数理解析研究所講究録 1345 巻 2003 年 148-155

148

On a bounding problem of Calabi-Yauthr eefolds
大阪大学大学院理学研究科

大野浩司 (Koji Ohno)

Department of Mathematics Graduate School of ScienceOsaka University
Ab st ract
In this rport, we shall give a tentative argument on bounding
probelm of Calabi Yau ’ the study of which was started by B.Huntand M.Gross.
$\mathrm{s}$1Introduction

Many families of projective smooth Calabi-Yau threefolds defined over thecomplex number field have been found, but as far as I know, they are allbounded ( , parametrized by quasi projective schemes over the complex
number field. It is natural to ask how many families of projective smoothCalabi-Yau threefolds exitst. So far, specialists seem to think that their deforrnation types are maybe finite or maybe not. Reid’s fantasy asserts that there
are essentially just one family which corresp onds to the fact (conjectured byA.Weil-A.Andreotti and solved affirmatively by K.Kodaira) that eveiy $K3$
surface is a deformation of a non-singular quartic surface in a projective
3-spaces. If Reid’s fatasy is true, even if there are non finitely many families of projective smooth Calabi-Yau threefolds, they are “transformed” to abounded family and one may get feedback. For example, let us recall thatK.Kodaira showed that any $K3$ surface can be deformed to an elliptic $K3$
surface in a one family which implies the conj ture of A.Weil-A.Andreottis true. In this direction, B.Hunt ([9]) asserted the Euler numbers of pro
jective smooth Calabi-Yau threefolds with a fiber structure are bounded but
unfortunately his proof based on the theory of variation of Hodge structurescontains many crucial gaps. Obviously more geometric information seems
to be needed. Later, M.Gross ([7]) showed, extending the Ogg-Shafarevich
theory, the surprizingly strong result that elliptic Calabi-Yau threefolds with$\mathrm{i}.\mathrm{e}.$$\mathrm{i}$143a rational base is birationally bounded. In this report, we shall consider theother cases of Calabi-Yau threefolds with a fiber structure, that is, the cases
where a general fibre is a surface with trivial canonical bundle and the baseis a projective line.

2Bimeromorphic invariants of degenerationsof surfaces of Kodaira dimension zeroIn this section, we define bimeromorphic invariants of degenerations of surfaces of Kodaira dimension zero.
variety and let
be a normal
be a boundary
Definition 2.1 Let
ison $X$ such that
. Assume thatcanonical and that
$K_{X}+2’$ isis said to be moderately $Og$ canonical uith-Cartier.
if for any exceptional prime divisor $E$ of the function field
respect$X$$ai(E;X,\mathrm{x})$$>1$ holds.
$=0,$ the inequalityofwith$\log$$(X, \mathrm{X})$$\triangle’\leq\triangle$$\triangle’$$(X, \mathrm{x})$$\log$$(X, \mathrm{X})$$\mathrm{Q}$$K_{X}+\triangle’$$a_{l}(E\mathrm{i}X, \triangle’)$: $Xarrow B$ be a proper connected morphism from a normal QLet
factorial variety defined over the complex number field $C$ (resp. a normal-factorial complex analytic space) $X$ onto a smooth projective curve (resp.
such that a general fibre $f^{*}(p)$ (resp.
a unit disk $V$ $:=$ $\{z\in C;|z|< 1\}$any fibre $f^{*}(p)$ where is not the origin) is a normal algebraic variety with(resp. the origin 0)
only terminal singularity. Let
be a set of points insuch that the fibre
is not a normal algebraic variety with only terminal. We shall consider thesingularity. Put
and7
following three birational ( or bimeromorphic ) models.
$f$$\mathrm{Q}$$)$

$B$

$p$

$B$

$\Sigma$$f^{*}.(p)$$\ominus_{p}:=$$7$’$($$)_{\mathrm{r}\mathrm{e}\mathrm{d}}$$\mathrm{O}-:=\sum_{p\in\Sigma}\Theta_{p}$Definition 2.2 We define the following three birational ( imeromorphi )mo dels.
$\mathrm{b}$$\mathrm{c}$(1): $Xarrow B$ is called a minimal fibration (resp. degeneration) if $X$ hasis /-nef ( , the intersection number
only terminal singularity andofand any complete curve contained in a fibre of is non-negative).
$f$$)$$K_{X}$$f$$K_{X}$(2)$\mathrm{i}.\mathrm{e}.$: $Xarrow B$ is called a Ogarithmic minimal (or abbreviated, 0 minimal)
terminal andfibration (resp. degeneration) if $(X, \Theta)$ is divisorially$K_{X}+\ominus$ is
f-nef.
$f$$g$$\log$(3): $Xarrow B$ is called a strictly Ogarithmic minimal (or abbreviated,isstri ctly 0 minimal) fibrati on (resp. de general on) if,
being both /-nef.
canonical with
$f$$(X, \ominus)$$g$$K_{X}$$\mathrm{O}-$$\log$

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Given a rations (resp. degenerations) of algebraic surfaces with Ko
daira dimension zero over a smooth projective curve (resp. 1-dimensionalunit disk) , one can get the model (1), (2) by using MMP and LMMP.
The model (3) can be constructed also by LMMP starting from the model(2). The model (3) constructed as above enjoyies also the following property;. Moreover, the model
is moderately
canonical with respect to(3) is good in the following sense.
$B$$K_{X}$$\log$$(X, \ominus)$: $X_{i}^{s}arrow B(i=1,2)$ beProposition 2.1 ( . [11], Theorem 4.9) Let
trvo strictly $Og$ minimal fibration (or degeneration) of surfaces with Kochirawhich are birationally equivalent to eachdimension zero projective overmoderately $Og$ canonical with respect
other over B. Assume that:are connected
andfor $i=1,2$ . Then :
$Og$
$p\in
B_{f}$
that is, there
flips over a neighbourhood of
by a sequence ofexist birational morphisms be rween normal threefolds over a neighbourhoodof $p\in B$ which are isomorphic in codimension me:$\mathrm{c}.\mathrm{f}$$f_{i}^{s}$$B$$(X_{i)}^{s}\ominus_{\dot{\iota}p}^{g})\dot{\mathfrak{B}}$$t(jK_{X_{i}^{s}}$$X_{1}^{s}arrow B$$f_{1}^{s}$$X_{2}^{s}" i$$f_{2}^{s}$$B$$X_{1}^{s}:=X^{(0)}arrow Z^{(0)}arrow X^{(1)}arrow Z^{(1)}\cdotsarrow X^{(n)}=:X_{2}^{s}$where $X^{(k)}u$\dot$\mathrm{Q}$-factorial for $k=0,1$ ,$\ldots$$n$,.

Let: $X^{s}arrow B$ be a strictly
minimal fibration ( or degeneration ) of’)surfaces with Kodaira dimension zero projective oversuch thatnumericallyis moderatelyiscanonical with respect to Kx-. Since
trivial over , there exists a positive integer
such that$fs$ ’(p)red: $Yarrow X^{s}$ be a minimal
for any $p\in$ B, whereLet, that is, is a projective birational morphism from a normal
model overis /x-nef.
-factorial
with only terminal singularity tosuch thatBy running the minimal model program overstarting from the induced$arrow B$morphism $g:=f^{s}\circ\mu$ : $Yarrow B$ , we obtain a minimal fibration :and a dominating rational map A: $Y-arrow Z$ over . Since
has onlyonsingularity,
terminal
there exists an effective -divisor
withsuch that$\log$$f^{s}$$B$$\log$$K_{X^{s}}$

$B$

$\mu$

$\cdot$

$\mu$

$Y$

$\mathrm{Q}$$f^{s*}(p)=\ell_{p}^{*}\ominus_{p}^{s}$$\ell_{p}^{*}\in N$$\ominus_{p}^{s}:=$$X^{s}$$(X^{s}\Theta))$$X^{s}$$K_{Y}$

$B$

$h$

$B$

$Z$

$X^{s}$$\triangle$$\mathrm{Q}$$\log$$\lfloor 2$$\rfloor=0$$Y$$K_{Y}+\triangle=\mu^{*}K_{X^{s}}$.

Since $K_{Y}+$ :s,andare all numerically trivial overexists a non-negative rational umber
such that$K_{Z}+\lambda_{*}\triangle$$K_{Z}$$\lambda_{*}\triangle_{p}=\mu_{p}^{*}h^{*}(p)$denotes the restriction of$p\in B.$ Put$\triangle_{p}$, there$\mu_{p}^{*}\in Q$$\mathrm{n}$

where

$B$

$\triangle$

(2.1)

,

in a neighbourhood of the fibre over$\mathrm{t}\mathrm{s}_{p}^{*}:=b(\frac{\ell_{p}^{*}-1}{l_{p}^{*}}-\mu_{p}^{*})$,$c_{p}^{*}:=\mu^{*}P\mathit{7}_{p}^{*}$.

151

Proposition 2.1 gives the following:

Corollary 2.1
andcrre birational (or bimeromorphic) invariants of germs of singular fibres over $p\in B$ and hence so areand$P_{p}^{*}\in N$$\mu_{p}^{*}\in Q$$s_{p}^{*}$$c_{p}^{*}$.

Let : $Xarrow B$ be a minimal fibration of surfaces with Kodaira dimension
zero projective over a projective smooth connected curve defined over thecomplex number field. The above invariants fit into the canonical bundle
formula as foUows:$f$$B$$K_{X}=f^{*}(K_{B}+ \frac{1}{b}L_{X/B}^{ss}+\sum_{p\in B}(\frac{l_{p}^{*}-1}{\ell_{p}^{*}}-\mu_{p}^{*})p)$

where$\frac{1}{b}L_{X/B}^{ss}$.

(2.2)

is a $Q$ -divisor on $B$ defined in [4].

Example 2.1 For degenerations of elliptic curves, one can define invariants,in the same way and it can be checked thatandcoincides withthe multiplicty if the singular fibre is of type I or otherwise, with the orderof the semisimple part of the monodromy group around the singular fibre.
We can also obtain the following table:$4_{p}^{*}$$\mu_{p}^{*}$$l_{p}^{*}$$s_{p}^{*}$$m$

TableIIIr$b$$\mathrm{V}$$m$I$\mathrm{I}\mathrm{I}^{*}$$\mathrm{I}_{b}^{*}$$b$

IIIIIIIII’IV$\mathrm{I}\mathrm{V}^{*}$

Here we are using the Kodaira’s notation ([10]). See also [6].

Seeing the above table, naively one may expect that the invariatnt
is theone determined by the variation of Hodge structure around , but involves
the index of $Kxs$ which is usually a minimal model theoretic invariant not
the Hodge theoretic one. So, in higher dimensional cases,
seems to be
extremely complicated, but still one may hope;$c_{p}^{*}$$p$$c_{p}^{*}$$c_{p}^{*}$

Conjecture 2.1 For any degeneration of algebraic surfaces with Kodairadimension zero over a one- imensional complex disk, the number of possible
values of is finite$c_{p}^{*}$For the above conjecture, we have a partial affirmative answer.

152Theorem 2.1 For any degeneration of abelian surfaces overcomplex disk, we have$a$

one-dimensional$c_{p}^{*}\in\{0,1/5,1/4,1/3,2/5,1/2,2/3, 1, 3/2, 2, 3, 4, 5, 6\}$ .

3

Bounding the number of singular fibres ofAbelian Fibred Calabi-Yau threefolds overa projective line

In [9], there is no argument on the number of singular fibres which is important to get the certain finiteness result. For, recalling theG.FaLtings
theory which was developed for solving Shafarevich conjecture, smooth families over fixed base tends to have a certain finiteness property. The study of
ber of singular fibres was started by Oguiso ([12]).
Let: $Xarrow B$ is a projective connected morphism from a normal Qfactorial projective varicty $X$ with only canonical singularity onto a . We
define the subset of closed points of ,by$\mathrm{A}^{\backslash }\mathrm{a}\mathrm{k}\mathrm{e}1\mathrm{o}\mathrm{v}$$\mathrm{t}1_{1}\mathrm{e}$$\mathrm{n}\mathrm{u}\mathrm{m}$

$f$

$B$

$B$

$\Sigma_{f}:=$$\Sigma_{f}${ $p\in B|f$ is not smooth over a neighbourhood of }.
$p$

Definition 3.1 Let
be the set of all the triple $(X, f, B)$ where $X$
is a normal projective threefold $X$ with only canonical singularity whose
canonical divisor: $Xarrow B$ is a projective
is umerically trivial andconnected morphism onto
whose geometric generic fibre is an ab eliansurface.
$C\mathcal{Y}_{B,\mathrm{a}\mathrm{b}}^{3}$$K_{X}$$f$$\mathrm{n}$$B\simeq P^{1}$From Theorem 2.1 : we can deduce the following theorem using G.FaLtings
theory and Zarhin’s trick.
Theorem 3.1 There exists $s\in$ N, such that for any triple $(X, f, B)\in$$\mathrm{C}\mathcal{Y}_{B,\mathrm{d})^{\rangle}}^{3}$$s_{f}:=$

Card$\Sigma_{f}\leq s,$that is, the number of singular fibres of Abelian Fibred Calabi-Yau threefolds
er a $pro|ectivehne$ is bounded frarn above by a universal constant.
$\sigma v$

4Further tentative argument

To get more results on Abelian Fibred Calabi-Yau threefolds over a projective
line, we should study albanese fibration, which is defined from$\mathrm{A}1\mathrm{b}^{0}(X_{\eta})$153(see, [8] and also [3]). Albanese fibration plays a role of jacobian fibrationfor elliptic fibrations. Since jacobian fibrations of eliiptic fibered Calabi-Yau
threefolds are again a birationaHy Calabi-Yau threefolds( [7] ), It is naturalto hope;

Conjecture 4.1 Albanese fibration of Abelian Fibred Calabi-Yau are again
Calabi-Yau.
For the above conjecture, we have:

Proposition 4.1 Let : $Xarrow B$ be a projective connected morphism fromvariety $X$ with only canonical singularity onto
normal -factorial
whose geometric generic fiber is an abelian variety arid $tt$ : $Aarrow B$ be
the Albanese group scheme associated to . Let :be a projective$A$ oversmooth
whichbirational.
atofibration from
$f$

$a$

$\Psi\dot{q}ective$$\mathrm{Q}$

$B$

$p$

$f$

$\overline{A}arrow B$$\overline{\varphi}$$B$$\overline{A}$$\uparrow s$(i)we have(ii)moreover, if we assume that $\dim X=3$ and that : $Xarrow|$ $B$ admits
an analytic Ocal section in a neighbourhood of any clos $ed$ pcint.c; $p\in B,$, wherethen we havearid
areandthe analytic Ocal bimeromorphic invariantsof the fibres of and 2
over $p\in B$ respectively. In particular, the Kodaira dimensions of $X$andare the same.

$L_{X/B}^{ss}\sim_{Q}L_{A/B}^{s_{\frac{s}{}}}$

an

$d$

$f$

$s_{p}^{*}(f)$$s_{p}^{*}(f)=s_{p}^{*}(\overline{\varphi})$$\Sigma_{f}=\Sigma_{\overline{\varphi}}$$s_{p}^{*}(\overline{\varphi})$$f$$s_{p}^{*}$$\overline{A}$One may also hope;
Conjecture 4.2 Abelian Fibred Calabi-Yau has local sections everywhere.

For farther investigation, R-R will be useful.
Proposition 4.2 Let : $Xarrow B$ be a (where $X$ is a Qfibroli
factorial normal variety with on canonical singularity and $Bs$ a connectedsmooth projective curve defined over the carnplex number field. Then,$f$$\Psi$$\dot{\eta}ective$$on_{\rangle}$$ly$$\sum_{q}(-1)^{q}\deg R^{q}f_{*}\mathcal{O}(K_{X/B})=(-1)^{\dim X}\{\chi(\mathcal{O})-\chi(\mathcal{O}_{X_{\eta}})\chi(\mathcal{O}_{B})\}$

In pcrti $cular_{f}$ If $\dim X=3,$ we $have$$\deg f_{*}\mathcal{O}(K_{X/B})-\deg$.

$R^{1}f_{*} \mathcal{O}(K_{X/B})=\frac{c_{2}(X_{\overline{\eta}})}{24}\deg(L_{X/B}^{ss}+\sum_{p\in B}s_{p}^{*}.)-\sum_{P_{\alpha}}\frac{r_{\alpha}^{2}-1}{247_{\alpha}}$(see [14] for notation).

154

If $q(X_{\eta})=0$ (for example, $K3$ fibred case), we have$\deg f_{*}\mathcal{O}(K_{X/B})=\frac{c_{2}(X_{\overline{\eta}})}{24}\deg(L_{X/B}^{ss}+\sum_{p\in B}s_{p}^{*})-\sum_{P_{\alpha}}\frac{r_{\alpha}^{2}-1}{24r_{\alpha}}$.

(for ex[9] asserts that boundedness results follows if we fix
ample, $\deg f_{*}\mathcal{O}(K_{X/B})=2$ if $X$ is Calabi-Yau) not using any other propert of Calabi-Yau. But seeing the above formula, it seems that fixing
is geomertrically nonsense.
$\deg f_{*}\mathcal{O}(K_{X/B})$$\mathrm{y}$$\deg f_{*}\mathcal{O}(K_{X/B})$

Refer ence$\mathrm{s}$[1] G. Faltings, Arokelov ’s theorem for Abelian Varieties, Invent.math., 73,(1983), pp. 337-347.
[2] G. Faltings and -L. Chai, Degeneration of Abelian Varieties, Ergeb.
Mat . Grenzgeb (3), 22, Spriger-Verlag, Berlin-Heidelberg-New YorkLondon-Paris-Tokyo Hong Kong-Barcelona, (1990).
$\mathrm{C}$$\mathrm{h}$[3] A. Fujiki, Relative Algebraic Reductio7b and Relative Abanese Map for
Fiber Space in , Publ. RIMS, Kyoto Univ. 19, (1983), pp. 207-236.

$a$$\mathrm{C}$[4] O. Fujino and S. Mori, A Canonical Bundle Formula, J.Differential Geometry 56, (2000), pp.167-188.

[5] T. Fujita, Fractionally Ogarithrnic canonical rings of algebraic surfaces,
J. Fac. Sci. Univ. Tokyo Sect. IA 30, (1984), pp. 685-696.
[6] T. Fujita, Zariski $kc\sigma mp$ osition and canonical rings of elliptic threefolds,J. Math. Soc. Japan 38, NO.1, (1986), pp. 19-37.

[7] M. Gross, On Finiteriess Theorem for $El$ lhptic Calabi-Yau
Duke Math. Jour. Vol. 74, N0.2, (1994), pp. 271-299.

Threefolds[8] A. Grothcndi ,ndati $om$ de & Geometrie Analytique [Extrait duS\’erninaire Bourbaki 1957-1962], Paris 1962.
$\mathrm{e}\mathrm{c}\mathrm{k}$$F\sigma u$$\mathrm{s}$[9] B. Hunt, A
on the Euler number for certain Calabi-Yau 3-folds, J.
reine angew. Math. 411, (1990), pp.137-170.
$b\sigma ur\iota d$[10] K. Kodaira, On compact analytic surfaces $II$, Ann. of Math. 77, (1963),
pp.563-626.

[11] J. Kollar, Hops, Nagoya Math. J. 113 (1989), pp.1536.

155[12] K. Oguiso, A Note
Moderate Abelian Fibrations, Contemporary
Math. 207, (1997), pp.101-118.
$\sigma rb$[13] K. Ohno, The Euler Characteristic Formula for Logarithmic Minimal
Degenerations of Surfaces with Kodaira Dimension Zero and its application to Calabi-Yau Threefolds with a pencil, preprint.
[14] M. Reid, Young person’s guide to canonical singularities, Proc. Symp.
in Pure Math. 46, 1987, pp.345-414.
[15] W. Shokurov, 3-fold $Log$ Flips, Russian Acad. Sci. Izv. Mat . 40,(1993), pp.95202.
$\mathrm{h}$[16] Yu. G. Zarhin, Endomorphisms of abelian varieties and paints of finite
order in characteristic , Math. Notes of the Academy of Sciences ofthe USSR, A translation of Mathematicheskie Zametki, 21, (1977) pp.
$p$415-419.
Department of Mathematics Graduate School of Science,
Osaka University,-0043, Japan
Toyonaka, Osaka
E-mai address : koji@math.sci. osaka-u.ac.j$\mathfrak{X}0$$l$$\mathrm{p}$