数理解析研究所講究録
第 1897 巻 2014 年 21-30

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MODULI OF FANO VARIETIESVIA KAHLER-EINSTEIN METRICS
YUJI ODAKA

ABSTRACT. In this short notes, we give an introduction to ourrecent approach to moduli of Fano manifolds via K\"ahler-Einsteingeometry.

CONTENTS
1.
2.

Introduction
Algebro-geometric history of moduli of varieties
3. Some metric geometry
3.1. Hausdorff distance3.2. Gromov-Hausdorff distance3.3. Pointed Gromov-Hausdorff limits
3.4. Collapse and non-collapse
4. Moduli of Fano varieties

References

1
2

4
4
4
5
6

7
8

1. INTRODUCTION
The purpose of this survey is to explain recent progresses on moduli
of Fano varieties on special focus on the relation with K\"ahler-Einsteingeometry and stability. It benefits by the recent equivalence theorem of-stability and K\"ahler-Einstein metric existence [CDSI], [Tia2]. Personally speaking, the study was originally motivated by [Odl], [Od3]on the study of general type case and differential geometric backgroundsuch as [Fuj], [FS], [Donl], [Don3] for smooth case and the result on
Gromov-Hausdorff limit [DS].
Regarding general theory of constructing moduli of varieties, after the ground-breaking work of Mumford [GIT] resolving the caseof smooth projective curves, as well as his introduction of the abstract
method to attack those construction problems “Geometric Invariant
Theory”, there has been a tons of great efforts poured to get extension$K$

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YUJI ODAKA

to higher dimensional case as well as singular cases, especially to get
compactification of the moduli.
In the next section, we detail a little more those historical development of moduli of varieties from algebro-geometric perspective, withpossible biases due to my personal perspective.

2. ALGEBRO-GEOMETRIC HISTORY OF MODULI OF VARIETIES

After [GIT] constructed the moduli of mooth projective curves, thefamous [DM] established its geometric compactification as a moduli
of nodal curves with canonical ample class-nowadays called (DeligneMumford) stable curves. Note that paper’s main construction was as analgebraic stack which is not technically hard from modern perspective.
(Indeed their main result is not the construction but the algebraic proofof irreducibility of the moduli as well as its properness, and the notionof algebraic stack they introduced is of course of great importance. )turned out to have a coarse moduli
After a while, the moduli stack
which is projective $[KnM]$ , [Mum2], [Gie2]. The proofs ofvarietyprojectivity were via Geometric Invariant Theory again. (The more
direct proof was recently provided by Li-Wang [LW].
Higher dimensional generalizations of the above story has been facing many difficulties and not fully settled yet. The GIT stability ofcanonical models in 2-dimensional case was proved by Gieseker [Giel].
However those surfaces do not have compact moduli so to get compactification one needs to discuss what was the right degenerations to put
on the boundaries. Shepherd-Barron observed that we can put somesuitable degenerations on the boundary via birational geometric methods while they are not necessarily asymptotically GIT stable. Thisbirational geometric idea was systematically studied and put in moregeneral context by Koll\’ar-Shepherd-Barron [KSB] and later by Alex$\mathcal{S}$$\overline{\mathcal{M}}_{g}$$\overline{M}_{g}$$)$eev in higher dimensions [All]. Nowadays this construction of moduli
of “general type” (more precisely of semi-log-canonical varieties withcanonical ample classes) is often being called KSBA (Koll\’ar-ShepherdBarron-Alexeev) construction. The construction benefits recent breakthrough [BCHM] of the Minimal Model Program and its some continuations.
Unfortunately the idea of KSBA works only for varieties whose
is ample (which is in particular of “general type” ifcanonical
the variety is smooth). On some class of variety $X$ we have natural boundary divisor $D$ (on $X$ ) and putting that boundary to form apair $(X, D)$ , we regard the pair as having ample $(\log!)$ -canonical class$clas\mathcal{S}$$K_{X}+D$ so that we can apply KSBA’s idea.

The works of Alexeev

23
MODULI OF FANO VARIETIESVIA K\"AHLER-EINSTEIN METRICS[A12], Hacking-Keel-Tevelev [HKT], Laza [Laz] follow this line. However, in general, those boundaries $D$ are additional informations (i.e.
there is no canonical choice of such boundaries) so that it does not givea desired compactification, as even raising the dimension of the originalmoduli and changing the problem.
So it would be wonderful if we get some generalization for otherclass of varieties (without putting boundaries as additional information), which is the main point of my idea of using stability notions
coming from differential geometric quest of canonical K\"ahler metrics
such as K\"ahler-Einstein metrics or constant scalar curvature K\"ahler
metrics. Indeed, the following theorem holds.
Theorem 2.1 ([Odl],[Od3]). For projective variety $X$ with -Cartier, the followings are equivalent.
$X$ is KSBA
tablei. $X$ has only semi-log-canonical singularities.
$(X, K_{X})$ is $K$-stable.
$(X, K_{X})$ is $K$-semistable.
$\mathbb{Q}$$K_{X}$$\bullet$$e.$$\mathcal{S}$$\bullet$$\bullet$

Recall that the -stability [Tial],[Don2] is a closely related version of
the classical GIT stability notion for projective varieties [GIT], [Mum2].
Their motivation was to formulate the following conjecture:
Conjecture 2.2 (Yau-Tian-Donaldson conjecture). For a polarized
projective manifold $(X, L)_{Z}$ the existence of constant scalar curvature K\"ahler metric in the K\"ahler classis equivalent to the $K$$(poly)$ stability of $(X, L)$ .
About the theorem 2.1, note the original stability works for one sideof the claim “semistable implies semi-log-canonicity” [Odl]. The mainidea of the proofs of Theorem 2.1 is to use the MMP or basic discrepancy arguments to test configurations (which encodes one parameter
subgroups) and see the behaviour of Donaldson-Futaki invariants after (equivariant) Riemann-Roch type formula for the invariants. The
above theorem 2.1 is recently linked back to differential geomery byBerman-Guenancia [BG] as follows.
Theorem 2.3 (Berman-Guenancia [BG]). In the same setting as inequivalent to the following metric existence
Theorem 2.1, they are
condition.
$(*)$ There is a K\"ahler-Einstein metric onwhich extends as acurrent to $X$ and volume
This can be seen as another case of the Yau-Tian-Donaldson correspondence. More details on the general conjectural pictures and backgroundrelated to moduli of -stable varieties can be found in [Od4].
$K$$c_{1}(L)$$al_{\mathcal{S}}o$$X^{reg}$$(K_{X})^{n}$

$K$

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YUJI ODAKA

3. SOME METRIC GEOMETRYThis section is some crash course on metric geometry to help somealgebraic readers to understand the rest of the article better, and obviously most of the contents are just classical and basically copied fromstandard references.
The point of bringing metric perspective into moduli of varieties is,giving that additional structures, we naturally get Hausdorff property
of moduli as well as the canonical limits as Gromov-Hausdorff limits
or its versions.
Note that the word “metric” can mean two things-either Riemannian metrics on manifolds or the distance structure on general sets or
topological spaces. We refer the details to the textbook [BBI] on which
my understanding also heavily rely.

3.1. Hausdorff distance. Given an ambient metric space $M$ and two
subsets (with induced metrics) $X,$ $Y\subset M$ , we define the Hausdorff
distance $d_{H}(X, Y;M)$ as$\inf\{r>0|\forall x\in X, \exists y\in Y, d(x, y)<r, \forall y\in Y, \exists x\in X, d(x, y)<r\}.$If $M$ is obvious from the context, we omit it. This gives a pseudo
distance between subsets of $M.$
Example 3.1. Consider a subset $Z\in M$ which satisfies $d_{H}(Z, M;M)<$. Then $Z$ is called -net. If $M$ is a Riemannian manifolds, we can see
as a discrete subset.
, we can take -net
that for any$\epsilon$$\epsilon$$\epsilon>0$

Example 3.2.
of in $M.$$\epsilon$$d_{H}(S,\overline{S})=0$$Z_{\epsilon}$

for any $S\subset M$ , where$\overline{S}$denotes the closure$S$

Set. $C(X)$$:=${closed subsets ofThen the followings are known.

$X$} with Hausdorff distance$d_{H}.$(1) $C(X)$ is complete if so is $X.$
Theorem 3.3.
(ii) $C(X)$ is compact if so is $X$ (Blaschke).

3.2. Gromov-Hausdorff distance. We now pass the distance structure among subsets of given metric space to abstract metric spaces
after the idea of M. Gromov.
are both metric spaces.
Definition 3.4. Let $X,$
Hausdorff distance of those two are defined as:$Y$$d_{GH}(X, Y)$ $:= \inf$

The Gromov-{ $d_{H}(X, Y;M)|X,$ $Y\subset Mwith$ distances preserved.}.

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MODULI OF FANO VARIETIES

Using this notion, we can see the moduli of compact Riemannian
manifolds are, in particular, Hausdorff (see also [Ebi]).
About the (pre)compactness of set of metric spaces, there is a famous
useful criterion due to Gromov. Let us prepare the following basic
definition.

Definition 3.5.
is said to beclass of compact metric spaces
uniformly totally bounded if it satisfies the following two conditions.
(i) $\exists D>0$ such that diam(X)for any(ii)such thathas -net of cardinality.
less than$A$$\mathcal{X}$$\leq D$$\forall\epsilon>0,$$X\in \mathcal{X}.$$X\in \mathcal{X}$$\exists N(\epsilon)\in \mathbb{Z}_{>0}$$\epsilon$$N(\epsilon)$

Then the Gromov precompactness theorem is then this.
Theorem 3.6 (Gromov). Any uniformly totally bounded class of compact metric spaces form precompact moduli via Gromov-Hausdorff dis-

tance.

The core idea of the known proof is, given a sequencein , toregard eachas a limit of -net where tends to zero. Then we
cook up the limit as some quotient of a set of certain sequences ofA famous example is the set of compact Riemannian -dimensional
manifolds whose Ricci curvature is at least $(n-1)k$ and diameter at
most $D$ , for so me fixed $D>0$ . It is fairly riontrivial to confirm thisdoes work as an example but it can be proved by constructing maps
from each metric space to the space form of curvature . We apply thisin combination with Myers theorem for Kahler-Einstein Fanos, whichasserts the boundedness of their diameters later.
$\{X_{i}\}_{i}$$X_{i}\in \mathcal{X}$$\epsilon$$\mathcal{X}$$\epsilon$$X_{i}.$

$n$

$k,$

$k$

3.3. Pointed Gromov-Hausdorff limits. For non-compact spaces,.
we need some modification of the notion because there is certain examples of sequences of compact metric spaces whose diameter diverges
even though it “converges” to a metric space with infinite diameter at
least intuitively. Here are some examples:
Example 3.7. Consider the intervals $[-a, a]$ with $a>0$ . If goes toinfinity, this should be regarded as converging to the whole line$a$$\mathbb{R}.$

Example 3.8. Recall that (Deligne-Mumford) stable curve which is not
smooth also has hyperbolic metric like the case of smooth hyperbolic
curve. As it is algebro-geometrically alimit of smooth hyperbolic curve,
it should be regarded as a “limit” of those smooth hyperbolic curves(with the hyperbolic metrics while preserving the curvature).
To justify these convergence, the following notion is useful.

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YUJI ODAKA

sequence of metric spaces with base points
Definition 3.9.
as the pointed Gromov-Hausdorff limit if forhasin the usual Gromovconverges to
all $r>0$ , the balls
Hausdorff sense.
$A$$p_{i}(i\in \mathbb{Z}_{>0})$$X_{i}\ni$$X_{\infty}\ni p_{\infty}$$B_{r}(p_{\infty})$$B_{r}(p_{i})$Then we expect the following, in relation with Koll\’ar-ShepherdBarron-Alexeev stable varieties.
Example 3.10. Recall that very recently Berman-Guenancia [BG] constructed singular K\"ahler-Einstein metrics on KSBA stable varieties ex-

tending the example 3.8.
Consider the -Gorenstein deformation (flat family) of KSBA stablevarieties $f:\mathcal{X}arrow C$ over a curve with a section $s:Carrow \mathcal{X}$ which
does not intersect with singularities of fibers. Then we expect thatthe pointed Gromov-Hausdorff limit of the singular K\"ahler-Einstein, while $p_{i}arrow p\in C$ and preserving the volume, is just
metrics on
again.
their singular K\"ahler-Einstein metric on$\mathbb{Q}$$\mathcal{X}|_{p_{i}}$$\mathcal{X}_{p}$3.4. Collapse and non-collapse. An interesting feature of GromovHausdorff limit is that sometimes the dimension can jump down. Consider the following example.
$ti>$ with usual
Example 3.11. If we consider elliptic curve. Instead, if
Euclidean metric where $tarrow 0$ converges to a circle$t^{-1}i>$ with $tarrow 0$ , then it converges to a line
as a
we takepointed Gromov-Hausdorff limit.
$\mathbb{C}/<1,$$\mathbb{R}/\mathbb{Z}$$\mathbb{R}$$\mathbb{C}/<t,$These convergence with dimensions jumping down is called coldoes not collapse if
lapse. Pointed Riemannian manifolds$vol(B_{r}(p_{i}))>c\cdot r^{n}$ for uniform $c>0$ where
and ranges all over.
We end the section by seeing some K\"ahler-Einstein situations wherecollapse never occurs. For that we need to recall the following two(even more) classical results:$M_{i}\ni p_{i}$
$i$

$r$

Theorem 3.12 (Myers). If a compact -dimensional Riemannian
uniform
manifold $M$ has Ricci curvature at least $(n-1)k$ withconstant $k>0$ , then the diameter is at most$n$$\mathcal{S}ome$$\pi/\sqrt{k}.$Theorem 3.13 (Bishop-Gromov inequality). Suppose a pointed Riemannian manifolds $M\ni p$ has Ricci curvature at least $(n-1)k$ with$:=vol(B_{r})$ be the volume of radiusconstant $k>0$ . And letball in the space form with Ricci curvature $(n-1)k.$Then the ratio $vol(B_{r}(p))/V_{r}(k)$ with does not increases when in$\mathcal{S}ome$$V_{r}(k)$

$r$

$r$

$crea\mathcal{S}e\mathcal{S}.$Following these two theorems, we see easily that

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MODULI OF FANO VARIETIESVIA K\"AHLER-EINSTEIN METRICS

Proposition 3.14. Any sequence of K\"ahler-Einstein Fano -manifolds
with volume preserved has non-collapsed Gromov-Hausdorff limit.
$n$Indeed the diameter should be bounded above by Myers theorem sothat we can apply the Gromov precompactness theorem to show wehave a Gromov-Hausdorff limit. Furthermore, by the Bishop-Gromov
inequality it cannot collapse.
We can show the same thing for a sequence of Ricci flat manifolds if
the diameter converges to a positive finite number. It is the case when
it is a sequence of smooth Calabi-Yau varieties degenerating to aterminal Calabi-Yau variety (see Tosatti [Tos]).
$\log$

4. MODULI OF FANO VARIETIES
Finishing the preparatory background in the previous section, now
we can discuss the main issue. As we wrote in the end of the section 2,
the KSBA projective moduli of semi-log-canonical varieties with ample
canonical classes turned out to correspond to -stability [Odl], [Od3].
Then a natural question is to extend the moduli of more general
stable varieties as we first discussed in [Od2].
Meanwhile the following result was proved.
$K$

$K$ -

Theorem 4.1 (Donaldson-Sun [DS]). Gromov-Hausdorff limits ofK\"ahler-Einstein Fano -dimensional manifolds are -dimensional
Fano varieties $(i.e$ . singular Fano varieties with only $log$ terminal singularities) with singular K\"ahler-Einstein metrics.
$n$

$n$

$\mathbb{Q}-$The key idea is to show lower boundedness of (asymptotic) Bergman
kernel-called “partial
estimates” -to give identifying map between
the Gromov-Hausdorff limit and a limit in Hilbert scheme. Originallythe theorem was proved for the existence problem of K\"ahler-Einstein
metrics-to construct a destabilizing test configuration of Fano manifolds without K\"ahler-Einstein metrics. Indeed the extension to conicalsingular situation is the key to [CDSI] and [Tia2]. Nevertheless, thetheorem implied some possible application to moduli as well whichmatches to the author’s algebraic results mentioned above.
Motivated by two of these aspects, the followings are proved.
$C^{0}$

Theorem 4.2 ([Od4]). K\"ahler-Einstein Fano manifolds with discrete
automorphism groups form Hausdorff moduli algebraic space. Furthermore, it is an orbifold $i.e$ . has only quotient singularities.

We expect that putting Gromov-Hausdorff limits on infinity, we getprojective geometric compactification and this is the moduli of(semi)stable Fano varieties ( -moduli”). In del Pezzo surfaces case,
$K$ -

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YUJI ODAKA

in [OSS], we have described the moduli and degenerations explicitly
whose rough statement is as follows. Mabuchi-Mukai [MM] pioneered
this direction.

Theorem 4.3 ([OSS]). The Gromov-Hausdorff compactification ofmoduli of $del$ Pezzo surfaces are some algebro-geometric (explicit, \’etale
locally $GIT$) compactification which is at least compact algebraic space.
(Other than degree 1 case, we confirmed projectivity as well.)We note that the Gromov-Hausdorff convergence we consider in theabove theorem is in a slightly different sense since we also concern thecontinuity of complex structures as well, whose definition is naturally
done in [DS] (also we recommend [Sp ). We consult [OSS] for more
precise statement of Theorem 4.3 which includes formulation at thestacky level, as well as the explicit construction of the moduli in thedel Pezzo case. Another reference we can refer to is [Od4] whose lasttwo sections gives a review of general conjecture picture (not sticking
to Fano case) as well as partial results on canonical limits.
$0]$

Acknowledgments. The author thanks Professor Daisuke Matsushita
very much for inviting him to the conference “Recent progress on Fano
manifolds” in Kyoto to give the opportunity as well as for his organizing. This notes is a proceeding for the conference.

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$K$$\log$$K$$K$$\mathbb{Q}$$K$

DEPARTMENT OF MATHEMATICS, KYOTO UNlVERSITY, OIWAKE-CHO, KlTASHIRAKAWA, SAKYO-KU, KYOTO CITY, KYOTO, 606-8285, JAPAN-mail address: yodaka@math.kyoto-u.ac.jp$E$