Flops connect minimal models
7 Flops connect minimal models
Yujiro Kawamata
October 30, 2018
Abstract
A result by Birkar-Cascini-Hacon-McKernan together with the bound-
edness of length of extremal rays implies that different minimal models
can be connected by a sequence of flops.
A flop of a pair (X,B) is a flip of a pair (X,B′) which is crepant for
KX + B where B
′ is a suitably chosen different boundary. We prove the
following:
Theorem 1. Let f : (X,B) → S and f ′ : (X ′, B′) → S be projective
morphisms from Q-factorial terminal pairs of varieties and Q-divisors such
that KX+B and KX′ +B
′ are relatively nef over S. Assume that there exists
a birational map α : X 99K X ′ such that α∗B = B
′, where the lower asterisk
denotes the strict transform. Then α is decomposed into a sequence of flops.
More precisely, there exist an effective Q-divisor D on X such that
(X,B +D) is klt and a factorization of the birational map α
X = X0 99K X1 99K · · · 99K Xt = X
which satisfy the following conditions:
(1) αi : Xi−1 → Xi (1 ≤ i ≤ t) is a flip for the pair (Xi, Bi +Di) over S,
where Bi and Di are strict transforms of B and D, respectively.
(2) αi is crepant for KXi−1 + Bi−1 in the sense that the pull-backs of
KXi−1 +Bi−1 and KXi +Bi coincide on a common log resolution.
We remark that the boundary B need not be assumed to be big as in
[1] Corollary 1.1.3. For example, a birational map between Calabi-Yau man-
ifolds can be decomposed into a sequence of flops. The number of marked
http://arxiv.org/abs/0704.1013v1
minimal models which are birationally equivalent to a fixed pair is finite if
B is big ([1] Corollary 1.1.5), but it is not the case in general (cf. [4]), where
a marked minimal model is a pair consisting of a minimal model and a fixed
birational map to it. If we relax the condition for the pairs to being klt, then
we should allow crepant blowings up besides flops.
The theorem was already proved in the case dimX = 3 and B = 0; first
in [2] assuming the abundance which was proved afterwards, and later in [5]
without assumption.
Proof. It is well-known that α is an isomorphism in codimension 1 because
(X,B) and (X ′, B′) are terminal and KX + B and KX′ + B
′ are relatively
nef (cf. [2]). We recall the proof for reader’s convenience. Let µ : V → X
and µ′ : V → X ′ be common log resolutions. We write
KV = µ
∗(KX +B)− µ
B + E = (µ′)∗(KX′ +B
′)− (µ′
)−1B′ + E ′
where E and E ′ are effective divisors whose supports coincide with the excep-
tional loci of µ and µ′, respectively, because (X,B) and (X ′, B′) are terminal.
Assume that there is a prime divisor on V which is contracted by µ but not
by µ′. Then it is an irreducible component of E but not of E ′. We set
F = min{E,E ′}, Ē = E − F and Ē ′ = E ′ − F . By the Hodge index theo-
rem, there exists a curve C on V which is contracted by µ and is contained
in Supp(Ē) but not in Supp(µ−1
B + Ē ′) and such that (Ē · C) < 0. Since
B ≥ (µ′
)−1B′, we have
((µ′)∗(KX′ +B
′) + µ−1
B − (µ′
)−1B′ + Ē ′) · C) ≥ 0.
But this is a contradiction to
(µ∗(KX +B) + Ē) · C) < 0.
The case where there is a prime divisor on V which is contracted by µ′ but
not by µ is treated similarly.
Let L′ be an effective f ′-ample divisor on X ′, and L its strict transform
on X . There exists a small positive number l such that (X,B + lL) is klt.
If KX + B + lL is f -nef over S, then α becomes a morphism by the base
point free theorem, hence an isomorphism since X is Q-factorial. Therefore
we may assume that KX +B + l
′L is not f -nef over S for any 0 < l′ ≤ l.
Let H be an effective divisor on X such that (X,B + lL+ tH) is klt and
KX + B + lL + tH is f -nef for some positive number t. We shall run the
MMP for the pair (X,B + l′L) over S with scaling of H for some l′. Since α
is an isomorphism in codimension 1, there are only flips in this MMP. The
following lemma shows that we can choose extremal rays such that the flips
are crepant with respect to KX +B.
Let k be a positive integer such that k(KX +B) is a Cartier divisor. We
set e = 1
2k dimX+1
Lemma 2. (1) There exists an extremal ray R for (X,B + lL) over S such
that ((KX +B) ·R) = 0.
(2) Let
t0 = min{t ∈ R | ((KX +B + lL+ tH) · R) ≥ 0 for all extremal rays R
for (X,B + lL) over S s.t. ((KX +B) ·R) = 0}.
Then KX +B + elL+ et0H is f -nef, and there exists an extremal ray R for
(X,B+elL) over S such that ((KX+B+elL+et0H)·R) = ((KX+B)·R) = 0.
Proof. (1) Since KX +B + elL is not nef, there exists an extrenal ray R for
(X,B+ elL) over S. Then R is also an extremal ray for (X,B+ lL) because
(X,B) is f -nef. Since the pair (X,B+ lL) is klt, R is generated by a rational
curve C, which is mapped to a point on S, such that
0 > ((KX +B + lL) · C) ≥ −2 dimX
by [3].
We claim that ((KX +B) ·C) = 0. Indeed we have otherwise ((KX +B) ·
C) ≥ 1/k, hence
((KX +B + elL) · C)
2k dimX + 1
((KX +B + lL) · C) +
2k dimX
2k dimX + 1
((KX +B) · C)
2k dimX + 1
(−2 dimX + 2dimX) = 0
a contradiction.
(2) If KX + B + elL + et0H is not f -nef, then there exists an extremal
ray R for (X,B + elL + et0H) over S. Then R is also an extremal ray for
(X,B+ lL+t0H) because (X,B) is f -nef. Since the pair (X,B+ lL+t0H) is
klt, R is generated by a rational curve C such that ((KX+B+lL+t0H)·C) ≥
−2 dimX by [3]. Then we have
((KX +B + elL+ et0H) · C)
2k dimX + 1
((KX +B + lL+ t0H) · C) +
2k dimX
2k dimX + 1
((KX +B) · C)
2k dimX + 1
(−2 dimX + 2dimX) = 0
a contradiction. Therefore KX +B + elL+ et0H is f -nef.
Since B + lL is f -big, the number of extremal rays for (X,B + lL) over
S is finite. Hence there exists such an R that ((KX +B + lL+ t0H) · R) =
((KX +B) · R) = 0.
We note that we can deduce (1) from only the finiteness of extremal rays,
but not (2). The point is that the number e stays independent of t0 during
the MMP.
We run the MMP for (X,B+elL) with scaling ofH . We take an extremal
ray R such that ((KX +B+ elL+ et0H) ·R) = ((KX +B) ·R) = 0. The flip
exists by [1] Corollary 1.4.1. Since (lL+ t0H) ·R) = 0, the pair (X,B+ lL+
t0H) remains to be klt after the flip. We also note that k(KX+B) remains to
be a Cartier divisor after the flip by the base point free theorem. Therefore
we can continue the process. By the termination theorem of directed flips
([1] Corollary 1.4.2), we complete our proof.
References
[1] Caucher Birkar, Paolo Cascini, Christopher D. Hacon, James McK-
ernan. Existence of minimal models for varieties of log general type.
math.AG/0610203.
[2] Kawamata, Yujiro. Crepant blowing-up of 3-dimensional canonical
singularities and its application to degenerations of surfaces. Ann.
of Math. 127 (1988), 93–163.
[3] Kawamata, Yujiro. On the length of an extremal rational curve. In-
vent. Math. 105 (1991), 609–611.
[4] Kawamata, Yujiro. On the cone of divisors of Calabi-Yau fiber
spaces. Internat. J. Math. 8 (1997), 665–687.
http://arxiv.org/abs/math/0610203
[5] Kollár, János. Flops. Nagoya Math. J. 113(1989), 15–36.
Department of Mathematical Sciences, University of Tokyo,
Komaba, Meguro, Tokyo, 153-8914, Japan
kawamata@ms.u-tokyo.ac.jp
