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Definition 3.4.1. Let (X, B) be a klt pair. For a nonzero ideal a ⊂ O_X, the log canonical threshold of a with respect to (X, B) is given by

lct(X, B; a) = sup{c ∈ Q^∗₊| J ((X, B), a^c) = O_X}.

This invariant measures the singularities of the subscheme cut out by a. Note that lct(X, B; O_X) = +∞. By convention, we set lct(X, B; a) = 0, when a is the zero ideal.

Similarly, if a• is a graded sequence of ideals on X, then log canonical threshold of a•

is given by

lct(X, B; a•) := sup{c ∈ Q^∗₊| J ((X, B), a^c_•) = O_X}.

When the choice of the pair (X, B) is clear, we will often simply write lct(a) and lct(a•).

Proposition 3.4.2. If (X, B) is a klt pair and a ⊂ OX is a nonzero ideal, then lct(X, B; a) = inf

v∈DivVal_X

A_X,B(v)

v(a) = inf

v∈Val^∗_X

A_X,B(v) v(a) .¹

1We use the convention that if either AX,B(v) = +∞ or v(a) = 0, then AX,B(v)/v(a) = +∞. Similarly, if a_•is graded sequence of ideals, we set AX,B(v)/v(a_•) = +∞ if either AX,B(v) = +∞ or v(a_•) = 0.

Furthermore, if π : Y → X is a log resolution of (X, B, a) and a · O_Y = O_Y(−F ), then the above infimum is achieved by a valuation of the form ord_E where E is a prime divisor in the support of F .

Proof. By Proposition 3.3.12, J ((X, B), a^c) = O_X if and only if v(1) > cv(a) − A_X,B(v) for all v ∈ DivVal_X (resp., v ∈ Val^∗_X). Noting that v(1) = 0 for all v ∈ Val_X yields the desired formulas. Using a similar argument and the definition of J ((X, B), a^c) in terms of a log resolution implies the last statement of the proposition.

Proposition 3.4.3. If (X, B) is a klt pair and a_• a graded sequence of ideals on X, then lct(X, B; a•) = inf

Proof. The proof is the same as the proof for the similar statement in Proposition 3.4.2, but uses Proposition 3.3.9 rather than 3.3.8.

Proposition 3.4.4. If a• is a graded sequence of ideals on X, then lct(X, B; a•) = lim

M (a•)3m→∞m · lct(X, B; a_m) = sup

m∈M (a•)

m · lct(X, B; a_m).

Proof. Following the argument in [Mus02], we begin by showing lim_{M (a•)3m→∞}m·lct(X, B; a_m) exists and equals sup_{m∈M (a•)}m · lct(a_m). Since a_m· a_p ⊂ a_m+p, we see

v(a_m+p)

AX,B(v) ≤ v(a_m)

AX,B(v)+ v(a_p) AX,B(v). for all v ∈ DivVal_X. By Proposition 3.4.3, it follows that

1

lct(a_m+p) ≤ 1

lct(a_m) + 1 lct(a_p).

for all m, p ∈ M (a•). Applying [JM12, Lemma 2.3] yields the desired statement.

We now move on to show lct(X, B; a•) = lim_{M (a•)3m→∞}m · lct(X, B; a_m). Fix c ∈ Q^∗₊, and note that c < lct(a•) if and only if J ((X, B), a^c_•) = O_X. Since J ((X, B), a^c_•) = J ((X, B), a^c/mm ) for all m-divisible enough, the latter condition is equivalent to c <

m · lct(X, B; a_m) for all m divisible enough. Thus, lct(a•) = lim_{M (a•)3m→∞}m · lct(X, B; a_m) and the proof is complete.

Proposition 3.4.5. Let (X, B) be a klt pair and a• a graded sequence of ideals. The graded sequence a• is nontrivial if and only if lct(X, B; a•) < +∞.

Proof. Recall that a• is nontrivial if and only if there exists a divisorial valuation v on X such that v(a_•) > 0. Therefore, the statement is an immediate consequence of Proposition 3.4.3.

Definition 3.4.6. If (X, B) is klt pair and D a Q-Cartier Q-divisor on X, we can also make sense of log canonical threshold of D. Pick m ≥ 1 so that mD is a Cartier divisor, and set lct(X, B; D) = m · lct(X, B; OX(−mD)). It is straightforward to check that this definition is independent of the choice of m and lct(X, B; D) = sup{λ ∈ Q^∗₊| (X, B + cD) is klt}.

The following statement follows immediately from the above definition and Proposition 3.4.2.

Proposition 3.4.7. If (X, B) is a klt pair and D is a Q-Cartier divisor on X, then lct(X, B; a) = inf

v∈DivValX

A_X,B(v)

v(D) = inf

v∈Val^∗_X

A_X,B(v) v(D) .

Furthermore, if π : Y → X is a log resolution of (X, B + D), then the above infimum is achieved by a valuation of the form ord_E where E is a prime divisor in the support of π^∗D.

3.4.1 Valuations computing the log canonical threshold

Definition 3.4.8. Let (X, B) be a klt pair and a• a graded sequence of ideals on X. In light of Proposition 3.4.3, we say that a valuation v∗ ∈ Val^∗_X computes lct(X, B; a•) if lct(a•) = A_X,B(v∗)/v∗(a•).

Lemma 3.4.9. If (X, B) is a klt pair and v ∈ Val^∗_X, then lct(X, B; a•(v)) ≤ AX,B(v) and equality holds if and only if v computes lct(X, B; a•(v)).

Proof. The statement is an immediate consequence of Proposition 3.4.3 and the fact that v(a•(v)) = 1 (Lemma 3.3.11).

The following theorem generalizes [JM12, Theorem A] to klt pairs. Our proof is similar in technique to the proof in [JM12].

Theorem 3.4.10. If (X, B) is a klt pair and a• a graded sequence of ideals on X, then there exists a valuation v∗ ∈ Val^∗_X computing lct(X, B; a•).

Proof. If lct(X, B; a•) = +∞, then any valuation v ∈ Val^∗_X computes lct(X, B; a•). Thus, we may assume lct(X, B; a•) < +∞ and set c := lct(X, B; a•).

Fix a normalizing subscheme N of (X, B) such that N contains the zero locus of a_m⁰ for some m⁰ ∈ M (a_•). We claim

lct(X, B; a•) = inf

v∈Val^N_X

A_X,B(v) v(a•) . Indeed, since v 7→ ^AX,B_v(a^(v)

•) is invariant under scaling, it is sufficient to show that if v(a•) > 0, then v ∈ R^∗₊· Val^N_X. Now, if v(a•) > 0, then v(a_m⁰) > 0. Since N was chosen to contain the zero locus of a_m⁰, v(a_m⁰) > 0 implies v(I_N) > 0. This completes the claim.

Next, by Proposition 3.1.7, we may choose B ∈ R₊ so that v(a•) ≤ B for all v ∈ Val^N_X. Fix ε > 0, and note that if A_X,B(v)/v(a_•) < c + ε, then A_X,B(v) ≤ (c + ε)B. Therefore,

lct(X, B; a_•) = inf

v∈W

A_X,B(v)

v(a•) , (3.2)

where W = Val^N_X∩{AX,B ≤ (c + ε)B}.

Let φ : Val^N_X → R ∪ {+∞} be the function defined by φ(v) = A_X,B(v) − c · v(a•).

By (3.2), inf_v∈W φ(v) = 0. Since W is compact (Proposition 3.2.6) and φ is lower semicontinous (Proposition 3.1.7 and Theorem 3.2.1), there exists v∗ ∈ W such that φ(v∗) = A_X,B(v∗) − c · v∗(a•) = 0. The latter implies v∗ computes lct(X, B; a•).

3.4.2 Log canonical thresholds in families

In this section we prove well known results on the behavior of the log canonical threshold along a family of ideals. See [Kol96] and [Amb16] for related statements.

We consider the following setup. Let (X, B) be a klt pair and T a variety. Write p : X × T → T for the second projection map. Set X_t:= p⁻¹(t) and B_t = B × {t}. If a is an ideal on OX×T, we write at:= a · O_X×{t} for each t ∈ T .

Proposition 3.4.11. If a ⊂ O_X×T is a nonzero ideal, then there exists a nonempty open set U ⊂ T such that U is smooth and

lct(X × U, B × U ; a|_X×U) = lct(X_t, B_t; a_t) for all closed points t ∈ U .

Proof. Since we may shrink T , we may assume T is smooth. Hence, (X × T, B × T ) is a

where each E_i is a distinct prime divisor on Y . Shrinking T further, we may also assume each E_i dominates T .

By generic smoothness, there exists a nonempty open set U ⊂ T such that Y → T is smooth over U and Exc(π) + ^B × T +P E_i has relative simple normal crossing over U .

i for all t ∈ U . Since the latter minimum is precisely lct(X × U, B × U ; a|_X×U), the proof is complete.

Proposition 3.4.12. Assume T is a smooth curve and 0 ∈ T is a closed point. If a⊂ O_X×T is a nonzero ideal such that V (a) is proper over T , then there exists an open is log canonical, [KM98, Theorem 5.50] implies there exists an open set W ⊂ X such that X₀ ⊂ W and the restriction of (X, B + X₀, a^c₀) to W is log canonical. Therefore, (X, B, a^c₀) restricted to W is log canonical as well.

Now, set

V := {t ∈ T | V (a) ∩ X_t⊂ W }.

The set V is nonempty and open in T . Indeed, V contains 0. Additionally, V is the complement of π(V (a) \ W ) and V (a) is proper over T .

Next, note that (X × V, B × V, a|^c_X×V) is log canonical. By Proposition 3.4.11, we may find a nonempty open set V⁰ ⊂ V such that

lct(X × V⁰, B × V⁰; a|X×V⁰) = lct(Xt, Bt : at)

for all t ∈ V⁰. Therefore, lct(Xt, Bt; at) ≥ c for all t ∈ V⁰. Setting U := V⁰∪ {0} completes the proof.