The K\"ahler-Ricci flow on projective bundles

Abstract

We study the behaviour of the K\"ahler-Ricci flow on projective bundles. We
show that if the initial metric is in a suitable K\"ahler class, then the
fibers collapse in finite time and the metrics converge subsequentially in the
Gromov-Hausdorff sense to a metric on the base.

Full text

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arXiv:1107.2144v1 [math.DG] 11 Jul 2011
THE K¨AHLER-RICCI FLOW
ON PROJECTIVE BUNDLES1
Jian Song∗, G´ abor Sz´ ekelyhidi∗∗and Ben Weinkove∗∗∗
Abstract
We study the behaviour of the K¨ ahler-Ricci flow on projective bundles. We show
that if the initial metric is in a suitable K¨ ahler class, then the fibers collapse in finite
time and the metrics converge subsequentially in the Gromov-Hausdorff sense to a
metric on the base.
1Introduction
Let X be a projective bundle over a smooth projective variety B. This means that
X = P(E), where π : E → B is a holomorphic vector bundle of rank r, say. We
study the behavior of the solution ω = ω(t) of the K¨ ahler-Ricci flow on X starting
at a K¨ ahler metric ω0:
∂
∂tω = −Ric(ω),ω|t=0= ω0.(1.1)
The solution ω(t) develops a singularity after a finite time. Indeed from [TZ] a
maximal smooth solution to (1.1) exists on [0,T) where T > 0 is given by
T = sup{t > 0 | [ω0] + tc1(KX) > 0}.(1.2)
T is finite since F ·c1(−KX)r−1> 0 for every fiber F. For t ∈ [0,T), ω(t) lies in the
class [ω0]+tc1(KX). We will assume that the limiting K¨ ahler class [ω0]+Tc1(KX)
satisfies
[ω0] + Tc1(KX) = [π∗ωB],(1.3)
for some K¨ ahler metric ωBon B.
Our main result shows that a sequence of metrics along the flow converges sub-
sequentially to a metric on B in the Gromov-Hausdorff sense as t → T.
Theorem 1.1 There exists a sequence of times ti → T and a distance function
dB on B (which is uniformly equivalent to the distance induced by ωB), such that
(X,ω(ti)) converges to (B,dB) in the Gromov-Hausdorff sense.
This implies that the fibers of X collapse. In order to prove this theorem we
establish (Lemmas 2.2 and 2.4) the following estimates for a uniform constant C
and for all t ∈ [0,T):
(i) ω(t) ≤ Cω0, and
1The first-named author was supported in part by an NSF CAREER grant DMS-08-47524 and a Sloan
Fellowship; the second-named author by the NSF grant DMS-09-04223; the third-named author by the
NSF grants DMS-08-48193, DMS-11-05373 and a Sloan Fellowship.
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(ii) diamω(t)F ≤ C(T − t)1/3, for every fiber F.
That is, the metrics ω(t) are uniformly bounded from above along the flow and
the diameters of the fibers tend to zero as t → T.
To illustrate how Theorem 1.1 ties in with the existing literature on the K¨ ahler-
Ricci flow we discuss the example of X = P2blown up at one point, which is a P1
bundle over P1. As t → T the behavior of ω(t) in the sense of Gromov-Hausdorff
depends on the point at which the K¨ ahler classes [ω(t)] hit the boundary of the
K¨ ahler cone. Write α = [ω0] + Tc1(KX). Then one of the following holds:
(1) α = 0.
(2) α is the pull-back of a K¨ ahler class from the base P1(the setting of this paper).
(3) α is the pull-back of a K¨ ahler class from P2via the blow-down map p : X → P2.
In each case we have a map f : X → M to a manifold M (of dimension 0, 1 and 2
respectively) and α = f∗β for β a K¨ ahler class on M. Feldman-Ilmanen-Knopf [FIK]
made a number of conjectures about the behavior of (1.1) which they established
for self-similar solutions. In particular they conjectured that in each of these three
cases the flow should converge in the Gromov-Hausdorff sense to a metric on M.
We now briefly describe some progress on these conjectures.
In case (1) it is an immediate consequence of a result of Perelman [P] (see [SeT])
that (X,ω(t)) converges in the Gromov-Hausdorff sense to a point. When the initial
metric is invariant under a U(2) symmetry it was shown in [SW1] that: in case (2),
(X,ω(t)) converges in the Gromov-Hausdorff sense to the base P1with the Fubini-
Study metric; and in case (3), (X,ω(t)) converges in the Gromov-Hausdorff sense
to (P2,d) where d is a metric on P2inducing the usual topology. In [SW2] the same
behavior for (3) was established without assuming symmetry of the initial data.
Thus the case of (2) without symmetry assumption was still open.
Although we have limited our discussion here to the case of P2blown up at one
point, the results of [SW1] and especially [SW2] apply to a much larger class of
manifolds. Moreover, these results fit into a general program of understanding the
K¨ ahler-Ricci flow on algebraic varieties and the analytic minimal model program
[Ts, SoT3, T2] (see also [EGZ, SoT1, SoT2, TZ, Zha1, Zha2]).
Our Theorem 1.1 is concerned with the setting of (2) for a general projective
bundle over an algebraic variety B. Even in the case of a product B × Pr−1with
arbitrary initial metric ω0, the result of Theorem 1.1 is new.
Of course, the [FIK] conjectures predict that the convergence should occur with-
out taking subsequences. A difficulty in (2) compared to the setting (3) is that the
metric ω(t) becomes singular at every point of the manifold X.
A further interesting and difficult question is to analyze the singularity at time
T by a rescaling procedure (see the discussion in [FIK]). In case (1) above, rescaling
so that X has fixed volume, one obtains a compact K¨ ahler-Ricci soliton [WZ, Zhu,
TZhu]. For more general manifolds with c1(X) > 0 this is related to a question
of Yau [Y2] regarding stability in the sense of geometric invariant theory (see for
example [CW, D, T1, PS, PSSW, Sz, T1, To]). In case (2), rescaling so that the
fibers have fixed volume, one would expect to obtain a product P1× C (see the
recent preprint [Fo] in the case of U(2) symmetry). In case (3) the appropriate
rescaling should yield the shrinking soliton constructed in [FIK].
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In Section 2, we establish the key estimates (i) and (ii) mentioned above. In
Section 3 we prove Theorem 1.1.
2 The main estimates
In this section we establish the main estimates. Assume X has complex dimension
n and X = P(E) where E is a rank r holomorphic vector bundle over B. We will
often write g for the K¨ ahler metric with K¨ ahler form ω. We use C, C′or Cito
denote a uniform constant, which may differ from line to line.
First, we have a lower bound for ω(t) from the parabolic Schwarz Lemma, as in
[SoT1].
Lemma 2.1 There exists a uniform constant c > 0 such that
ω ≥ cπ∗ωB.(2.4)
Proof This estimate is well-known to hold. Indeed the argument is almost identical
to the proof of Lemma 2.2 in [SW2] (see also [TZ]). We provide a brief sketch
for the reader’s convenience. First, we reformulate (1.1) as a parabolic complex
Monge-Amp` ere equation. Define a family of reference metrics ˆ ωt∈ [ω(t)] by
ˆ ωt=1
T((T − t)ω0+ tπ∗ωB),(2.5)
and let Ω be the unique volume form on X satisfying
√−1
2π
∂∂ logΩ =∂
∂tˆ ωt=1
T(π∗ωB− ω0) ∈ c1(KX),
?
X
Ω = 1. (2.6)
If ϕ = ϕ(t) solves the parabolic complex Monge-Amp` ere equation
∂ϕ
∂t= log(ˆ ωt+
√−1
Ω
2π∂∂ϕ)n
,ˆ ωt+
√−1
2π
∂∂ϕ > 0,ϕ|t=0= 0,(2.7)
then ω = ˆ ωt+
on [0,T), we can extract the unique solution ϕ = ϕ(t) of (2.7) for t ∈ [0,T).
Since c0(T − t)nΩ ≤ ˆ ωn
elementary maximum principle argument shows that ϕ is uniformly bounded (see
[TZ] or Lemma 2.1 of [SW2]). We then compute using the parabolic Schwarz lemma
computation of [SoT1] that
√−1
2π∂∂ϕ solves (1.1). Conversely, given the solution ω = ω(t) of (1.1)
t≤ C0Ω for uniform positive constants c0 and C0, an
?∂
∂t− ∆
?
logtrω(π∗ωB) ≤ Ctrω(π∗ωB),(2.8)
for a uniform constant C. On the other hand, for a uniform constant c′> 0,
∆ϕ = trω(ω − ˆ ωt) ≤ n − c′trω(π∗ωB).(2.9)
The result follows by applying the maximum principle to the quantity
Q = logtrω(π∗ωB) − Aϕ(2.10)
for A sufficiently large and using the fact that ϕ is uniformly bounded.
?
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We will make use of this in the key estimate:
Lemma 2.2 There exists a constant C > 0 such that for t in [0,T),
trω0ω(t) ≤ C.(2.11)
Proof For any line bundle L, P(E) = P(E ⊗ L). Hence by replacing E by E ⊗ A−1
for some sufficiently ample line bundle A, we can assume that the dual bundle E∗
is generated by global sections.
Fix an arbitrary p ∈ B. Choose r sections s1,...,sr of the dual bundle E∗,
which are linearly independent at p. Write
f = s1∧ s2∧ ... ∧ sr
for the corresponding section of the line bundle?rE∗. Let U ⊂ B be the set where f
does not vanish. On this set s1,...,srgive a biholomorphism Φ : π−1(U) → U×Pr−1
such that the diagram
π−1(U)
U × Pr−1
✑
✑
✑✰ pr1
U
◗◗◗
π
s
✲
Φ
(2.12)
commutes, where pr1is the projection map onto the first factor.
Let ωprod= (pr1)∗ωB+(pr2)∗ωFSbe the product metric on U ×Pr−1, where ωFS
is the Fubini-Study metric on Pr−1. Define ωsing= Φ∗ωprod. Then ωsingis a smooth
K¨ ahler metric on π−1(U), which we can also think of as a singular metric on X.
Fix a Hermitian metric h on E, and write h for the induced metric on E∗,?rE∗
as well. We claim that there is a constant C such that
trωsingω0≤ C|f|−2
h.(2.13)
This can be seen as follows. The metric ω0is uniformly equivalent to the metric
ωloc, say, we obtain by locally, over a small ball V ⊂ B, taking (non-holomorphic)
sections σ1,...,σrof E∗which are pointwise orthonormal with respect to h, and
then using these sections to pull back the product metric on V × Pr−1. Thus to
compare the metrics ωsingand ωlocalong each fiber of P(E), we need to compare
the Fubini-Study metrics on a given projective space constructed using two different
Hermitian metrics. On a given fiber P(Eq) the metric ωlocis constructed using the
Hermitian metric h, whereas ωsingis constructed using the metric on Eqin which
s1,...,sngive an orthonormal basis of E∗
linear map Eq→ Crgiven by s1,...,sr, and λ1≤ λ2≤ ... ≤ λrfor the eigenvalues
of A∗A, where the adjoint is formed by using the metric h on Eqand the standard
metric on Cr. Lemma 2.3 below then implies that ωsing ≥ Cλ1λ2λ−2
eigenvalue is bounded above uniformly since s1,...,srare defined over the whole of
B, so λ1λ2is bounded below by the determinant. The determinant of A∗A is given
by |f|2
To prove the lemma we will apply the maximum principle to the quantity
q. At each point q let us write A for the
rω0. Each
h, so it follows that ωsing≥ C|f|2
hω0.
H = log?|f|3
htrωsingω(t)?,
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(2.14)

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on the set π−1(U). By the claim above, H tends to negative infinity along X\π−1(U)
and hence a maximum must occur in π−1(U) at each fixed time.
We recall the well-known evolution inequality [C] (see also [Y, A]) for ω = ω(t)
solving (1.1):
?∂
∂t− ∆
?
logtrˆ ωω ≤ −
1
trˆ ωωgkℓˆR
ji
kℓgij,(2.15)
where ˆ ω is any fixed K¨ ahler metric on X, andˆR
We will apply this with ˆ ω = ωsing.
First note that the curvature tensor of ωprodhas a lower bound
ji
kℓ
are its curvature components.
Rijkℓ(ωprod) ≥ (pr∗
2gFS)ij(pr∗
1gB)ij(pr∗
2gFS)kℓ+ (pr∗
2gFS)iℓ(pr∗
2gFS)kj
− c(pr∗
1gB)kℓ
(2.16)
for a constant c depending only on the curvature of gB, where the inequality of
tensors is meant in the sense of Griffiths. It follows that
gijgkℓ(R
ℓk
ij
(ωsing)) ≥ −C(trωsingω)(trωπ∗ωB), (2.17)
for a uniform constant C. Then compute, using (2.15), (2.17) and the fact that the
curvature of the metric h on?rE∗is bounded by some multiple of π∗ωB,
?∂
∂t− ∆
?
H ≤ Ctrωπ∗ωB≤ C′.(2.18)
where we used Lemma 2.1 for the second inequality.
It follows from the maximum principle that H is bounded from above. This
gives the bound
trωsingω ≤ C|f|−3
It follows that for some open subset U′⊂⊂ U containing p, we have a bound
trω0ω ≤ C on π−1(U′), where C depends on p. We can repeat the argument at each
point of B, and by compactness of B, a finite number of the open sets U′cover B.
Taking the largest of the corresponding constants C we obtain the required result.
?
h. (2.19)
We have used the following lemma in the proof:
Lemma 2.3 Let A : Cr→ Crbe an invertible linear map, and write ψ : Pr−1→
Pr−1for the induced map of projective spaces. Let ωFSbe the standard Fubini-Study
metric on Pr−1. Then
ψ∗ωFS≥λ1λ2
where 0 < λ1≤ λ2≤ ... ≤ λrare the eigenvalues of the matrix A∗A.
Proof Let ξ be a tangent vector of type T1,0at x ∈ Pr−1with |ξ|2
A by UA for U a unitary transformation we may assume that x is the point [1 : 0 :
... : 0]. Choose holomorphic coordinates zi= Zi+1/Z1for i = 1,...,r − 1 where
Z1,...,Zr are the homogeneous coordinates on Pr−1. Applying another unitary
λ2r
ωFS,
ωFS= 1. Replacing
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