinct$M_{i}$

be compact Riemann surfaces ofgenus$g_{i=}\mathit{1}>(l==<_{i}<_{\mathit{1}l)}$$i\Pi ed_{U}cib]_{G}$$e_{i^{>_{\mathit{2}}}}=(l\leqq j=^{k})<$distinct
Let $M$ be an -dimensional complex torus andhypersurfaces on M. Then, for any integers $e_{j}\geqq \mathit{2}(l=j<<k=)$, there is solvable
covering $f:Xarrow M$ branching at $D=e,D_{f}+\cdots+e_{k}D_{\mathrm{A}}$

Theorem 4.

$D_{f},\ldots,$$n$$D_{k}$$i\Pi educible$$a$5. Theorem 3 or 4 is deduced from the following propositions respevtively.
same as is stated inare
Proposition 5.1 ([11]). Suppose that $M$ and$f:xarrow M$ branching atcovering,afiniteGaloisthere
isfflen,
any
integers
for
Theorem 3.
$D=e(D_{\mathrm{f}}+\cdots+D_{k})$ whose Galois group is isomorphic to a meta-abelian groupequippedsequence:with the following exact$D_{(},\ldots,$$D_{h}$$t\mathrm{A}e$$e_{=}>\mathit{2}$$G$$\mathit{1}arrow Z/eZarrow Garrow(Z/eZ)2(q|+\cdots+\oint’\iota^{)}arrow \mathit{1}$(Here, $Z$ denotes the ring of integers)$\mathrm{p}_{\mathrm{r}\mathrm{o}_{\mathrm{P}}}\mathrm{o}\mathrm{S}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{o}\mathrm{n}5.2$. Suppose that $M$ and$D_{l},\ldots,$$D_{k}$.

are the same as is stated in$7B_{Cot}em\mathit{4}$.

branching at, there is a finite Galois
Then, for any integers
equipped$D=e(D_{\mathrm{f}}+\cdots+D_{h})$ whose Galois group is isomorphic to a meta-abelian group
following exact sequence:with$coveri\mathit{1}\iota gf:xarrow M$$e\underline{\geq}\mathit{2}$$G$$t\Lambda e$$\mathit{1}arrow Z/eZarrow Garrow(Z/eZ)^{z}arrow \mathit{1}n$.

a brief outIine of the proof of Proposition 5.1. (For details, see [11].): $M_{i}arrow M_{i}(1\leqq i\leqq n)$ be the unramified coverings corresponding to the kernels of$Z/eZ$ -Hurewicz homomorphism:
. Put
.
:
Let

We

$\mathrm{g}\mathrm{i}\mathrm{v},\mathrm{e}$$\phi_{i}$$\pi_{\int(M_{i})arrow}>\mathrm{H}_{\mathrm{I}}(M_{i}, Z/ez)$$\phi=\phi_{\iota^{\mathrm{X}}}\cdots \mathrm{x}\phi n$$M’=M_{l}’\cross\cdots\cross M_{\hslash}^{J}arrow M$

77
We denote by $[D]$ the line bundle on $M$ associated to $D=D_{l}\cup\cdots\cup D_{k}$ . Let$\emptyset^{*}[D]$be the pull,

we see thatfor any 2-dimensional cycle, wheremeans the cap product. From this it follows that, which implies that there is aline bundle onthat
as -line bundles. As a consequence, we canconstructcovering $\mu:Xarrow M^{J}$ branching at. Moreover, we see that eachcovering transformation of lifts to a biholomorphism of $X$ up to covering transformations$\overline{C_{1}(\Phi^{\lambda}\lfloor D\rfloor)}\cap\sigma=0$$\sigma\in \mathrm{H}_{l}(M’, z/_{eZ})$$\cap$$\overline{c_{t}(\Phi*\lfloor D\rfloor)}=0$$L$$L^{\Phi \mathrm{e}}=\emptyset*[D]$$M’\mathrm{S}\mathrm{u}\mathrm{c}\mathrm{h}$$C^{\infty}$$Z/ez_{-}$$e(\emptyset^{*}D)$$\emptyset$. Thus, the composite
is a finite Galois covering having the properties
in the statement of Proposition 5.1.
Proposition 5.2 is proved in a similar way.
of

$f=\emptyset \mathrm{o}_{lt:xarrow}M$$\mu$

6. We now give some examples of projective manifolds with ample canonical bundles andcalculate their Chern numbers. We denote by
the canonical divisor of a compact complex
manifold $M$ and by(resp. $c_{2}(M)$ the first (resp. second) Chern class of the tangent
bundle of $M$.
For Il-dimensional projective manifolds $M$ with
ample, the inequality$K_{M}$$c_{\mathrm{f}}(M)$$K_{tt}$$(-|)^{\nu\iota_{C_{l}(\mathit{1}}}\mathrm{n}\}f\leq(-\mathrm{t})_{\frac{2(t\iota\cdot tl)}{r\iota}}\mathrm{n}C_{i}^{\mathrm{n}-}\iota \mathrm{c}_{\lambda}(\mathrm{r}\mathrm{t})$holds, where the equality holds if and only if the universal cover of $M$ is the unit ball (see Yau[12], also Miyaoka [8] for surfaces of general type).

be a compact Riemann surface of genus
Example 1.
andthe diagonal ofC.
By Proposition 5.1, we have a meta-abelian coveringof degreebranching$X$ is nonsingular surface.
at. Since is a nonsingular curve on
By Hurwitz,, whereformula
we have$g\geqq 2$$\mathrm{I}x\mathrm{t}\mathrm{C}$$\triangle$$e\triangle$$\triangle$$\mathrm{C}\cross$$e^{*\theta^{+}} \int$$f:Xarrow \mathrm{c}\cross \mathrm{c}$$\mathrm{C}\cross \mathrm{C},$$K_{X}=f^{*}\overline{K}$$(\mathrm{e}.\mathrm{g}., [1])$$\overline{K}=K_{\mathrm{C}}\cross C+\frac{\mathrm{e}-\dagger}{e}\triangle$

$=$

Then,

( $2g-2$ points).

$\mathrm{X}\mathrm{C}+\mathrm{C}\cross(2g-2\mathrm{p}\mathrm{o}\mathrm{i}\mathrm{n}\iota \mathrm{S})+\frac{\mathrm{e}-\dagger}{e}\triangle$$c_{l\mathrm{X}}^{Z}=K^{2}=(\deg D\overline{K}^{2}$$=e^{*}i^{+\mathfrak{l}}(K \mathrm{C}^{1}\mathrm{x}c+2\frac{\mathrm{e}-l}{\mathrm{e}}K_{C}\chi C. \triangle+\frac{\mathrm{t}e-1J}{\mathrm{e}^{l}}z\triangle)$$=e^{\mathrm{t}g-}(g-1)(8\neq g^{-}2(G+1)^{\iota})$.

Note that.
For every irreducible curve $D$ on $X$, we have.
criterion
implies thatis ample.
The second Chern class , which is equal to the Euler numberin two-dimensional
cases, can be calculated by the cardinality(see, Hirzebruch [4], [5]).
Indeed,$K_{\mathrm{X}}^{L}>0$$\mathrm{K}_{\mathrm{X}}\cdot D=\overline{K}\cdot f_{*}D>0$$(\mathrm{e}.\mathrm{g}., [1])$$\mathrm{N}\mathrm{a}\mathrm{k}\mathrm{a}\mathrm{i}_{\mathrm{S}}^{\mathrm{I}}$$\mathrm{K}_{\mathrm{X}}$$\mathrm{e}(X)$$c_{x_{\dagger 1}}$$\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{C}\mathrm{i}\mathrm{p}\mathrm{l}\mathrm{e}|\dagger$$c_{l^{=}}\mathrm{e}(X)$$=e^{*_{3^{*}}}=( \mathrm{d}\mathrm{e}\ovalbox{\tt\small REJECT}((D(2-\mathrm{e}(\mathrm{c}\mathrm{x}2g)^{\mathrm{z}}-\frac{-\mathrm{e}-}{\mathrm{e}}\mathrm{C})\frac{e-1}{1\mathrm{e}(}2-2\mathrm{e}(\triangle g))))$.

$=e^{*\uparrow-}\mathrm{f}(g-1)(4e^{Z}g^{-}2e(e+1))$

Thus we get$\frac{C^{1},}{\mathrm{C}_{2}}=\frac{4\theta^{-(f+\dagger}/\mathrm{e})}{2\#-(f+|/e)}\mathrm{z}$(Remark 6)

The composite X$\mathrm{L}\mathrm{C}\cross \mathrm{C}7_{arrow}^{\tau \mathrm{o}i}$.
$\mathrm{C}$gives a Kodaira fibration in the sense of [1].

Example 2. Let be the compact Riemann surface defined by the equation,
with genus two. The meromorphic function-fold cyclic
covering ramified completely at the three point
over , l,and (Figure 7).
$y^{\mathit{5}}=_{X}(x-1)$$\mathrm{C}$$(x, y)\in \mathrm{C}\vdash\# x\in \mathrm{P}^{1}\mathrm{g}\mathrm{i}\mathrm{V}\mathrm{e}\mathrm{s}\mathit{5}$$p_{\mathit{0}},$$p_{1},$$pw\in \mathrm{C}$$\mathit{0}$$\infty$78$F_{|t^{\downarrow r\mathrm{e}}}^{\backslash }\{7$$\mathbb{P}^{1}$\"i$0^{\overline{\mathrm{r}}}$$\infty..-$, we put. For
be a generator of the Galois group. ( is nothing but diagonal , see Figure 8).
of degree 9 branching at
By Proposition 5.1, we have a mata-abelian covering

Let$0\leqq i\leqq 4$$(^{\cong}Z/\mathit{5}z)$$\sigma$$\triangle_{i}=\{(p, \sigma^{i}(p))\in \mathrm{C}\mathrm{X}\mathrm{C}|p\in \mathrm{C}\}$$\triangle$$\triangle \mathit{0}$$f:Xarrow \mathrm{C}\cross \mathrm{C}$5$(\triangle_{0}+\triangle_{\mathrm{I}S^{+}}+\triangle f+\triangle\triangle_{t})$.

and $(p_{\nu}, p_{w})$ ,We have singularities on the fibres of over three points
.
at
the 3
of
up
blow
be
the
Let
as
follows.
which can be resolvedtheand$E_{i}(i=0,1,\infty)$bypoints,the3
toassociatedcurvestheexceptional
We denote by(Figure 9).
proper transforms of$f$$(p_{\delta}, pa),$$(p_{1}, p_{l})$$\mathrm{C}\cross \mathrm{C}$$Marrow \mathrm{C}\cross \mathrm{C}$$\mathrm{p}_{\sim}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{s}$$\triangle_{i}$$\triangle_{i}(0\leqq i\leqq 4)$$f^{\neg}\mathfrak{l}^{\backslash }\theta^{\iota\{\mathrm{r}}e(f$Let : $X^{/}arrow M$ be the pull-back ofof degree
we get a meta-abelian covering :branches at 5in Proposition 5.1,$(i=0,1, \infty)$ .
Therefore is a nonsingular surface.
, whereWe have$f:Xarrow \mathrm{C}\cross \mathrm{C}$$f^{\mathfrak{l}}$$\mathrm{Y}arrow M$

$\pi$

and

$5^{?}$

the normalization of $X’$ . Thus
As we see from the construction, unramified along$\mathrm{Y}$.

$(\triangle+\sim_{\mathit{0}t}g+x_{z}+\triangle\sim ff+\triangle+)\sim$$\pi$$E_{i}$$\mathrm{Y}$$\pi^{*}\overline{K}$$K_{\mathrm{Y}}=$$\overline{K}=K\eta+4/5(^{\sim\sim}\triangle \mathit{0}+\triangle|\sim\wedge+\triangle \mathrm{Z}^{+}\triangle s+\sim\ )$.

It is easy to check that there is no curve isomorphic to the projective line on
ample. We obtain by a calculation similar to that in Example 1,$c_{t}=K_{\mathrm{Y}}^{Z}=5\rho$

We have
Hirzebruch [4].)$C^{\chi},/CZ^{=3}$’.

$4\mathit{5}$,$c_{l}=\mathrm{e}(\mathrm{Y})=5^{?}\cdot 15$$\mathrm{Y}$As a result,$K_{1’}\mathrm{i}\mathrm{s}$.

which implies the universal cover of Yis the unit ball.

(Comparebe an -dimensional complex abelian variety and $D$ a nonsingular hyperExample 3.
$M$, we have a meta-abelian covering
surface on . By Proposition 5.2, for any integer$eD$$f:Xarrow M$ of degree. Since the tangent bundle of $M$ is trivial, we havebranching at
is also ample divisor.
. Therefore, if is ample,$\mathrm{I}x\mathrm{t}M$$n$$e\geqq 2$$e^{zn+1}$$K_{\mathrm{X}}= \frac{\mathrm{e}\sim l}{\mathrm{e}}f^{*}D$$D$$K\chi$

79
We have$c_{t}^{l\iota}=(^{-}K\mathrm{x})^{\prime \mathrm{L}}$$=(-1=(^{-}1)’\gamma\iota\iota(eeD(_{\frac{\mathrm{e}-1}{\underline{\mathrm{e}}_{1)^{\nu\iota}}})^{\mathrm{n}}}r\mathrm{t}+|(\deg f)\hslash D^{\mathrm{h}}$In order to calculate , we need the following
manifold and a nonsingular hypersurface
Lemma 6.1. Let $M$ be an -dimensionalon M. Suppose that $f:Xarrow M$ is a ffiite Galois covering branching at D. $nen$ we have$c_{\chi}$$D$$F^{IO}jeC\dot{a}ve$$I\mathrm{J}$$e$$C_{l(x)}=f*(C_{k}(M)- \frac{\mathrm{e}-l}{\mathrm{e}}C4^{(M})\cdot D+\frac{\mathrm{e}-1}{\mathrm{e}}D^{l})$This lemma can be proved by induction on .
By Lxmma 6.1, we see that.

$I\mathrm{J}$$C \chi(X)=\frac{\mathrm{e}-|}{\mathrm{e}}D^{2}$$c_{1\mathrm{a}}^{\prime \mathrm{L}}-\mathrm{z})\hslash c=(^{-}1(z-1)-\mathrm{n}- G\mathrm{t}\mathrm{n}eD^{\mathrm{n}}*z$

Thus, we get.

Hence we have$\frac{\grave{C}_{\mathrm{f}}^{\gamma \mathrm{t}}}{C_{1}^{\mathrm{n}-Z}\mathrm{C}_{z}}=\frac{\mathrm{e}-1}{\mathrm{e}}$(Remark 7)

Hirzebruch [5] constructed a surface with ample canonical bundle with the universal

cover the unit ball as a branched covering of a blow-up of certain abelian surface with complex
with the same properity
multiplication. Can we construct a projective manifold of dimension
as a branched covering of a blow-up of some abelian variety?
$\geqq 3$

References
[1]
[2]
[3]
[4]
[5]
[6]
[7]

[8]
[9]

[10]

Barth, W., Peters, C., Van de Ven, A.: Compact complex surfaces, Springer Verlag

1984.

index in F-groups,
Bundgaard, S., Nielsen, J.: On normal subgroups with
1951,
B,
p. 56-58.
Mathematisk Tidsskrift
Fox, R.H.: On Fenchel’s conjecture about -groups, Mathematisk Tidsskrift B, 1952,$B\dot{\mathit{1}}Iite$$F$

p. 61-65.

Hirzebruch, F.:andangements of lines and algebraic surfaces, inBirkhauser,
II,
1983.
Geometry, Vol.
Hirzebruch, F.: Chern numbers of algebraic surfaces, Math. Ann. 266, p. 351-356(1984)
Kato, M.: On the existence offinuteprincipal UlIifolmizgtiollS of
wcigBtedline configulations, Mem. kyushu Univ. Ser. A. 38, p. 127-131 (1984)
Matsuno, T.: The problem of Fenchel about $F$-groups, Bulletin of Osaka prefectura.lcollege of technology 33, p. 53-56 (1999).
Miyaoka,Y.: On the Chern numbers of surfaces ofgeneral type, Inv. Math. 42, p.
225-237 (1977).
Namba, M.: Branched coverings and algebraic functions, Research Notes in Math.
161, Longman, 1987.
Ueno, R.: Which properties of coverings of the Riemann sphere minus three points
can be also in higher dimensional cases? (RIMS doctoral thesis, 1997), Jour,.al of
economics and management 42, No. 2, p. 177-199 (1998).
]
theorem, NUCB joumal
Ueno, R.: A generalization of the..
of economics and manag.ement 44, No. 2, p. 183-188 (2000). .
Yau, S.-T.: Calabi’s con] eCfurG and some new results in algebraic geometry, Proc. Nat.
Acad. Sci. USA 74, p. 1798-1788 (1977).
$A\acute{\Pi}$$\mathrm{A}\dot{\mathrm{n}}\mathrm{t}\mathrm{h}\mathrm{m}\mathrm{e}\iota \mathrm{i}\mathrm{C}$$CP^{1}e\mathit{1}ong$$\mathrm{n}$

[11]

[12]

$F_{CI\mathit{1}}cpe$$-BuI\mathit{1}dgaeId- NiC\mathit{1}se\mathit{1}\mathit{1}$Nagoya University of Commerce and Business Administration
4-4 Sagamine Komenoki-cho Nisshin-shi Aichi Japan-mail address: ryoichi\copyright nucba.ac.jp$\mathrm{E}$