数理解析研究所講究録1033 巻 1998 年 1-6

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THE WORLD OF RATIONAL GORENSTEIN SINGULARITIESYUKARI ITO
Department of Mathematics
Tokyo Metropolitan UniversityHachioji, 192-03 Tokyo, Japan\S 1. An entrance to the world.
We are going to the world of rational Gorenstein singularities from the view pointof classification theory of algebraic varieties. In the classification theory of 3-folds,canonical singularities and terminal singularities are very important. We recall thedefinitions here.
Definition. A normal variety $X$ has canonical (resp. terminal) singularities if thefollowing two conditions hold:(1) There exists a integer(2) For a resoluion

$f$

$r$

such that the Weil divisor $rK_{X}$ is Cartier divisor.

: $Yarrow X$ and the exceptional prime divisors$E_{i}\mathrm{s}$, thefollowing formula holds.
$rK_{Y}=f^{*}(rKx)+\Sigma a_{i}E_{i}$where$a_{i}\geq 0$(resp. $a_{i}>0$ ).

Definition. In the above definition, we call the smallest number
discrepancy at$E_{i}$$r$index and$a_{i}$.

Remark. These singularities are very familier with well known singularities in twodimension. If the dimension of the variety $X$ is two, then terminal singularity is
non-singular and canonical singularities are same as rational double points of type
$A_{n},$$D_{n}$

and$E_{n}$.

Now we will see three dimensional case. Terminal singularities are classified completely by Mori [Mo, cf.Rel]. If the index $r=1$ , then they are isolated hypersurface
Typeset by$A_{k\mathrm{I}}s_{-}\mathrm{I}\mathrm{E}\mathrm{X}$2compound$\mathrm{o}\mathrm{f}_{-}\mathrm{t}\mathrm{h}\mathrm{e}$$\mathrm{D}\mathrm{u}\mathrm{V}\mathrm{a}\mathrm{l}$singularities. If$r$is greater than 2, then they are cyclic quotient

above singularities. At this moment, there is no classifications for canonicalsingularities. But we have the following fact:
Theorem. In any dimension, canonical singularity of index 1 is rational Gorenstein.
It is very convenient to understand canonical singularity in some sense.

\S 2. Resolution of singularities.
Now we know the existence of the resolution in general and we will introduce
special resolution here:
Definition. The resolution of the singularities$f$:$\mathrm{Y}arrow X$

is crepant if and only ifthere is no discrepancy at any exceptional prime divisors.

From this resolution, we can obtain terminal singularities in$Y$. Naturally if thesingularity is two dimensional rational double point, then the crepant resolution isminimal resolution.

We have too many rational Gorenstein singularities in general, then we will see
only quotient singularities here. For this, we have following fact:
Fact. The quotient singularity $X=\mathbb{C}^{n}/G$ has rational Gorenstein singularities if

and only$\dot{i}f$the group$G$

is a finite subgroup of$SL(n, \mathbb{C})$We don’t know the existence of the crepant$\mathrm{r}\mathrm{e}\mathrm{S}\mathrm{o}$}without quasi refrections.

$\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$in general. Morover wewill consider the following conjecture which came from Vafa’s formula in superstring
theory:$\mathrm{C}_{\mathrm{o}\mathrm{n}}\mathrm{j}\mathrm{e}\mathrm{C}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}[\mathrm{H}\mathrm{H}]$

. Let

$X$

be the quotient of$\mathbb{C}^{n}$by the finite subgroup$G$of$SL(n, \mathbb{C})$and $f:Yarrow X$ crepant resolution. Then the topological Euler number is the numberof the conjugacy classes of the group .
$G$

Remark. This conjecture is for local topology, but we can see similarly for globaltopology. Moreover we can consider mathematical meaning of the above formula for
Euler numbers. In two dimesional case, we can see it as$\mathrm{M}\mathrm{c}\mathrm{K}\mathrm{a}\mathrm{y}$correspondence andin higher dimensional case we have some results for the explanation.

3To consider this conjecture we will see some examples.
(1) $n=2$ The singularities are rational double points and they have minimalresolutions. The conjecture holds for them. We can also see this from$\mathrm{M}\mathrm{c}\mathrm{K}\mathrm{a}\mathrm{y}$correspondence and we have the following formula from it:$h_{2}(Y)=\#${$\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{j}\mathrm{u}\mathrm{g}\mathrm{a}\mathrm{c}\mathrm{y}$class of$G$

but not identity}$h_{0}(Y)=1=\#\mathrm{f}^{\mathrm{i}\mathrm{d}\mathrm{t}}\mathrm{e}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{y}\}$As the global example, we have Kummer surface which is obtained as aminimal resolution of the quotient by the involution of an Abelian surface $M$ .
Original quotient has 16 singularities of type

$A_{1}$. By the minimal resolution,

we get the Euler number is 24 and it is also computed as the orbifold Euler
number:$e_{V}(x)=\Sigma_{ghg}h=e(M^{g}\mathrm{n}M^{h})$where the summation runs over the pair $(g, h)$ in the acting group$G$

which iscommutative and $e(M^{g}.\cap M^{h})$ is the topological Euler number of the common
component of two fixed parts.

By the way, if you consider the finite linear group which is not subgroupof$SL(2, \mathbb{C})$, then the conjecture does not hold.

(2) $n=3$ In this case, the singularities are canonical but not terminal. The

existence of the crepant resolution is shown by some people from 1987 to1996. ($\mathrm{c}\mathrm{f}.[\mathrm{M}\mathrm{a}\mathrm{l}]$[Rol] [Ma2] [Ma3] [Ro2] [Itl] [It2] [Ro3]) These proofs weredepend on the classification of the finite subgroups in$SL(3, \mathbb{C})$and there

are some papers on this conjecture. And there is no complete proof for theexistence without the classification.
And for Betti numbers are computed as follows:$h_{2i}=\#${$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{j}\mathrm{u}\mathrm{g}\mathrm{a}\mathrm{c}\mathrm{y}$class of age$\dot{i}$}for $\dot{i}=0,1$ and 2. The age of the elememnt is computed from the eigen value.
For precise definition, see the paper by Reid and the author [IR].
Three dimensional global example is a Calabi-Yau 3-fold. If you take aelliptic curve$C$with complex multiplicity of order 3 and the the finite group4

which is isomorphic to the cyclic group of order three. Let $M$ be$C\mathrm{x}C\cross C$.

Then we obtain a Calabi-Yau 3 fold as a crepant resoution of the quotient

space $M/G$ . The original quotient space has 27 fixed points which are isolated
sigularities of type $1/3(1,1,1)$ . And the Euler number and also orbifold Euler
number are 27.
(3) $n=4$ Some of these singularities are canonical and terminal, then they don’t’

have any crepant resolutions. Moreover if the singularity has two crepantmorphism, the topological type of the terminal singularities are not same ingeneral. And we have no classification of these subgroups.
(4)

$n$

general(i) The action of the group is diagonal as follows$(x_{1}, x_{2}, \cdots , x_{n})arrow(\epsilon x_{1}, \epsilon x2, \cdots , \epsilon x_{n})$where$\epsilon$

is n-th root of unity. Then we have a crepant resolution and theunique exceptional divisor is isomorphic to(ii) $X=\mathbb{C}^{2n}/S_{n}$ where$S_{n}$$\mathrm{P}^{n-1}$

and the conjecture also holds.

is symmetric group, that is, n-th symmetric. The crepant resolution is obtained by Hilbert-Chow morphism
and it is Hilbert scheme of points on . We will see these fact in the next

product of$\mathbb{C}^{2}$$\mathbb{C}^{2}$$n$section. And the conjecture is true for them.[G\"o](5) If you assume the existence of the crepant, the conjecture holds$\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}_{\dot{\mathrm{O}}\mathrm{n}}$for any case. It was proved by Batyrev and Dais [BD] in the case of abeliangroups and by Batyrev and Kontsevich in general.

\S 3. Canonical resolution.
In the case $\dim X=2$ , we can construct minimal resolution without classificationof the finite subgroups of$SL(2, \mathbb{C})$. First, we recall some properties of Hilbert schemeof -points on the smooth projective surface
$n$

$S$.

The Hilbert scheme Hilb $n(S)$ is a projective scheme parametrizing O-dimensional
subschemes of length
Fact 1.

$[F_{\mathit{0}}]$

If

$S$

$n$

of$S$.

is smooth projective surface, then the Hilbert-Chow morphism$\pi$: $H_{\dot{i}}lb^{N}(S)arrow Symm^{n}s$5

is a resolution of singularities.

From this fact we can see the Hilbert scheme is smooth and irreducible.
Fact 2. [$Fu(n=\mathit{2})\mathit{1}lBe\mathit{1}$

The Hilbert-Chow morphism is a crepant resolution.

Using these fact, we will get the following theorem:
Theorem. [IN] If the group
is$n$

$G$

is the finite subgroup of

and the order of$G$, then$\phi$:$H_{\dot{i}}lb^{c_{(}}\mathbb{C}^{2}$) $arrow Symm^{n}(\mathbb{C}^{2})c$

is a minimal resolution of rational double point, wheremensional irredusible component of$\mathbb{C}^{2}$$SL(2, \mathbb{C})$dominating$\mathbb{C}^{2}/G$$G$$H_{\dot{i}}lb^{G}(\mathbb{C}^{2})$is unique two

fixed part of the Hilbert scheme of$n$$d_{\dot{i}-}$points on.

For the proof of this theorem, we have to consider the restriction to the G-fixedpart of the Hilbert-Chow morphism. If we take care of the holomophic symplectic
form, then we obtain the result.

\S 4. Recent progress.
The construction of the minimal resolution with Hilbert scheme does not dependon the classifiation of the finite subgroups in$SL(2, \mathbb{C})$. So we can obtain it canonically.

If we can do same things in higher dimension, we will be very happy, but it is not soeasy because the Hilbert scheme of -points on$n$$\mathbb{C}^{n}$

is not smooth in general.

In spite of this difficulty, Nakamura proved that we can construct a crepant resolution with Hilbert scheme if the group$G$

is abelian in$SL(3, \mathbb{C})$[Na] [Re2].

On the other hand, we have another construction of minimal resolution by Kronheinler [Kr] and it is releted with the construction with Hilbert scheme in the senceof Kronheimer and Nakajima [KN]. Moreover we have a result by Sardo-Infirri [Sa].

which is 3 dimensional generalization of the construction by Kronheimer.
Recently Nakajima and the author showed that there is a similar description as[KN] for 3 dimensional case and which coincides with the result of [Sa] with a particular parameter. And they will also show you 3 dimensional
The paper is in preparation now.

$\mathrm{M}\mathrm{c}\mathrm{K}\mathrm{a}\mathrm{y}$correspondence.

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quivers, University of Oxford, Thesis (1994).
$\mathbb{C}^{3}/H_{168}$

$G$

$G$

$\mathrm{a}$–,Currently : Lehrstuhl f\"ur Mathematik VI,Universit\"at Mannheim,68131 Mannheim, Germany$E$ -mail address:
yulcari@math.metro-u.ac.jp