KAWAMATA LOG TERMINAL AND LOG CANONICAL PAIRS
The goal of this note is to introduce the basic definitions of klt and lc singularities.
Why are Kawamata log terminal and log canonical pairs important?
• In contrast to projective surfaces, the classification of higher dimensional
algebraic varieties cannot be accomplished within the category of smooth
varieties. We need to work with mildly singular varieties to construct minimal models.
• Even if one is interested only in proving statements for smooth varieties, one
still often constructs log canonical pairs as an intermediate step in various
proofs (see e.g. a remark about Angehrn-Siu theorem).
Definition 0.1. We call (X, ∆) a log pair, if X is normal and ∆ is an effective
Q-divisor such that KX + ∆ is Q-Cartier1.
Note that, for any birational morphism f : Y → X, there exists a unique Qdivisor ∆Y such that
KY + ∆Y = f ∗ (KX + ∆), and
f∗ ∆Y = ∆.
This follows from the fact that the relative canonial divisor KY /X is uniquely defined
as a Q-divisor2. We call ∆Y the log pull-back of ∆. Note that ∆Y need not be
effective.
We say that f : Y → X is a log resolution of (X, ∆) if f∗−1 Supp(∆) ∪ Exc(f ) is
a simple normal crossing divisor on a smooth variety Y .
Definition 0.2. We say that a log pair (X, ∆) is kawamata log terminal (klt), if
there exists a log resolution f : Y → X such that each irreducible component of ∆Y
occurs with coefficient lower than one.
Definition 0.3. We say that a log pair (X, ∆) is log canonical (lc), if there exists a
log resolution f : Y → X such that each irreducible component of ∆Y occurs with
coefficient at most one.
One can show that ”exists” in the above definitions can be replaced by ”for any
birational morphism” (not necessary a log resolution).
The leading example of a klt pair is: X smooth, and effective ∆ is such that
Supp ∆ is simple normal crossing and b∆c = 0.
Let us state a few easy properties of klt/lc singularities:
(1) If (X, ∆) is klt, then b∆c = 0
(2) If (X, ∆) is lc, then each irreducible component of ∆ has coefficient at most
one
(3) if (X, ∆) is klt, then it is lc
1A Q-divisor is Q-Cartier if some multiple of it is a Cartier divisor
2Recall that one defines a Weil divisor K to be the closure of a canonical divisor on the
X
smooth locus. It is not uniquely defined. Having said that, formally, one should define ∆Y as
∆Y := −KY /X + f ∗ ∆. In this way, there is no ambiguity in the choice of KY and KX .
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KAWAMATA LOG TERMINAL AND LOG CANONICAL PAIRS
(4) If (X, ∆) is klt (/lc) and 0 ≤ ∆0 ≤ ∆, then (X, ∆0 ) is klt (lc, respectively)
(5) if (X, ∆) is klt and D is an arbitrary effective Q-Cartier divisor, then (X, ∆+
D) is klt for all small enough 0 < 1.
(6) If X is klt in characteristic zero, then X has rational singularities (this is
not easy)
The item (5) is not true for lc pairs. One can think of lc pairs as being ”limits” of
klt pairs (it’s not a formal statement).
Remark 0.4. If X is smooth, then for any birational morphism f : Y → X, we have
that the boundary on Y has non-positive coefficients, i.e. ∆Y ≤ 0.
Remark 0.5. In the definition of a log pair we usually assume that ∆ ≥ 0. If we
hadn’t assumed this, then we could’ve expressed the definition of klt pairs in the
following way:
• First, we say that (X, ∆) is klt simple normal crossing, if X is smooth,
Supp ∆ is simple normal crossing, and all coefficients in ∆ are lower than
one.
• We say that (X, ∆) is klt, if there exists a log resolution Y → X such that
(Y, ∆Y ) is klt simple normal crossing, where:
KY + ∆Y = f ∗ (KX + ∆), and
f∗ ∆Y = ∆.
The above remark applies also to lc pairs.
1. Philosophical statements
The definition of klt/lc sounds reasonable when there is no boundary, but it is
usually confusing to people what it means for a pair to be ”mildly singular”.
A philosophical statement (from the perspective of X) A pair (X, ∆) is
klt if X is klt, and the singularities of X along ∆ are even milder than klt (closer
to being smooth). The higher the coefficients of ∆ are, the milder the singularities
of X along ∆ must be.
Another way of looking at it is the following: for (X, ∆) to be klt/lc means that:
X is close to being smooth, and ∆ is close to begin simple normal crossing Q-divisor
with coefficients smaller (/smaller or equal) than one.
A philosophical statement (from the perspective of ∆) If X is smooth,
then (X, ∆) is klt, if ∆ has mild singularities (close to being simple normal crossing). The lower the coefficients of ∆ are, the more singular Supp ∆ can be. If X is
not smooth, then ∆ must have even milder singularities.
2. Power of klt/lc
The reason why klt/lc is so powerful comes from the following theorem:
Theorem 2.1 (Kawamata-Viehweg vanishing theorem). Let (X, ∆) be a klt pair,
and let A be an ample Q-Cartier Q-divisor such that KX + ∆ + A is Cartier. Then
H i (X, KX + ∆ + A) = 0
for any i > 0.
KAWAMATA LOG TERMINAL AND LOG CANONICAL PAIRS
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The theorem was first proven in the case when X is smooth, ∆ is effective simple normal crossing, and b∆c = 0 3 . The breakthrough observation of Kawamata/Mori/Reid/Kollar/etc. was that this can also work when X is not smooth
and ∆ is not simple normal crossing.
The main trick used over and over in birational geometry is the following: assume
we want to show that a nef line bundle L is semiample (some power of it is base
point free). Then we try to represent L in the form KX +S+∆+A, where (X, S+∆)
is lc, and S is an irreducible divisor. If this is possible, then
H 0 (X, L) → H 0 (S, L|S )
is surjective, because H 1 (X, KX + ∆ + A) = 0. Thus, we can proceed inductively
on the dimension.
Remark 2.2. Similar idea is used in the proof of Angehrn-Siu theorem: if X is a
smooth projective variety
of dimension n and L is ample, then KX + mL is very
.
Again,
we find ∆ ∼Q nL such that (X, ∆) is lc, and there
ample for m > n+1
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exists a ”log canonical centre” (a generalization of a divisor with coefficient one, as
in the above). Then, we write:
KX + mL ∼ KX + ∆ + (m − n)L,
restrict to a log canonical centre, and do induction on dimension (one can show
that a minimal log canonical centre is smooth).
What I find really cool here, is that we are solving a problem for smooth varieties
using log canonical pairs and log canonical centres (notions coming from the theory
of singularities).
3. Klt/lc in positive characteristic
The reason why klt, lc are still useful notions in positive characteristic (even
though we don’t have Kawamata-Viehweg vanishing theorem) comes from the following theorem:
Theorem 3.1. If X is a Q-Gorenstein F -pure singularity, then X is lc. If X is a
Q-Gorenstein strongly F -regular singularity, then X is klt.
Moreover, in the case of surface singularities in characteristic p > 5, being
strongly F -regular is equivalent to being klt.
Further, note that from the reduction modulo p perspective klt singularities and
strongly F -regular singularities coincide. More precisely: a reduction mod p 0
of a klt singularity in characteristic zero, is always strongly F -regular.
Lastly, let me say what’s happening in the global setting:
Theorem 3.2. Let X be a Q-Gorenstein F -split variety. Then, there exists an
effective Q-divisor ∆ such that (X, ∆) is log Calabi-Yau, i.e.
• (X, ∆) is log canonical
• KX + ∆ ∼ 0.
It is easy to define ∆, when X is smooth. In this case, the F -splitting induces a
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section D ∈ H 0 (X, (1 − p)KX ), and one sets ∆ := p−1
D.
3It is easy to see why KV-vanishing for klt (X, ∆) should follow from this ”smooth” version.
Indeed, we can take a log resolution f : Y → X of (X, ∆). Then H i (X, KX + ∆ + A) = H i (Y, KY +
∆Y + f ∗ A). If ∆Y ≥ 0, then we are done. Otherwise, there is some technical work to be done
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KAWAMATA LOG TERMINAL AND LOG CANONICAL PAIRS
4. Other stuff
Some other important stuff:
(1) Adjunction: if (X, S + ∆) is lc with S being an irreducible divisor, then
(S, ∆S ) is lc, where ∆S is defined by adjunction:
(KX + S + ∆)|S = KS + ∆S
(2) There are other notions of singularities: terminal (all exceptional divisors of
f : Y → X have negative coefficients in ∆Y ), canonical (∆Y has non-positive
coefficients), and some technical notions between klt and lc, just used in
proofs (plt, dlt). There is also a notion of semi-log-canonical singularities (a
generalization of lc to non-normal singularities). It is important for studying
compactifications of moduli spaces of varieties.
(3) When dealing with lc singularities, a very important notion is a log canonical
centre. A vague definition is the following: a subvariety V ⊆ X is a log
canonical centre of (X, ∆), if there exists a divisor E on Y with coefficient
one in ∆Y such that f (E) = V .