K\"ahler-Ricci Flow on Projective Bundles over K\"ahler-Einstein Manifolds

We study the K\"ahler-Ricci flow on a class of projective bundles $\mathbb{P}(\mathcal{O}_\Sigma \oplus L)$ over compact K\"ahler-Einstein manifold $\Sigma^n$. Assuming the initial K\"ahler metric $\omega_0$ admits a U(1)-invariant momentum profile, we give a criterion, characterized by the triple $(\Sigma, L, [\omega_0])$, under which the $\mathbb{P}^1$-fiber collapses along the K\"ahler-Ricci flow and the projective bundle converges to $\Sigma$ in Gromov-Hausdorff sense. Furthermore, the K\"ahler-Ricci flow must have Type I singularity and is of $(\C^n \times \mathbb{P}^1)$-type. This generalizes and extends part of Song-Weinkove's work \cite{SgWk09} on Hirzebruch surfaces.


Introduction
The Ricci flow was introduced by Hamilton in his seminal paper [H1] in 1982, proving the existence of constant sectional curvature metric on any closed 3-manifold with positive Ricci curvature. Since then, the Ricci flow has been making breakthroughs in settling several long-standing conjectures. Just to name a few, based on a program proposed by Hamilton, a complete proof of the Poincaré conjecture was given by Perelman [P1-P3] around 2003. See also [CZ, KL, MT]. Furthermore, the Differentiable Sphere Theorem was proved by Brendle-Schoen [BS] in 2007, giving an affirmative answer to a conjecture about differential structures of quarter-pinched manifolds proposed by Berger and Klingenberg in 1960s. In the realm of Kähler geometry, the Kähler-Ricci flow was introduced by Cao in [Cao1], which proves the smooth convergence towards the unique Kähler-Einstein metric in the cases c 1 < 0 and c 1 = 0.
There has been much interest in understanding the limit behavior and singularity formation of the Ricci flow in both Riemannian and Kähler settings. Hamilton introduced in [H3] a method of studying singularity formation of the Ricci flow by considering the Cheeger-Gromov limit of a sequence of rescaled dilated metrics. The singularity model obtained, which is often an ancient or eternal solution, captures the geometry of the singularity formation near the blow-up time of the flow. For closed 3-manifolds, the study of ancient κ-solutions formed by the dilated sequence limit in Hamilton-Perelman's works (e.g. [H3, P1]) leads to a solid understanding of singularity formation of closed 3-manifolds.
Another way of interpreting singularity formation is by the Gromov-Hausdorff limit, regarding the manifold as a metric space. This notion was employed recently in the study of algebraic varieties by Song, Tian, Weinkove et. al in [ST1,T,ST3,SW2,SW3,SSW]. The unified theme of these works is the conjecture that the Kähler-Ricci flow will carry out an analytic analogue of Mori's minimal model program which is about searching for birationally equivalent models "minimal" in some algebraic sense. Like Hamilton-Perelman's work, a surgery may need to be performed in continuing the flow if necessary. To this end, the Gromov-Hausdorff convergence provides a bridge to continue the relevant geometric data.
For a better understanding of singularity formation of the Kähler-Ricci flow, one could study some algebraically concrete spaces and explore their flow behavior and possible singularity types and models. In the work by Feldman-Ilmanen-Knopf [FIK], Cao [Cao2] and Koiso [Koi], gradient Kähler-Ricci solitons were constructed on the O(−k)-bundles over P n . Their work employs the U (n + 1)/Z ksymmetry introduced by Calabi in [C] which reduces the Kähler-Ricci flow equation to a PDE with one spatial variable. Assuming Calabi's symmetry, Song-Weinkove [SW1] characterized the limit behavior (in the Gromov-Hausdorff sense) of the Hirzebruch surfaces P(O ⊕ O(−k)) and their higher dimensional analogues, which are P 1 -bundles over P n . In their paper, it was proved that the Kähler-Ricci flow exhibits three distinct behaviors: (1) collapsing along the P 1 -fibers; (2) contracting the exceptional divisor; or (3) shrinking to a point. This trichotomy is determined by the triple (n, k, [ω 0 ]) where [ω 0 ] is the initial Kähler class. Later in [SW2] by the same authors, case (2) is much generalized and the assumption on the symmetry is removed. The Calabi symmetry assumption is removed in case (1) by a recent preprint [SSW] by Song, Székelyhidi and Weinkove. The purpose of this paper is two-fold. For one thing, we generalize Song-Weinkove's work [SW1] on Hirzebruch surfaces to a class of projective bundles over any compact Kähler-Einstein manifold. We will employ an ansatz, known as the momentum construction, which coincides with Calabi's U (n + 1)/Z k -symmetry on Hirzebruch surfaces where the base manifold has the Fubini-Study metric. The idea of the momentum construction of projective bundles was introduced and studied in the subject of extremal Kähler metrics by Hwang-Singer in [HS] and by Apostolov-Calderbank-Gauduchon-(Tønnesen-Friedman) in [ACGT]. We will show that under this momentum construction, one can give a cohomological criteria under which the Kähler-Ricci flow will collapse the P 1 -fiber near the singularity similar to the Hirzebruch surface cases in [SW1]. Secondly, we study the singularity model of these projective bundles (including Hirzebruch surfaces) via the techniques developed by Hamilton in [H3]. We show that these collapsing projective bundles equipped with momenta will all exhibit C n × P 1 -singularities, and also that the Ricci flow solution has a Type I singularity. Here is the summary of our results: Main results. Let M = P(O Σ ⊕ L) be a projective bundle where (Σ, ω Σ ) is a compact Kähler-Einstein manifold such that Ric(ω Σ ) = νω Σ for some ν ∈ R, and L → Σ is a holomorphic line bundle that admits a Hermitian metric h such that the Chern curvature is given by F ∇ = −λω Σ , λ > 0. Let ω 0 be a Kähler metric on M constructed by a U (1)-invariant momentum profile with Kähler class then along the Kähler-Ricci flow ∂ t ω t = −Ric(ω t ), t ∈ [0, T ), we have • (M, g(t)) converges in Gromov-Hausdorff sense to (Σ, ω Σ ) (Theorem 5.4); • the associated ancient κ-solution is C n × P 1 (Theorem 7.3); • the Ricci flow solution must have a Type I singularity (Theorem 7.4). This paper is organized as follows. Sections 2 and 3 are the preliminaries which define our projective bundles and construct Kähler metrics using momentum profiles. We will see that the Kähler-Ricci flow is equivalent to a heat-type equation for the evolving momentum profile. Section 4 explains the trichotomy of blow-up exhibited by different choice of the triples (Σ, L, [ω 0 ]) via the calculation of Kähler classes and Chern classes. Section 5 is a variation on the theme of Song-Weinkove's work [SW1] on Hirzebruch surfaces (the collapsing case). We show that similar limiting behavior can be observed in our projective bundles. Sections 6 and 7 are about singularity analysis using rescaled dilations. We show in Section 6 that the ancient κ-solution obtained from the Cheeger-Gromov limit must split into a product. We will classify their singularity type and the curvature blow-up rate in Section 7.
We also acknowledge that Kähler-Ricci solitons on this category of bundles (and their variants) were studied and constructed in [DW, Yg, Li]. his advisor Professor Richard Schoen for all his continuing support and many productive discussions. The author would also like to thank Yanir Rubinstein for arousing his interest in this topic and for many helpful ideas, and also Ziyu Zhang for informing him of some algebraic aspects related to this study.

Projective Bundles
In this section, we will define and elaborate on the projective bundles under consideration in this paper. We first start with a compact Kähler-Einstein manifold Σ n with dim C = n. A Kähler manifold is called Kähler-Einstein if it admits a Kähler form ω Σ whose Ricci form is a real constant multiple of ω Σ , i.e. Ric(ω Σ ) = νω Σ , ν ∈ R. Clearly a necessary condition for a compact Kähler manifold to be Kähler-Einstein is that the first Chern class c 1 has a definite sign. It is well-known by results of Aubin [A] and Yau [Y] that when c 1 < 0 or = 0 Kähler-Einstein metric always exists. However, if c 1 > 0 (i.e. Fano manifolds), Kähler-Einstein metrics do not exist in general. For compact Riemann surfaces, i.e. dim C = 1, Kähler-Einstein metric must exist according to the classical uniformization theorem. See also Cheng-Yau's work [CY] on pseudoconvex domains in the complete non-compact case.
In this article, we will not go into the detail of existence issues of Kähler-Einstein metrics, but we will start with a compact Kähler manifold Σ n which is equipped with a Kähler-Einstein metric ω Σ , such that the Ricci form is given by Ric(ω Σ ) = νω Σ where ν ∈ R. We take this Kähler-Einstein manifold to be our base manifold, and build a projective P 1 -bundle upon it. Precisely, we construct our projective bundles as follows: Here O Σ is the trivial line bundle, and L → Σ is a holomorphic line bundle which is equipped with a Hermitian-Einstein metric h such that √ −1∂∂ log h = λω Σ , λ ∈ R. Here P denotes the projectivization of the holomorphic rank-2 bundle O Σ ⊕ L over Σ. The local trivialization (z, u) of this rank-2 bundle has transition functions of the form (z α , u α ) ≈ (z β , η αβ u α ) for some η αβ ∈Ȟ 1 (Σ, O * Σ ). Passing to the projectivization quotient, every element under this trivialization can be expressed as either [1 : u z ] for z = 0 or [0 : 1] and we may regard [0 : 1] as the infinity. One can check easily that the projectivization factors through the identification by the transition functions O Σ ⊕ L. Therefore, one can regard the projectivization of O Σ ⊕ L as compactifying each fiber by adding an infinity point (x, [0 : 1]) and hence M can be regarded as a P 1 -bundle over Σ. We define Σ 0 to be the zero section {x : [1 : 0]} and Σ ∞ to be the infinity section {x : [0 : 1]}. It is easy to see that the zero section Σ 0 and the infinity section Σ ∞ are global over Σ.
The class of holomorphic line bundles over Σ with tensor product as the operation form a group which is known as the Picard group, denoted by Pic(Σ). For Σ = P n , it is well-known (see e.g. [GH]) that Pic(P n ) = Z and the line bundles over P n are given by O P n (k), k ∈ Z. In particular if (Σ, L) = (P 1 , O P 1 (−k)), k > 0, the projective bundles M = P(O P 1 ⊕ O P 1 (−k)) are called the Hirzebruch surfaces. When k = 1, the projective bundles is P 2 #(−P 2 ), i.e. P 2 blown-up at a point. When Σ = C/Λ, i.e. an elliptic curve or a 2-torus, the class of line bundles are classified by a classical result by Appell-Humbert (see [Mum]). In general for Riemann surface Σ g of genus g, the Picard group Pic(Σ g ) is isomorphic to J The projective bundle M under our consideration is characterized by the pair (Σ, L) where Σ is a compact Kähler-Einstein manifold and L a holomorphic line bundle over Σ which is equipped with a Hermitian metric h such that the Chern cuvature is of the form F ∇ = −λω Σ . In particular, the line bundles generated by det(T * M ) all fall into this category. Moreover, we will only focus on line bundles L with λ > 0, since projective bundles P(O Σ ⊕ L) is biholomorphic to its dual cousin P(O Σ ⊕ L * ). Since c 1 (L) = −c 1 (L * ), one can replace L by L * in case c 1 (L) is negative. We do not discuss the case of flat bundles, i.e. λ = 0, in this paper.

U(1)-invariant Kähler metrics
Let's first recapitulate the construction of the category of projective bundles we concerned about in the rest of this article. We let M = P(O Σ ⊕ L), where (Σ, ω Σ ) is a Kähler-Einstein manifold such that Ric(ω Σ ) = νω Σ , ν ∈ R. Suppose L is a holomorphic line bundle over Σ such that it equips with a Hermitian metric h whose Chern curvature is of the form F ∇ = −λω Σ , λ > 0. In particular, such a Hermitian metric h must exist if ω Σ is a compact Riemann surface.
We will discuss the construction of U (1)-invariant Kähler metrics on these projective bundles in this section. Regard the circle group U (1) as {e iθ : θ ∈ [0, 2π)}. The U (1)-action defined by Clearly, the action factors through the transition functions of the bundle, and fixes the zero and infinity sections.
Recall that ω Σ be the Kähler-Einstein form on the manifold Σ and we have Ric(ω Σ ) = νω Σ for some ν ∈ R. Using the Hermitian-Einstein metric h described above, one can define a height parameter ρ on M \(Σ 0 ∪ Σ ∞ ) given by h . Note that ρ = −∞ corresponds to the zero section and ρ = ∞ corresponds to the infinity section. Our next step is to define Kähler metrics on M which is invariant under the circle action defined above. We start by looking for possible Kähler classes that M can have. We denote [Σ 0 ] and [Σ ∞ ] as the Poincaré duals (with respect to a fixed background volume form) of Σ 0 and Σ ∞ in H 2 (M, R) respectively, i.e.
We look for Kähler metrics whose Kähler classes have In order to define a Kähler metric in the Kähler class . The idea of this momentum construction comes from the works [HS] by Hwang-Singer and [ACGT] by Apostolov-Calderbank-Gauduchon-(Tønnesen-Friedman) on extremal and constant scalar curvature Kähler metrics. Together with a pair of asymptotic conditions given below, one can extend the metric induced by f to the whole manifold M . Here is the detail: Let f (ρ) : R → (a, b) be a strictly increasing function. We define a Kähler metric Remark 3.1. If we let u(ρ) be the anti-derivative of f , i.e. u ρ = f , then one can In order for the Kähler metric to be defined on M , we require the following asymptotic conditions: (1) There exists a smooth function Note that f has to be a strictly increasing function, so we have a < f (ρ) < b for ρ ∈ R, and

The Kähler class [ω] can be easily seen to be
Under this construction, the Kähler form depends only on the height parameter ρ. We can see immediately that these Kähler metrics are invariant under the U (1)action defined earlier, since the action preserves ρ: e iθ u h = u h for any section u ∈ Γ(Σ, L).
Next we derive the local expression of the Kähler metric ω constructed by the above momentum profile as well as its Ricci curvature. Let (z 1 , . . . , z n , ξ) be local holomorphic coordinates of M where z = (z 1 , . . . , z n ) are the base coordinates and ξ is the fiber coordinate. Recall that the height parameter is defined to be ρ = log · 2 h . Let φ(z) be a positive function such that ξ 2 = |ξ| 2 φ(z) for any (z, ξ) in the local coordinate chart. Then we have Using this, one can easily check ρ ξξ = ρ iξ = ρ ξī = 0 for any i = 1, 2, . . . , n. Moreover, Hence, the Kähler metric in (z, ξ) coordinates is given by Let g be the metric associated to the Kähler form ω, and g Σ be that of ω Σ . The determinant of the metric g and its logarithm are given by Using this, one can then compute the Ricci form − √ −1∂∂ log det(g): In the computation of the Ricci form, we used the fact that ω Σ is Kähler-Einstein so that − √ −1∂∂ log det(g Σ ) = νω Σ . Observing that the ω and Ric have similar linear-algebraic expressions when ω is constructed by a momentum profile f , one can see easily that the Kähler-Ricci flow on M is equivalent to a parabolic equation that evolves the momentum profile. In other words, the Kähler-Ricci flow preserves the momentum construction. Precisely, we have or equivalently,

Kähler classes under Kähler-Ricci Flow
From now on, we will consider the Kähler-Ricci flow ∂ t ω t = −Ric(ω t ) on M which satisfies the aforesaid U (1)-symmetry and admits evolving momenta f (ρ, t). We say T is the blow-up time of the Ricci flow if [0, T ) is the maximal time interval for the Ricci flow to exist. For Ricci flow on compact Kähler manifolds, the blow-up time is completely determined by the initial Kähler class and the first Chern class. Namely, we have the following theorem proved by Tian-Zhang: Theorem 4.1 (Tian-Zhang, [TZ]). Let (X, ω(t)) be an (unnormalized) Kähler-Ricci flow ∂ t ω t = −Ric(ω t ) on a compact Kähler manifold X n . Then the blow-up time T is given by Note that the Kähler class [ω t ] at any time t is given by . In order to work out the evolving Kähler classes and the blow-up time, one needs to understand the first Chern class of K X , which can be computed by the adjunction formula.
Given any smooth divisor D of compact Kähler manifold X, the adjunction formula relates K X and K D by where N M/D is the normal bundle of D in M . Using (4.1), one can easily work out c 1 (K M ) by taking D = Σ 0 , Σ ∞ in turn. For example, taking D = Σ ∞ , we have Since Σ is Kähler-Einstein such that Ric(ω Σ ) = νω Σ , we then have Similarly, one can also show by taking D = Σ 0 in (4.1) (now N M\D = L) to show Therefore, the first Chern class of the canonical line bundle K M is given by: Hence, under the Kähler-Ricci flow ∂ t ω t = −Ric(ω t ) with initial class , therefore the Kähler class can also be expressed as Hence, by Theorem 4.1, the maximal time is characterized by λ and ν in the following way: • Case 1: ν ≤ λ In this case, [ω t ] ceases to be Kähler when b t = a t , namely, at T := b0−a0 2 . The limiting Kähler class is given by This holds true for any given [ω t ] ceases to be Kähler when b t = a t . Likewise, the limiting Kähler class is given by [ω t ] is then proportional to c 1 (K −1 M ), i.e. canonical class. The flow stops at T = a0λ ν−λ and the limiting class [ω T ] = 0. It is well-known (see e.g. [ST2]) that in such case (M, g(t)) extincts and converges to a point in the Gromov-Hausdorff sense as t → T .
[ω t ] ceases to be Kähler when at T = a 0 , and the limit class is given by This trichotomy resembles that in Song-Weinkove's work [SW1] on Hirzebruch surfaces and Hirzebruch-type manifolds, i.e. (Σ, L) = (P n , O P n ⊕ O P n (−k)). In their work, from which our study was motivated, similar trichotomy of the blow-up time of the Kähler-Ricci flow with initial Kähler class [ω 0 ] was also exhibited as it is characterized by the triple (n, k, [ω 0 ]). It was shown in [SW1] assuming Calabi's U (n+1)/Z k -symmetry and in [SSW] assuming Σ is projective that in case of having limiting Kähler class a T [π * ω Σ ], the Kähler-Ricci flow collapses the P 1 -fiber of the projective bundle, which hereof converges to some Kähler metric of Σ as metric spaces in Gromov-Hausdorff sense.
Case 2(iii) is in reminiscence of Song-Weinkove's recent works [SW2] and [SW3] of contracting exceptional divisors, in which O(−k)-blow-up of arbitrary compact Kähler manifold X are considered. In their works, a cohomological condition is given on the initial Kähler class and the first Chern class, under which the blown-up manifold will converge in Gromov-Hausdorff sense back to X with orbifold singularity of type O(−k). There is no symmetry assumption in these works.
In our paper, we will only focus on Case 1 and Case 2(i) which exhibit collapsing of P 1 -fiber assuming the Kähler metric admits the aforesaid momentum construction.

Estimates of the Kähler-Ricci flow
From now on we assume that the triple (Σ, L, [ω 0 ]) satisfies Case 1 or Case 2(i) stated in the previous section, i.e. either ν ≤ λ, or Recall that ν is the Ricci curvature of the Kähler-Einstein manifold Σ and λ is the Chern curvature of the Hermitian-Einstein line bundle L, i.e.
Recall that the first Chern class of K M and the evolving Kähler class are given by: where a t and b t defined in (4.3) and (4.4).
Since pluripotential theory plays a very important role in Kähler-Ricci flow and in Kähler geometry in general, we would like understand the Kähler-Ricci flow ∂ t ω t = −Ric(ω t ) from potential functions viewpoint. To do so, we need a reference family of Kähler metrics {ω t } t∈[0,T ) whose Kähler class at each time t coincides with that of ω t , the Kähler-Ricci flow solution. We chooseω t to be the U (1)-invariant Kähler metric induced by the following momentum profile: This momentum profile gives the following Kähler metric: Clearly,f satisfies the asymptotic conditions for extendingω t to the whole M . Also, we have For simplicity, we denote Θ := e 2ρ 1+e 2ρ Note that [Θ] = [Σ ∞ ] and so ∂ωt ∂t = (−ν + λ)π * ω Σ − 2λΘ ∈ c 1 (K M ). Take Ω be a fixed volume form of M such that ∂ωt ∂t = √ −1∂∂ log Ω. Then the Kähler-Ricci flow ∂ t ω t = −Ric(ω t ) is equivalent to the following complex Monge-Ampère equation Working similarly as in [ST1,SW1,SW2,TZ] etc., one can derive the following estimates using maximum principles.
Hence we have, Note that a t ≥ a for any t ∈ [0, T ). Combining (5.3) and (5.4), we have As φ is uniformly bounded from (1), (5.5) implies a uniformly upper bound for Q.
Next, we will derive estimates on the Kähler-Ricci flow by assuming the metric is U (1)-invariant and admits a momentum profile f (ρ, t). First note that because f ρ (ρ, t) > 0 for any t and also lim ρ→− Note that a t and b t are both bounded away from zero as t → T , (2) in Lemma 5.1 implies f ρ is also uniformly bounded. Using these, one is able to derive the following estimates.
Lemma 5.2. There exists a constant C = C(n, ω 0 , ν, λ) > 0 such that Proof. As discussed above, (1) clearly holds because a t is bounded away from zero and b t is uniformly bounded above on [0, T ).
Lemma 5.2 implies the P 1 -fiber of our manifold M is collapsing along the flow. Precisely we have the following: Proposition 5.3. Assume (Σ, L, [ω 0 ]) satisfies the condition stated in Case 1 and Case 2(i) in P.8. Let V x ∈ T x M be a tangent vector of M at x ∈ M \(Σ 0 ∪ Σ ∞ ) which lies T x P 1 x . Here we denote P 1 x as the P 1 -fiber passing through x. Then we have V x g(t) → 0 as t → T .
Proof. It suffices to express V x g(t) in terms of f and f ρ . Since the metric g(t) is given by g(t) = f λπ * g Σ + f ρ ∂ρ ⊗∂ρ. Since V x is parallel to the fiber, we have π * V x = 0 and so π * g Σ (V x ,V x ) = 0. Hence Proof. The proof goes almost the same as in Song-Weinkove's paper [SW1] on Hirzebruch surfaces with Calabi ansatz. We will sketch the main idea here. For detail, please refer to Song-Weinkove's paper. The main ingredients of the argument are as follows: (1) the metric g(t) is degenerating along the fiber direction on compact subsets of M \(Σ 0 ∪ Σ ∞ ), (2) g(t) is bounded above uniformly g(0), and (3) for any 0 < α < 1, g(t) converges to a T π * ω Σ in C α -sense on compact subsets of M \(Σ 0 ∪ Σ ∞ ). We have proved (1) in Proposition 5.3. (2) can be proved by a uniform estimate on f ρ which can be obtained easily by the bound on the volume form ω n+1 g0 is a polynomial expression of f (ρ, t), f ρ (ρ, t) and f ρρ (ρ, t) where the coefficients are time-independent. Lemma 5.2 then shows for any compact subset K ∈ M \(Σ 0 ∪ Σ ∞ ), so we have sup K×[0,T ) ∇ g0 g(t) 2 g0 ≤ C K for some time independent constant C K > 0. It proves (3).
To show the Gromov-Hausdorff convergence, first fix a leave of Σ in M \(Σ 0 ∪ Σ ∞ ). We denote it by σ(Σ). Using (2), one can choose a sufficiently small tubular neighborhood of Σ 0 and Σ ∞ such that their complement contains σ(Σ). Then given any two points x 1 , x 2 ∈ M , we project them down to the base Σ via the bundle map π. Consider the length of the geodesic γ joining π(x 1 ) and π(x 2 ), by lifting the geodesic up by σ, we know that the lifted γ has length arbitrarily close to the a T ω Σ -length by (3). Finally, using (1), one can show x i is arbitrarily close to σ • π(x i ) as t → T . Using triangle inequality, one can then prove the g(t)-distance between x 1 and x 2 are is arbitrarily close to the a T ω Σ -distance as t → T .

Splitting Lemma
In the singularity analysis of closed (real) 3-manifolds as in [H3] and [P1], one often consider a rescaled dilation, which is a rescaled sequence of metrics g i (t) = K i g(t i +K −1 i t) where K i are chosen such that K i = Rm(x i ) g(ti) → ∞ and Rm gi(t) gi(t) ≤ C for some uniform constant C > 0 independent of i. By Hamilton's compactness [H3] and Perelman's local non-collapsing theorem [P1], one can extract a subsequence, still call it g i (t), such that (M, g i (t), x i ) → (M ∞ , g ∞ (t), x ∞ ) on compact subsets in Cheeger-Gromov sense. The convergence is in C ∞ -topology because once the curvature tensor is uniformly bounded, Shi's derivative estimate in [Shi] asserts all the higher order derivatives of Rm are uniformly bounded. The limit obtained is often called a singularity model. According to the curvature blowup rate (Type I or II), a singularity model may be an ancient or eternal solution, and is κ-non-collapsed by Perelman's result. These singularity models encode crucial geometric data near the singularity region of the flow.
We will show that under our momentum construction and our assumption on the triple (Σ, L, [ω 0 ]), the singularity model M ∞ obtained by the aforesaid rescaled dilations splits isometrically into a product N ×L, where dim C N = n and dim C L = 1.
Let (z 1 , . . . , z n , ξ) be local holomorphic coordinates where z = (z 1 , . . . , z n ) are the base coordinates and ξ is the fiber coordinate. Then λπ * ω Σ = √ −1ρ ij (z)dz i ∧ dzj, the the Kähler metric defined by momentum profile f (ρ, t), its inverse and the Ricci tensor are locally written as From the local expressions of g and g −1 , one can easily derive local expressions of the Christoffel symbols which we will need for deriving our splitting result.
Lemma 6.1. The Christoffel symbols of the Kähler metric g on M constructed by momentum profile f are given by Remark 6.2. Recall that for Kähler manifolds, the only (possibly) non-zero Christoffel symbols are those with indexes of either all (1, 0)-type or all (0, 1)-type. For succinctness, please excuse us for omitting those which are vanishing or conjugate to one of the above.
Remark 6.3. We will see that the vanishing of Γ i ξξ is crucial when dealing with the curvature tensor in the blow-up analysis in the next section. Moreover, we only need the first four Christoffel symbols in order to obtain the splitting lemma.
Proof. Using the formula Γ γ αβ = g γδ ∂ α g βδ for Kähler manifolds, one can compute the Christoffel symbols directly: Let's state and prove our splitting lemma.
Then, after passing to a subsequence, (M n+1 , g i (t), x i ) converges smoothly in pointed Cheeger-Gromov sense to a complete ancient Kähler-Ricci flow (M ∞ , g ∞ (t), x ∞ ) whose universal cover is of the form Proof. By the uniform boundedness condition of Rm gi(t) over M × [−β i , α i ], the subsequential Cheeger-Gromov convergence can be done by Hamilton's compactness theorem and Perelman's local non-collapsing theorem. See [CCG + , H3, P1], etc. Furthermore, we may assume the complex structure of J of M converges after passing to a subsequence to a complex structure J ∞ of M ∞ . That makes (M ∞ , J ∞ ) Kähler because ∇ g∞ J ∞ = lim i→∞ ∇ gi J = 0.
We will use the well-known de Rham's holonomy splitting theorem, which asserts that if the tangent bundle T M ∞ admits an irreducible decomposition k i=1 E i under the holonomy group action, i.e. parallel translation, then the universal cover of M ∞ splits isometrically as ( Note that in the Kähler case where the holonomy group is a subgroup of the unitary group, each N i is also Kähler. Suppose (M ∞ , g ∞ (t), x ∞ ) is the pointed Cheeger-Gromov limit obtained above. We would like to show it (precisely, the universal cover) splits isometrically into a product. According to the nature of the collapsing of the P 1 -fiber, it is natural to guess that one factor of the split product should correspond to the base and the other should correspond to the fiber. Based on these, we define the following unit vector fields Then we have Z j gi(t) gi(t) = Ξ gi(t) gi(t) = 1. After passing to a subsequence, they converge to vector fields Z j g∞(t) and Ξ g∞(t) in the limit M ∞ . We will show that the real distribution E ∞ = span R {ℜ(Ξ g∞(t) ), ℑ(Ξ g∞(t) )} is invariant under parallel translation. Here ℜ and ℑ denote denote the real and imaginary parts respectively. For simplicity, we will denote Z j gi(t) as Z j i and Ξ gi(t) as Ξ i for any i ∈ N ∪ {∞}.
It is worthwhile to note that which tends to 0 as i → ∞ using Lemma 5.2. Note that we have used g jj = f ρ jj + f ρ |ρ j | 2 ≥ f ρ jj . Since ρ = log |ξ| 2 + log φ(z) by (3.1), the term ρj √ ρ jj is independent of ξ, i and t, and hence is uniformly bounded near ρ = ±∞. Therefore Ξ ∞ is orthogonal to each of Z j ∞ , i.e. E ⊥ ∞ = span R {ℜ(Z j ∞ ), ℑ(Z j ∞ )} n j=1 . In order to show E ∞ is invariant under parallel translation, we need to show that by parallel translating Ξ ∞ along any vector field X on M ∞ , it stays inside E ∞ . We will prove it by showing ∇ ∞ X Ξ ∞ lies inside E ∞ , or equivalently, orthogonal to E ⊥ ∞ . We will make use of the Christoffel symbols calculated in Lemma 6.1, Finally, we have spans the whole T C M ∞ , the above calculations show that for any vector field X on (M ∞ , g ∞ (t)), one has ∇ X Ξ ∞ ,Z j ∞ g∞ = 0 for any j = 1, 2, . . . , n. Therefore, ∞ for any s. By the above calculation, we have ∇ γ ′ (s) V T (s) ∈ E ∞ for any s. Therefore, It implies that V ⊥ (s) ≡ V ⊥ (0) = 0 for any s. In other words, V (s) ≡ V T (s) ∈ E ∞ for any s. Therefore, E ∞ is invariant under parallel transport. By the de Rham's decomposition theorem, our splitting lemma follows.

Singularity Analysis
The splitting lemma in the previous section allows a dimension reduction for our singularity analysis. The ultimate goal of this section is to analyze the singularity formation of the Ricci flow on our projective bundles M = P(O Σ ⊕ L) whose P 1fiber collapses near the singularity. We are going to prove that the Kähler-Ricci flow (M, g(t)) must be of Type I (see definition below) and the singularity model is C n × P 1 , in a sense that one can choose a sequence (x i , t i ) in space-time in the high curvature region such that the universal cover of the Cheeger-Gromov limit of the rescaled dilated sequence is isometric to (C n × P 1 , dz 2 ⊕ ω FS (t)). Here ω FS (t) is the shrinking Fubini-Study metric.
According to the blow-up rate of the Riemann curvature tensor, the singularity type of a Ricci flow solution which encounters finite-time singularity is classified as in [H3].
Definition 7.1. Let (M, g(t)) be a Ricci flow solution ∂ t g(t) = −Ric(g(t)) on a closed manifold M which becomes singular at a finite time T . We call the Ricci flow encounters We would like to remark that although the Type I/II classification of finite-time singularity was proposed in the early 90's, surprisingly the first compact Type II solution was constructed by Gu-Zhu in [GZ] only recently in 2007.
In order to understand the singularity formation, we need to bring curvatures into the topic. Therefore, we will compute and analyze the Riemann curvature tensor of our projective bundle M which is equipped with momentum profile f . Recall that for Kähler manifolds, the Riemann curvature (3, 1)-tensor can be computed using the formula where A, B, C, D = 1, . . . , n or ξ. The non-zero components of the Riemann curvature tensor are given below. For the ease of inspection of the norm Rm later on, we will split the components into five groups according to the number of ξ-indexes.
Since the understanding of Rm is crucial in analyzing the singularity according their type (I or II), we need an organized expression of Rm that is written in terms of our momentum profile f . Obviously, it would take loads of unnecessary work. However, in order to study the singularity model in our class of manifolds, it suffices to understand the asymptotics of Rm 2 in terms of f and its derivatives.
. Therefore we have the following asymptotics The asymptotic of (log f ρ ) ρρ is not yet known because it involves the third ρderivative of f which we have not derived. Also, the local expressions of g and g −1 have the following asymptotics . We claim that the norm Rm 2 can be expressed in the following asymptotic form Lemma 7.2.
Proof. A generic term in Rm 2 can be expressed as where A, . . . , H ∈ {1, . . . , n, ξ}. From Lemma (5.2), we know f ρ = O(T − t) and so f −1 ρ is a bad term as it diverges as t → T . The only factor in (**) which can contribute to a f −1 ρ is g ξξ , and there are at most three g ξξ 's in (**). We are going to check that (1) whenever f −1 ρ appears in (**) exactly once, there must at least one factor of (log f ρ ) ρρ from the curvature components; (2) whenever f −2 ρ appears in (**), there must be a (log f ρ ) 2 ρρ factor from the curvature components; (3) it is impossible for f −3 ρ to appear in (**).
Combining these, it is not difficult to see Rm 2 satisfies the asymptotic form (7.1).
We start by arguing (1). Suppose there is exactly one f −1 ρ factor in (**), we can assume WLOG that either (C, D) = (ξ, ξ) or (E, F ) = (ξ, ξ). Suppose the former, we can check from the table of Riemann curvatures in P.20 that almost all R A ξF G terms have either asymptotics O(f ρ ) (which cancels out f −1 ρ ) or a (log f ρ ) ρρ factor.
There is only one exception: R ξ ξjk which has an O(1)-term from (log f ρ ) ρ . However, if both of R A CF G and R B DĒH are taken to be in this form, then (**) becomes where the g ξξ = O(f ρ ) cancels out the undesirable f −1 ρ factor, and end up with no f −1 ρ at all. Similar argument applies to the case (E, F ) = (ξ, ξ), and (1) is proved. For (2), since g ξξ is the only possible contribution to f −1 ρ , at least two of C, F, G (and their corresponding two of D, E, G) must be ξ. Check again the table of Riemann curvature components in P. 20, we see all the terms with two lower ξindexes must either of O(f ρ )-type or has a (log f ρ ) ρρ factor. It proves (2).
For (3), the only possible case for f −3 ρ to appear is that all of (C, D), (E, F ) and (G, H) are (ξ, ξ). The only possible choice for the curvature components are R l ξξξ and R ξ ξξξ . However, the former is 0. For the latter case, all indexes will be ξ and (**) becomes g ξξ g ξξ g ξξ g ξξ R ξ ξξξ R ξ ξξξ which can be computed easily as f −2 ρ (log f ρ ) 2 ρρ . Finally, we remark that g ξξ g ξξ g ξξ g ξξ R ξ ξξξ R ξ ξξξ is the only term that f −2 ρ (log f ρ ) 2 ρρ appears, thanks to the fact that R i ξξξ = 0. As a result, the leading term of (7.1) is f −2 ρ (log f ρ ) 2 ρρ with coefficient 1 which can be easily verified by computing g ξξ g ξξ g ξξ g ξξ R ξ ξξξ R ξ ξξξ .
Having understood the asymptotics of Rm 2 , we are in a position to study the singularity models. Let's first consider the Type I case: Theorem 7.3. Let M = P(O Σ ⊕L) be the projective bundle with the triple (Σ, L, [ω 0 ]) satisfying the conditions listed in P.2. Let (M, ω t ) be the Kähler-Ricci flow ∂ t ω t = −Ric(ω t ), t ∈ [0, T ) with initial Kähler class [ω 0 ]. Suppose the flow encounters Type I singularity, then choose (x i , t i ) in space-time such that K i := Rm(x i , t i ) g(ti) = max M Rm g(ti) and t i → T . Consider the rescaled dilated sequence of metrics g i (t) := K i g(t i + K −1 i t), t ∈ [−t i K i , (T − t i )K i ). Then the pointed sequence (M, g i (t), x i ) converges, after passing to a subsequence, smoothly in pointed Cheeger-Gromov sense to an ancient κ-solution (M ∞ , g ∞ (t), x ∞ ), whose universal cover splits isometrically as (C n × P 1 , dz 2 ⊕ ω FS (t)), where dz 2 is the Euclidean metric and ω FS (t) denotes the shrinking Fubini-Study metric.
Proof. Suppose C = C(n) is a constant depending only on n such that |R| ≤ C(n) Rm . Since the blow-up factor K i is defined by K i = max M Rm g(ti) = Rm(x i ) g(ti) , the scalar curvature at time t i satisfies |R(g(t i ))| ≤ CK i on M . One can compute the scalar curvature explicitly: R g(t) = Tr ωt Ric(ω t ) where F = log f ρ + n log f . Hence, Therefore, for any ρ ∈ [−∞, ∞] at t i , we have Recall that K i → ∞. Letting i → ∞ yields By considering the asymptotic expression of Rm 2 given by (7.1), we have for any ρ ∈ [−∞, ∞] at time t i , where equality is achieved at x i . Letting i → ∞ and using (7.2) and the fact that f ρ = O(T − t) from Lemma 5.2, we can deduce: (K i f ρ ) −2 (log f ρ ) 2 ρρ (xi,ti) = 1. (7.3) Recall that g i (t) = K i g(t i + K −1 i t), we then have Letting i → ∞, we have By strong maximum principle, the scalar curvature of every ancient solution must be either identically zero or everywhere positive. In our case, (7.3) and (7.4) together implies R g∞(0) = 1 and hence R g∞(t) > 0 on M × (−∞, 0]. By our splitting lemma 6.4, we know that the limit manifold M ∞ splits isometrically as a product N n 1 ×N 1 2 , such that T N n 1 = span R {ℜ(Z j ∞ ), ℑ(Z j ∞ )} n j=1 and T N 1 2 = span R {ℜ(Ξ ∞ ), ℑ(Ξ ∞ )}. As a result, the curvature tensors also split as Rm M∞ = Rm N n 1 ⊕ Ric N 1 2 . Next, we would like to compute the curvatures of each factor. Again, for simplicity we shrinking round 2-sphere.
To conclude, if the Kähler-Ricci flow (M, g(t)) is of Type I, then the universal cover of the limit solution (M ∞ , g ∞ (t)) of the rescaled dilated sequence g i (t) is isometric to (C n × P 1 , dz 2 ⊕ ω FS (t)).
Next, we will rule out the possibility of Type II singularity on (M, g(t)). We will show that by a standard point-picking argument for Type II singularity, one can form a rescaled dilated sequence of metrics which converges, after passing to a subsequence, to a product of the cigar soliton and a flat factor. By Perelman's local non-collapsing result, such limit model is not possible. Let's state this result and give its proof. (T i − (t i + K −1 i t)) Rm g(ti+K −1 i t) .
We denote K i = Rm (ρ i , t i ), then K i (T i − t i ) → ∞ by the Type II condition.