Pull-back of quasi-log structures

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by  Osamu Fujino

PULL-BACK OF QUASI-LOG STRUCTURES
OSAMU FUJINO
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Abstract. Oneofthemainpurposesofthispaperistomakethe
n
a theory of quasi-log schemes more flexible and more useful. More
J precisely, we prove that the pull-back of a quasi-log scheme by a

8 smooth quasi-projective morphism has a natural quasi-log struc-
ture after clarifying the definition of quasi-log schemes. We treat
]
G some applications to singular Fano varieties. This paper also con-
tains a proof of the simple connectedness of log canonical Fano
A
varieties.
.
h
t
a Contents
m
[ 1. Introduction 1

2. Preliminaries 3
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v 3. Pull-back of quasi-log structures 6
4
4. On quasi-log structures 9
5
8 5. Proof of the main theorem 15
1 6. Applications to quasi-log canonical Fano varieties 16
.
1 7. Simple connectedness of log canonical Fano varieties 19
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8. Appendix: Ambro’s original definition 20
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1 References 24
:
v
i
X
r 1. Introduction
a
In [A], the notion of quasi-log structures was introduced in order
to prove the cone and contraction theorem for (X,∆) where X is a
normal variety and ∆ is an effective R-divisor such that K + ∆ is
X
R-Cartier. Although the theory of quasi-log schemes is very powerful
and useful, it may look much harder than the usual X-method for
kawamata log terminal pairs. Moreover, the paper [F4] recovers the
main theorem of [A] without using the notion of quasi-log structures.
So the theory of quasi-log schemes is not yet popular. We note that the
Date: 2014/1/8,version 1.02.
2010 Mathematics Subject Classification. Primary 14E30;Secondary 14J45.
Keywordsandphrases. quasi-logstructures,Fanovarieties,fundamentalgroups.
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2 OSAMU FUJINO
framework of [F4] is more similar to the theory of algebraic multiplier
ideal sheaves than to the traditional X-method. Recently, the author
proved that every quasi-projective semi log canonical pair has anatural
quasi-log structure in [F7]. The theory of quasi-log schemes seems
to be indispensable for the study of semi log canonical pairs. Now
the importance of quasi-log structures is increasing. One of the main
purposes of this paper is to clarify the definition of quasi-log structures
andmakethetheoryofquasi-logschemes moreflexibleandmoreuseful.
The following theorem is the main theorem of this paper, which is
natural but missing in the literature. For the precise statement, see
Theorem 3.11 below.
Theorem 1.1 (Pull-backofquasi-logstructures). Let [X,ω]be a quasi-
log scheme and let h : X′ → X be a smooth quasi-projective morphism.
Then [X′,ω′], where ω′ = h∗ω ⊗ωX′/X with ωX′/X = detΩ1X′/X, has a
natural quasi-log structure induced by h.
Theorem 1.1 does not directly follow from the original definition
of quasi-log schemes. We have to construct a quasi-log resolution of
[X′,ω′] suitably. We make an important remark. We do not know
whether Theorem 1.1 holds true or not without assuming that h is
quasi-projective. As a useful special case of Theorem 1.1, we have:
Theorem 1.2 (Finite ´etale covers). Let [X,ω] be a quasi-log scheme
and let h : X′ → X be a finite ´etale morphism. Then [X′,ω′], where
ω′ = h∗ω, has a natural quasi-log structure induced by h.
As an easy application of Theorem 1.2 to singular Fano varieties, we
obtain:
Corollary 1.3. Let [X,ω] be a projective quasi-log canonical pair such
that −ω is ample, that is, [X,ω] is a quasi-log canonical Fano variety.
Then the algebraic fundamental group of X is trivial, equivalently, X
has no non-trivial finite ´etale covers.
By Corollary 1.3, it is natural to conjecture:
Conjecture 1.4. Let [X,ω] be a projective quasi-log canonical pair
such that −ω is ample. Then X is simply connected.
In general, there exists an irreducible projective variety whose alge-
braic fundamental group is trivial and whose topological fundamental
group is non-trivial (see Example 6.4). As a special case of Conjecture
1.4, we have:
Conjecture 1.5. Let (X,∆) be a projective semi log canonical pair
such that −(K +∆) is ample, that is, (X,∆) is a semi log canonical
X
Fano variety. Then X is simply connected.
PULL-BACK OF QUASI-LOG STRUCTURES 3
For the details of semi log canonical pairs, see [F7]. Note that every
quasi-projective semi log canonical pair has a natural quasi-log struc-
ture with only quasi-log canonical singularities. It is the main theorem
of [F7].
It is well known that Conjecture 1.5 holds when (X,∆) is kawamata
log terminal (see [T]). Kento Fujita pointed out that Conjecture 1.5
holds true when (X,∆) is log canonical (see Theorem 7.1 below). We
give Fujita’s proof in Section 7 for the reader’s convenience.
We summarize the contents of this paper. Section 2 collects some
basic definitions and results. In Section 3, we recall the definition of
quasi-log schemes and state the main theorem of this paper precisely
(see Theorem 3.11). Section 4 is the main part of this paper. Here we
discuss the basic properties of quasi-log schemes. The author believes
that Section 4 makes the theory of quasi-log schemes more flexible and
more useful than Ambro’s original framework in [A]. Section 5 is de-
voted to the proof of the main theorem (see Theorem 1.1 and Theorem
3.11). In Section 6, we treat some applications of the main theorem
to singular Fano varieties. We prove that the algebraic fundamental
group of a quasi-log canonical Fano variety is always trivial (see Corol-
lary 1.3). In Section 7, we prove that a log canonical Fano variety is
simply connected (see Theorem 7.1). The proof of Theorem 7.1 is inde-
pendent of the theory of quasi-log schemes. Section 8 is an appendix,
where we discuss Ambro’s original definition of quasi-log schemes.
Acknowledgments. TheauthorwaspartiallysupportedbytheGran-
in-Aid for Young Scientists (A) ♯24684002 from JSPS. He would like
to thank Takeshi Abe, Kento Fujita, Yuichiro Hoshi, and Tetsushi Ito
for answering his questions and giving him useful comments.
We will work over C, the complex number field, throughout this
paper. For the standard notation of the log minimal model program,
see, for example, [F4]. For the basic properties and results of semi log
canonical pairs, see [F7].
2. Preliminaries
In this section, we collect some basic results and definitions.
Notation 2.1. A pair [X,ω] consists of a scheme X and an R-Cartier
R-divisor (or R-line bundle) ω on X. In this paper, a scheme means a
separated scheme of finite type over SpecC.
Notation 2.2 (Divisors). Let B and B be two R-Cartier R-divisors
1 2
on a scheme X. Then B is linearly (resp. Q-linearly, or R-linearly)
1
4 OSAMU FUJINO
equivalent to B , denoted by B ∼ B (resp. B ∼ B , or B ∼ B )
2 1 2 1 Q 2 1 R 2
if
k
B1 = B2 +Xri(fi)
i=1
such that f ∈ Γ(X,K∗ ) and r ∈ Z (resp. r ∈ Q, or r ∈ R) for every
i X i i i
i. Here, K is the sheaf of total quotient rings of O and K∗ is the
X X X
sheaf of invertible elements in the sheaf of rings K . We note that
X
(f ) is a principal Cartier divisor associated to f , that is, the image of
i i
f by Γ(X,K∗ ) → Γ(X,K∗ /O∗ ), where O∗ is the sheaf of invertible
i X X X X
elements in O .
X
Let D be a Q-divisor (resp. an R-divisor) on an equi-dimensional
variety X, that is, D is a finite formal Q-linear (resp. R-linear) combi-
nation
D = XdiDi
i
of irreducible reduced subschemes D of codimension one. We define
i
the round-up ⌈D⌉ = ⌈d ⌉D (resp. round-down ⌊D⌋ = ⌊d ⌋D ),
Pi i i Pi i i
where every real number x, ⌈x⌉ (resp. ⌊x⌋) is the integer defined by
x ≤ ⌈x⌉ < x+1 (resp. x−1 < ⌊x⌋ ≤ x). The fractional part {D} of
D denotes D−⌊D⌋. We put
D<1 = XdiDi, D≤1 = XdiDi, and D=1 = XDi.
di<1 di≤1 di=1
We can define D≥1, D>1, and so on, analogously. We call D a boundary
(resp. subboundary) R-divisor if 0 ≤ d ≤ 1 (resp. d ≤ 1) for every i.
i i
Notation 2.3 (Singularities of pairs). Let X be a normal variety and
let ∆ be an R-divisor on X such that K + ∆ is R-Cartier. Let f :
X
Y → X be a resolution such that Exc(f)∪f−1∆, where Exc(f) is the
∗
exceptional locus of f and f−1∆ is the strict transform of ∆ on Y, has
∗
a simple normal crossing support. We can write
KY = f∗(KX +∆)+XaiEi.
i
We say that (X,∆) is sub log canonical (sub lc, for short) if a ≥ −1 for
i
every i. We usually write a = a(E ,X,∆) and call it the discrepancy
i i
coefficient of E with respect to (X,∆). It is well known that there
i
exists the largest Zariski open set U of X such that (U,∆| ) is sub log
U
canonical. If there exist a resolution f : Y → X and a divisor E on Y
such that a(E,X,∆) = −1 and f(E) ∩ U 6= ∅, then f(E) is called a
log canonical center (an lc center, for short) with respect to (X,∆). If
PULL-BACK OF QUASI-LOG STRUCTURES 5
(X,∆) is sub log canonical and ∆ is effective, then (X,∆) is called log
canonical (lc, for short).
We note that we can define a(E ,X,∆) in more general settings (see
i
[K2, Definition 2.4]).
Let us recall the definition of simple normal crossing pairs.
Definition 2.4 (Simple normal crossing pairs). We say that the pair
(X,D) is simple normal crossing at a point a ∈ X if X has a Zariski
open neighborhood U of a that can be embedded in a smooth variety
Y, where Y has regular system of parameters (x ,··· ,x ,y ,··· ,y )
1 p 1 r
at a = 0 in which U is defined by a monomial equation
x ···x = 0
1 p
and
r
D = Xαi(yi = 0)|U, αi ∈ R.
i=1
We say that (X,D) is a simple normal crossing pair if it is simple
normal crossing at every point of X. We say that a simple normal
crossing pair (X,D) is embedded if there exists a closed embedding
ι : X → M, where M is a smooth variety of dimX +1. We call M the
ambient spaceof(X,D). If(X,0) isa simple normal crossing pair, then
X is called a simple normal crossing variety. If X is a simple normal
crossing variety, then X has only Gorenstein singularities. Thus, it has
aninvertible dualizing sheafω . Therefore, we candefinethecanonical
X
divisor K such that ω ≃ O (K ). It is a Cartier divisor on X and
X X X X
is well-defined up to linear equivalence.
Let X be a simple normal crossing variety and let X = X be
Si∈I i
the irreducible decomposition of X. A stratum of X is an irreducible
component of X ∩···∩X for some {i ,··· ,i } ⊂ I.
i1 ik 1 k
Let X be a simple normal crossing variety and let D be a Cartier
divisor on X. If (X,D) is a simple normal crossing pair and D is
reduced, then D is called a simple normal crossing divisor on X.
Let (X,D) be a simple normal crossing pair. Let ν : Xν → X be
the normalization. We define Θ by the formula
KXν +Θ = ν∗(KX +D).
Then a stratum of (X,D) is an irreducible component of X or the ν-
image of a log canonical center of (Xν,Θ) (see Notation 2.3). When
D = 0, this definition is compatible with the above definition of the
strataofX. WhenD isaboundaryR-divisor, W isastratumof(X,D)
if and only if W is an slc stratum of (X,D) (see [F7, Definition 2.5]).
Note that (X,D) is semi log canonical if D is a boundary R-divisor.
6 OSAMU FUJINO
Notation 2.5. π (X) denotes the topological fundamental group of
1
X.
3. Pull-back of quasi-log structures
In this section, we give a precise statement of Theorem 1.1 (see
Theorem 3.11). First, let us recall the definition of globally embedded
simple normal crossing pairs in order to define quasi-log schemes.
Definition 3.1 (Globallyembedded simplenormalcrossing pairs). Let
Y be a simple normal crossing divisor on a smooth variety M and let
D be an R-divisor on M such that Supp(D + Y) is a simple normal
crossing divisor on M and that D and Y have no common irreducible
components. We put B = D| and consider the pair (Y,B ). We
Y Y Y
call (Y,B ) a globally embedded simple normal crossing pair and M
Y
the ambient space of (Y,B ).
Y
It is obvious that a globally embedded simple normal crossing pair
is an embedded simple normal crossing pair in Definition 2.4.
Let us define quasi-log schemes (see also Definition 8.2 below).
Definition 3.2 (Quasi-log schemes). A quasi-log scheme is a scheme
X endowed with an R-Cartier R-divisor (or R-line bundle) ω on X,
a proper closed subscheme X ⊂ X, and a finite collection {C} of
−∞
reduced and irreducible subschemes of X such that there is a proper
morphism f : (Y,B ) → X from a globally embedded simple normal
Y
crossing pair satisfying the following properties:
(1) f∗ω ∼ K +B .
R Y Y
(2) The natural map O → f O (⌈−(B<1)⌉) induces an isomor-
X ∗ Y Y
phism
I −≃→ f O (⌈−(B<1)⌉−⌊B>1⌋),
X−∞ ∗ Y Y Y
where I is the defining ideal sheaf of X .
X−∞ −∞
(3) The collection of subvarieties {C} coincides with the image of
(Y,B )-strata that are not included in X .
Y −∞
We simply write [X,ω] to denote the above data
X,ω,f : (Y,B ) → X
(cid:0) Y (cid:1)
if there is no risk of confusion. Note that a quasi-log scheme X is the
union of {C} and X . We also note that ω is called the quasi-log
−∞
canonical class of [X,ω], which is defined up to R-linear equivalence.
A relative quasi-log scheme X/S is a quasi-log scheme X endowed with
a proper morphism π : X → S.
PULL-BACK OF QUASI-LOG STRUCTURES 7
Remark 3.3. Let Div(Y) be the group of Cartier divisors on Y and
let Pic(Y) be the Picard group of Y. Let
δ : Div(Y)⊗R → Pic(Y)⊗R
Y
be the homomorphism induced by A 7→ O (A) where A is a Cartier
Y
divisor on Y. When ω is an R-line bundle in Definition 3.2,
f∗ω ∼ K +B
R Y Y
means
f∗ω = δ (K +B )
Y Y Y
in Pic(Y)⊗R. Even when ω is an R-line bundle, we use −ω to denote
the inverse of ω in Pic(X)⊗R (see Corollary 1.3 and Conjecture 1.4)
if there is no risk of confusion. If ω is an R-Cartier R-divisor on X in
Theorem 1.1,
h∗ω ⊗detΩ1
X′/X
means
δX′(h∗ω)⊗detΩ1X′/X
in Pic(X′)⊗R where δX′ : Div(X′)⊗R → Pic(X′)⊗R.
We give an important remark on Definition 3.2.
Remark 3.4 (Schemes versus varieties). A quasi-log scheme in Defini-
tion 3.2 is called a quasi-log variety in [A] (see also [F2]). However, X
is not always reduced when X 6= ∅ (see Example 3.5 below). There-
−∞
fore, we will use the word quasi-log schemes in this paper. Note that
X is reduced when X = ∅ (see Remark 3.10 below).
−∞
Example 3.5 ([A, Examples 4.3.4]). Let X be an effective Cartier
divisor on a smooth variety M. Assume that Y, the reduced part of
X, is non-empty. We put ω = (K +X)| . Let X be the union of
M X −∞
the non-reduced components of X. We put K +B = (K +X)| .
Y Y M Y
Let f : Y → X be the closed embedding. Then
X,ω,f : (Y,B ) → X
(cid:0) Y (cid:1)
is a quasi-log scheme. Note that X has non-reduced irreducible com-
ponents if X 6= ∅. We also note that f is not surjective if X 6= ∅.
−∞ −∞
Notation 3.6. In Definition 3.2, we sometimes simply say that [X,ω]
is a quasi-log pair. The subvarieties C are called the qlc centers of
[X,ω], X is called the non-qlc locus of [X,ω], and f : (Y,B ) → X
−∞ Y
is called a quasi-log resolution of [X,ω]. We sometimes use Nqlc(X,ω)
to denote X .
−∞
For various applications, the notion of qlc pairs is very useful.
8 OSAMU FUJINO
Definition 3.7 (Qlc pairs). Let [X,ω] be a quasi-log pair. We say
that [X,ω] has only quasi-log canonical singularities (qlc singularities,
for short) if X = ∅. Assume that [X,ω] is a quasi-log pair with
−∞
X = ∅. Then we simply say that [X,ω] is a qlc pair.
−∞
We give some important remarks on the non-qlc locus X .
−∞
Remark 3.8. Weput A = ⌈−(B<1)⌉andN = ⌊B>1⌋. Then we obtain
Y Y
the following big commutative diagram.
0 // f∗OY(A−N) // f∗OY(A) // f∗ON(A)
OO OO OO
α1 α2 α3
0 // f∗OY(−N) // f∗OY // f∗ON
OO OO OO
β1 β2 β3
0 // I // O // O // 0
X−∞ X X−∞
Note that α is a natural injection for every i. By an easy diagram
i
chasing,
≃
I −→ f O (A−N)
X−∞ ∗ Y
factors through f O (−N). Then we obtain β and β . Since α is
∗ Y 1 3 1
injective and α ◦β is an isomorphism, α and β are isomorphisms.
1 1 1 1
Therefore, we obtain that f(Y) ∩ X = f(N). Note that f is not
−∞
always surjective when X 6= ∅. It sometimes happens that X
−∞ −∞
contains some irreducible components of X. See, for example, Example
3.5.
Remark 3.9 (Semi-normality). By restricting the isomorphism
≃
I −→ f O (A−N)
X−∞ ∗ Y
to the open subset U = X \X , we obtain
−∞
≃
OU −→ f∗Of−1(U)(A).
This implies that
≃
OU −→ f∗Of−1(U)
because A is effective. Therefore, f : f−1(U) → U is surjective and has
connected fibers. Note that f−1(U) is a simple normal crossing variety.
Thus, U is semi-normal. In particular, U = X \X is reduced.
−∞
Remark 3.10. Ifthepair[X,ω]hasonlyqlcsingularities, equivalently,
X = ∅, then X is reduced and semi-normal by Remark 3.9. Note
−∞
that f(Y)∩X = ∅ if and only if B = B≤1, equivalently, B>1 = 0,
−∞ Y Y Y
by the descriptions in Remark 3.8.
PULL-BACK OF QUASI-LOG STRUCTURES 9
Let us state the main theorem of this paper precisely.
Theorem 3.11 (Main theorem). Let [X,ω] be a quasi-log pair as in
Definition 3.2. Let X′ be a scheme and let h : X′ → X be a smooth
quasi-projective morphism. Then [X′,ω′], where ω′ = h∗ω ⊗ ωX′/X
with ωX′/X = detΩ1X′/X, has a natural quasi-log structure induced by
h. More precisely, we have the following properties:
(i) (Non-qlclocus). There is a proper closed subscheme X′ ⊂ X′.
−∞
(ii) (Quasi-log resolution). There exists a proper morphism f′ :
(Y′,BY′) → X′ from a globally embedded simple normal cross-
ing pair (Y′,BY′) such that
f′∗ω′ ∼R KY′ +BY′,
and the natural map
OX′ → f∗′OY′(⌈−(BY<′1)⌉)
induces an isomorphism
IX−′∞ −≃→ f∗′OY′(⌈−(BY<′1)⌉−⌊BY>′1⌋)
where IX−′∞ is the defining ideal sheaf of X−′ ∞ and
IX−′∞ = h∗IX−∞.
(iii) (Qlc centers). There is a finite collection {C′} of reduced and
irreducible subschemes of X′ such that {C′} = {f−1(C)} and
that the collection of subvarieties {C′} coincides with the images
of (Y′,BY′)-strata that are not included in X−′ ∞.
Remark3.12. Forthedefinitionandbasicpropertiesofquasi-projective
morphisms, see [G, Chapitre II §5.3. Morphismes quasi-projectifs].
We will prove Theorem 3.11 in Section 5 after we prepare various
useful lemmas in Section 4.
We recommend the reader to see [F3] for some basic applications of
thetheoryofquasi-logschemes. The adjunctionandvanishing theorem
(see, for example, [F3, Theorem 3.6]) is a key result for qlc pairs.
4. On quasi-log structures
The following proposition makes the theory of quasi-log schemes
more flexible. It is a key result in this paper.
Proposition 4.1 ([F2, Proposition 3.50]). Let f : V → W be a proper
birational morphism between smooth varieties and let B be an R-
W
divisor on W such that SuppB is a simple normal crossing divisor
W
10 OSAMU FUJINO
on W. Assume that
K +B = f∗(K +B )
V V W W
and that SuppB is a simple normal crossing divisor on V. Then we
V
have
f O (⌈−(B<1)⌉−⌊B>1⌋) ≃ O (⌈−(B<1)⌉−⌊B>1⌋).
∗ V V V W W W
Furthermore, let S be a simple normal crossing divisor on W such that
S ⊂ SuppB=1. Let T be the union of the irreducible components of
W
B=1 that are mapped into S by f. Assume that Suppf−1B ∪Exc(f)
V ∗ W
is a simple normal crossing divisor on V. Then we have
f O (⌈−(B<1)⌉−⌊B>1⌋) ≃ O (⌈−(B<1)⌉−⌊B>1⌋),
∗ T T T S S S
where (K +B )| = K +B and (K +B )| = K +B .
V V T T T W W S S S
Proof. By K +B = f∗(K +B ), we obtain
V V W W
K =f∗(K +B=1 +{B })
V W W W
+f∗(⌊B<1⌋+⌊B>1⌋)−(⌊B<1⌋+⌊B>1⌋)−B=1 −{B }.
W W V V V V
If a(ν,W,B=1 + {B }) = −1 for a prime divisor ν over W, then we
W W
can check that a(ν,W,B ) = −1 by using [KM, Lemma 2.45]. Since
W
f∗(⌊B<1⌋+⌊B>1⌋)−(⌊B<1⌋+⌊B>1⌋)
W W V V
is Cartier, we can easily see that
f∗(⌊B<1⌋+⌊B>1⌋) = ⌊B<1⌋+⌊B>1⌋+E,
W W V V
where E is an effective f-exceptional divisor. Thus, we obtain
f O (⌈−(B<1)⌉−⌊B>1⌋) ≃ O (⌈−(B<1)⌉−⌊B>1⌋).
∗ V V V W W W
Next, we consider the short exact sequence:
0 → O (⌈−(B<1)⌉−⌊B>1⌋−T)
V V V
→ O (⌈−(B<1)⌉−⌊B>1⌋) → O (⌈−(B<1)⌉−⌊B>1⌋) → 0.
V V V T T T
Since T = f∗S −F, where F is an effective f-exceptional divisor, we
can easily see that
f O (⌈−(B<1)⌉−⌊B>1⌋−T) ≃ O (⌈−(B<1)⌉−⌊B>1⌋−S).
∗ V V V W W W
We note that
(⌈−(B<1)⌉−⌊B>1⌋−T)−(K +{B }+B=1 −T)
V V V V V
= −f∗(K +B ).
W W
Therefore, every associated prime of R1f O (⌈−(B<1)⌉−⌊B>1⌋−T) is
∗ V V V
thegenericpointofthef-imageofsomestratumof(V,{B }+B=1−T)
V V
(see, for example, [F4, Theorem 6.3 (i)]).