Convergence of general inverse $\sigma_k$-flow on K\"{a}hler manifolds with Calabi Ansatz

We study the convergence behavior of the general inverse $\sigma_k$-flow on K\"{a}hler manifolds with initial metrics satisfying the Calabi Ansatz. The limiting metrics can be either smooth or singular. In the latter case, interesting conic singularities along negatively self-intersected sub-varieties are formed as a result of partial blow-up.


Introduction
Geometric flows are powerful tools to study the metric, algebraic and topological properties of the underlying manifold. An important example is the Ricci flow introduced by Hamilton [H] three decades ago and its Kählarian version, the Kähler-Ricci flow [Cao]. Since then, both have developed into significant research fields. See [T,SW4] for more complete surveys and further references on the Kähler-Ricci flow.
In [FLM, FL], we have introduced the general inverse σ k -flow on compact Kähler manifolds, which is a generalization of the J-flow [Chen,Do,SW1]. In this paper, we study some concrete examples to explore several applications of the general inverse σ k -flow in algebraic geometry and fully non-linear partial differential equations.
We recall the definition of the general inverse σ k -flow. Let (X, ω) be a compact Kähler manifolds of dimension n, with a fixed Kähler form ω. Let χ be another Kähler form. For a fixed integer k ∈ [1, n], let c k = n k X χ n−k ∧ ω k X χ n , and define σ k (χ) = n k χ k ∧ ω n−k ω n .
Any stationary point of (1.1) corresponds to a metricχ ∈ [χ] satisfying the following Kählerian inverse σ k equation on X: Locally, (1.3) can be written as 2χ ij dz 1 ∧ dz j andχ −1 is the inverse of the matrix (χ ij ). Equation (1.4) leads to an obvious necessary condition for equation(1.3) to admit a smooth solution, which was first formulated in [SW1] for the J-flow. Define (1.5) such that nc k χ ′n−1 − n k (n − k)χ ′n−k−1 ∧ ω k > 0 }.
Note that for k = n, (1.5) holds for any Kähler class. Hence C n (ω) is the entire Kähler cone. If there exists a smooth metricχ ∈ [χ] solving (1.3), it is necessary that [χ] ∈ C k (ω). (1.6) Following our earlier work with Ma [FLM], we have shown that condition (1.6) is also sufficient for the existence of smooth solutions of (1.4) via the convergence of the flow (1.1) [FL].
Theorem 1.1. Let (X, ω, χ) be given as above, then the flow (1.1) has long time existence and converges to the critical metricχ satisfying (1.3) if and only if [χ] ∈ C k (ω).
It is easy to see that for k = 1 and F (x) = −x, (1.1) reduces to the J-flow. The smooth convergence of the J-flow has significant geometric implication on the properness of Mabuchi energy (see [SW1]). Another special case of the flow occurs when k = n, where the critical equation is a complex Monge-Ampère equation. If one takes F (x) = − log x, then the flow resembles Kähler-Ricci flow.
Same as the J-flow, the general inverse σ k -flow (1.1) always has long time existence (see [FL]). It is thus interesting to study convergence properties of the flow when the condition (1.6) fails to hold. While convergence to smooth metrics is no longer expected, due to the geometric set-up, blow-up behavior of the solution along proper subvarieties is expected (cf. [SW1]). In particular, it is our hope that the analytical behavior of the limit metric will reflect the algebro-geometric properties of the original Kähler manifold X.
From an analytical point of view, (1.3) deserves study in its own right. For k = n, it is a complex Monge-Ampère equation. If [χ] is Kähler, by Yau's renowned solution of Calabi conjecture [Y], (1.3) admits a smooth solution unique up to a constant. If [χ] lies on the boundary of Kähler cone, i.e., [χ] is nef not Kähler, then (1.3) becomes a degenerate complex Monge-Ampère equation. It is a subject of intensive study over past two decades following the pioneering work of Ko lodziej [K]. When χ is a big semi-positive form, the result in [EGZ] implies that (1.3) admits a bounded plurisubharmonic solution. Such solution of degenerate complex Monge-Ampère equation is used to produce singular Kähler-Einstein metrics on Kähler manifolds with indefinite anticanonical class [EGZ]. There have been Kähler-Ricci flow approaches for more general canonical singular Kähler-Einstein metrics. See [TZ,ST1,ST2] for details and further references.
For k = n, we refer (1.3) as the Kählerian inverse σ k equation. When [χ] lies on the boundary of C k (ω), this Monge-Ampère type equation degenerates in an intriguing way. Suggested by the convergence results we obtained on general inverse σ k flow, we conjecture an analogous result of [EGZ] on boundedness of the solution in pluri-potential sense still holds.
In this paper, we shall study the general inverse σ k -flow assuming certain symmetry of initial data. This is partially inspired by similar results on the Kähler-Ricci flow [SW2,SW3,SY] and on Kähler-Ricci solitons [L]. It is interesting to compare convergence behaviors of the general inverse σ k -flow with those of the Kähler-Ricci flow.
Theorem 1.2 (Main Theorem 1). Let X = P n #P n be P n blowing up at one point. Let E 0 and E ∞ be the exceptional divisor and the pull-back of the divisor associated to O P n (1) respectively. Assume that χ, ω are Kähler metrics on M satisfying Calabi Ansatz (see Section 2 for details) such that Let χ t be the solution of the flow (1.1), then the following convergence behavior of χ t holds: (2) If α k β n−k −1 Kähler metric that is smooth away from E 0 and has conic singularity at E 0 of angle π. Further more, there is a universal constant C such that the oscillation of the limiting potential ϕ ∞ satisfies and [E 0 ] is the current of integration along the exceptional divisor E 0 . As is in case (2), is also a singular Kähler metric with conic singularity with angle π transverse to E 0 . Remark 1.3. Note that in the case (3) of Theorem 1.2, χ ∞ can be also obtained as the limit of flow (1.1) from some smooth initial data (X, ω, χ) . This is an interesting example of partial blow-up in the sense of algebraic geometry using analytical tools. See [SW2,SW3,SY] for some corresponding results for the Kähler-Ricci flow.
Remark 1.4. For the J-flow on Kähler surface (k = 1, n = 2), the cone condition (1.5) reduces to a simple class condition: Donaldson [Do] noticed such condition holds for all Kähler classes [χ] and [ω] if there are no curves of negative self-intersection. He also conjectured if (1.8) fails to hold then one might expect the flow to blow up over some such curves. This was confirmed in [SW1] in the sense that the quantity |ϕ| + |∆ ω ϕ| blows up, where ϕ is the solution of the J-flow. In fact, their estimate is on any dimension n when the condition (1.5) is violated. They posed a question on improving these estimates. Our results give a partial answer to their question. In particular, case (2) of the theorem asserts that for the given X with initial metrics satisfying Calabi ansatz, |ϕ| stays bounded. = n−k n , which defines the boundary of C k (ω) inside of K(X). Therefore case (2) of Theorem 1.2 corresponds to [χ] lying on the upper boundary of C k (ω); case (3) of Theorem 1.2 corresponds to [χ] ∈ K(X) \ C k (ω). The limit χ ∞ of case (3) jumps to the class β[E ∞ ] − λ[E 0 ] which lies on the upper boundary of C k (ω). The relation (1.7) then follows. Note the constant α k β n−k −1 β n −1 in the Theorem 1.2 is in fact the topological constant Hence by Theorems 1.1 and 1.2, we obtain Corollary 1.5. Fix the notation as in Theorem 1.
In order to get examples involving singularities of higher co-dimension, we study the flow (1.1) on more complicated manifolds admitting the Calabi Ansatz.
Let X m,n = P(O P n ⊕ O P n (−1) ⊕(m+1) ) be a projective bundle over P n of total dimension m + n + 1. Let D ∞ be the divisor given by P(O P n (−1) ⊕(m+1) ) and D H be the pullback of the divisor on P n associated to O P n (1).
For future use, letX m,n be the blow up of X m,n along P 0 , where P 0 ⊂ X m,n is the projectivization of the section (1, 0, · · · , 0) ∈ O P n ⊕ O P n (−1) ⊕(m+1) . Note that P 0 is of dimension n. We denote the resulting exceptional divisor inX m,n by E.
Theorem 1.6 (Main Theorem 2). Let X m,n be as given as above. Assume that ω, χ are two Kähler metrics satisfying the Calabi Ansatz and (1) c k > n k : a singular Kähler metric with cone singularity of angle π transverse to P 0 as t → ∞; there is a universal contant C such that osc ϕ ∞ ≤ C.
(3) c k < n k : Let π be the blown up map: π :X m,n → X m,n . Then π * (χ t ) Note that in the case (3) above, when no confusion arises, D H and D ∞ are also used to denote the corresponding pulled-back divisors onX m,n .
Combining Theorem 1.1 and Theorem 1.6, we also obtain If ω satisfies the Calabi Ansatz, then C k (ω) is the entire Kähler cone whenever k > n; and when k ≤ n, Remark 1.8. For both case (2) of Theorems 1.2 and case (2) of Theorem 1.6, the critical equation (1.3) admits a bounded solution in the sense of pluri-potential theory. This indicates a general result should hold for this type of equations. Thus, the results of [K] and [EGZ] on solutions for degenerate complex Monge-Ampère equations may be extended to more general complex Monge-Ampère type equations. We would like to discuss this aspect in future works.
Remark 1.9. Similar to Remark 1.3, it is easy to construct a proper inverse σ k type of flow onX m,n such that the limiting metric under the flow coincides with that of case (3) listed above. The inverse σ k flow can thus be viewed as an analytical method to connect birationally equivalent varieties.
In this paper we only discuss some special Kähler manifolds with metrics satisfying strong symmetric conditions. However, the algebraic pictures revealed indicate that the geometric flows that we have studied can be used to transform between (possibly singular) algebraic varieties. In subsequent works, we will discuss general cases and their further applications to birational geometry.
The rest of paper is organized as follows. In Section 2, we study the J-flow on P n #P n as a prototype. In Section 3, we use the same method to treat the general inverse σ k -flow on P n #P n . In Section 4, we study the convergence behavior on X m,n , a class of more generality.
Acknowledgments: Both authors would like to thank Ben Weinkove, Jian Song and Lihe Wang for helpful discussion.
2. J-flow on P n #P n Let X = P n #P n be P n blowing up at one point. Denote [E 0 ] and [E ∞ ] the exceptional divisor and pull-back of the hyper-plane in P n respectively. We have H 1,1 (X, R) = span{[E 0 ], [E ∞ ]} and any Kähler class Ω on X is of the form: We recall Calabi Ansatz [Ca] on construction of rotational symmetric Kähler metrics on X. For notational convenience, let d = ∂ +∂ and In order to extend ω to a smooth Kähler metric on X, the following asymptotic properties of u are required: is extendable by continuity to a smooth function at r = 0, and is extendable by continuity to a smooth function at r = 0, and u ′ ∞ (0) > 0. It is easy to see, by the asymptotic behavior of u, that Moreover, since u ′′ (ρ) > 0, b > a. a and b characterize the Kähler class of ω in the following manner: In this section, we treat the J-flow, which is a special case of general inverse σ k flows. It is defined as follows: After normalization we may assume If both ω and χ satisfy the Calabi Ansatz, on the coordinate patch X

This leads to
It is easy to see that the J-flow preserves the Calabi Ansatz condition. Hence, we may assume that solution of the flow (2.2) is χ ϕ(·,t) = dd c v(ρ, t). Consequently, (2.2) is reduced to an evolution equation on v(ρ, t): The corresponding critical equation (1.3) is Taking one time derivative on (1.1), and applying maximum principle, we obtain the bound for two universal constants C 1 and C 2 depending only on the initial data.
In terms of potential u and v, (2.5) is To study flow (2.3), we regard (v ′ (ρ, t), u ′ (ρ)) as a family of parametric plane curves. Since u ′′ > 0 and v ′′ > 0, each curve is the graph of a strict monotone increasing function f (x, t), i.e., We are concerned with the corresponding evolution equation for f . Proof. Taking one time derivative of (2.7), we get Taking one spacial derivative of (2.3), we get Taking spacial derivatives of (2.7) we also have Plugging (2.10) and (2.11) in (2.9), we get . For a given u, since u ′′ > 0, then u ′ is monotone, it follows that we can write u ′′ as By the asymptotic behavior of u ′ (2.1), we have Q(1) = Q(α) = 0 and Q(f ) > 0 whenever 1 < f < α.
The boundary and initial conditions follow directly from the limit behavior of u ′ and v ′ .
To study the convergence behavior of (2.3), it suffices to study the convergence behavior of (2.8). This is a degenerate parabolic equation of one spacial dimension. A priori, we know the long time existence and we also have a uniform C 1 bound on f by (2.6), i.e., ∂f ∂x = u ′′ v ′′ ≤ C 2 . Therefore we have a uniform limit It follows that f ∞ (x) is a weak solution of the corresponding stationary problem: Equivalently, we can write (2.14) as subject to the boundary condition f (1) = 1 and f (β) = α, where Ψ . is the characteristic function of a set. Standard elliptic theory shows that f ∞ is a strong solution of (2.14) and (2.15).
It is straightforward to calculate all possible solutions of (2.15). Piece-wisely, they are either constant functions f = 1 and f = α, or solutions of the differential equation which can be written as Ax + B x n−1 , for appropriate constants A and B. Since f (x, t) is monotone for all t, limit f ∞ can only be of the form where g is the solution of (2.16) on [s, t].
Due to the degeneracy condition on Q, f ∞ may lose regularity at x = s and x = t. Nevertheless, on any compact subset in which {1 < f ∞ (x) < α}, (2.8) is uniform elliptic, and we obtain higher order estimates (cf. [W]) from general theory of nonlinear parabolic equations, consequently the convergence (2.13) is smooth in region {x|1 < f ∞ (x) < α}.
Consider the ODE with boundary value f (1) = 1 and f (β) = α, it has a unique solution: We now have the following Lemma 2.2. Let ϕ(x, t), ψ(x, t) ∈ P χ be two solutions of the flow (1.1) with initial values ϕ 0 and ψ 0 respectively. Let f t ,f t be their corresponding functions satisfying (2.8). Then there exists a universal constant C such that Proof. Taking difference of flows with two initial values, we get ϕ(x, t) − ψ(x, t) satisfies a parabolic equation where 0 < s t < 1. Inequality (2.20) then follows from the maximum principle.
For the second part of the lemma, we consider the corresponding limits for Thus, |v ∞ −ṽ ∞ | is not uniformly bounded unless s =s and t =t. We have thus proved the lemma.
Let g be the corresponding solution of (2.18) with g(λ) = 1 and g(β) = α, we have in this case To better understand the theorem, we illustrate initial and limit functions of four cases by Figure 2.

Proof.
Note that for the first three cases, 1 < f ∞ (x) < α if 1 < x < β. In Case 1, f ∞ (x) is a concave function. In Cases 2 and 3, f ∞ (x) is convex. We distinguish Case 2 and Case 3 by the fact that f ′ ∞ (1) > 0 in Case 2 and f ′ ∞ (1) = 0 in Case 3. Case 1: α > β By Lemma 2.2, we may choose a special initial value Claim 2.4. With conditions given as above, Proof of the claim: This is a simple application of the strong maximum principle. Since f ∞ is the solution of (2.18) on [1, β] Claim 2.5. ∂f ∂t ≥ 0. Proof of the claim: A direct computation shows that ∂f ∂t | t=0 > 0.
We can prove the claim by taking time derivative of (2.8), and applying the strong maximum principle.
Combining Claim 2.4 and Claim 2.5, we find that f (x, t) is monotone increasing to a limiting function f ∞ (x) = lim t→∞ f (x, t) and However, there is only one solution of the form (2.17) satisfying (2.21), which is exactlyf (x). Thus we have proved that f ∞ (x) =f (x) for Case 1.
For simplicity, we omit the proofs of these claims.
Since f (x, t) is monotone increasing with respect to t, and f ∞ (x) ≥f (x). f ∞ has to be the unique solution satisfying (2.17), which isf . We have thus finished the proof for Case 2.
By the characterization of λ and g(x), we obtain our conclusion.
We have proved Theorem 2.3.
Remark 2.8. It is interesting to remark thatg in Case 4 of Theorem 2.3 arises as a solution to an obstacle problem(cf. [C]). In fact, in the convex set The unique minimizer of E in K satisfying Remark 2.9. It worths pointing out that for all cases of Theorem 2.3, f ′ ∞ (x) is continuous. Further more, for Case 3 and Case 4, we have This is an important feature of the limiting function that will indicate geometric properties for the geometric flow.
The limiting behavior described for f (x, t) can be used to determine the convergence behavior of metrics following the inverse σ k flow.
Proof of Main Theorem 1: J-flow case. We will divide the proof into four cases, just as in the proof of Theorem 2.3.
Case 1: α > β First, we claim that there exists a uniform positive lower bound for ∂f ∂x . Suppose not, then there is sequence of (x n , t n ) such that Then there must be an accumulation point x ∞ in [1, β] for a subsequence of {x n }, which still denoted by {x n } for simplicity. On the other hand lim n→∞ ∂f ∂x (x n , t n ) = f ′ ∞ (x ∞ ) = 0. Thus this contradiction proves the claim. Therefore there exists a universal constant ǫ such that ∂f ∂x ≥ ǫ > 0. Consequently, there exists a universal constant C > 0 such that along the flow (2.2) Higher order estimates of χ ϕ follow from Evans-Krylov and Schauder estimates. Consequently χ t converges smoothly to χ ∞ , which solves the critical equation (1.3).
Proof. Suppose that the claim does not hold. Then there exists sequence of (x n , t n ) such that ∂f ∂x (x n , t n ) → 0, t n → ∞.
We also have that a subsequence of x n converges to x ∞ ∈ [1, β], for simplicity still denoted by we get a contradiction, thus we have proved the claim.
We continue our proof of Case 3, Main Theorem 1 for J-flow. For any compact subset K ⊂ X \ E 0 , there exists a constant ǫ > 0 such that K ⊂ u ′−1 ([1 + ǫ, α]). By Claim 2.10, for any t > 0, the corresponding f (x, t) defined by (2.7) satisfies f x (x, t) ≥ c(ǫ) for a given c(ǫ) > 0. We may conclude that χ t converges smoothly to χ ∞ , following arguments given in the proof of Case 1. We have thus established the smooth convergence of the J-flow away from E 0 .

The corresponding function
is thus a potential for a Kähler metric with singularity along E 0 . In this case, it is easy to see that dd c v ∞ can be extended as a smooth Kähler form to E ∞ .
On the other hand, near ρ = −∞, the resulting Kähler form is not smooth. In In the local coordinate patch (z 1 , · · · , z n ) centered at any p ∈ E 0 , where E 0 ∩ U = {z 1 = 0}, the metric is equivalent to where ω F S is the Fubini-Study metric on E 0 . Thus the metric is singular with cone angle π transverse to E 0 .
Case 4: α < β, αβ n−1 −1 β n −1 < n−1 n Similar to Case 3, we may prove that the flow is convergent smoothly away from E 0 . However, now we have ρ) and its resulting metric flow may be understood, from a geometric point of view, as two distinct behaviors: one is a δ-concentration on E 0 with coefficient (λ − 1); the other is a smooth convergence of metrics away from E 0 . For x ∈ (λ, β], it is easy to see that v ′ ∞ is the corresponding limit solution for the J-flow of the triple (X, ω, χ) with Since g ′ (λ) = 0, this corresponds to the Case 3. One can readily check by scaling We have thus finished the proof.

CONVERGENCE OF GENERAL INVERSE σ k -FLOW ON KÄHLER MANIFOLDS WITH CALABI ANSATZ 15
3. General inverse σ k -flow on P n #P n In this section, we discuss the general inverse σ k -flow (1.1) on X = P n #P n . We follow the discussion of Section 2, however the parabolic equation analogous to (2.8) is more complicated.
Let (X, ω, χ) be given as before. Assume both ω and χ safisfy Calabi Ansatz. The general inverse σ k -flow (1.1) for k = 1 can be written as Again suppose (v ′ (ρ, t), u ′ (ρ)) is a family of parametric curves implicitly given by then the evolution of f is By (1.2), F ′ < 0, the convergence behavior of (3.3) is same as that of (2.8).

General case (k > 1).
For general k > 1, (1.1) reduces to then f defined by (3.2) evolves as (3.5) Define g(x, t) := f k (x, t), then the evolution of g is Proof of the Main Theorem 1. As in Section 2, we need to discuss the corresponding ODE: with boundary values g(1) = 1 and g(β) = α k .
Let g be the corresponding solution of (3.7) with g(λ) = 1 and g(β) = α k . Then the unique limit in this case is By the definition of λ, we have g ′ (λ) = 0. Then it is easy to check that λ ∈ (1, β) is the unique solution of Finally, notice we still have the universal constants C 1 , C 2 > 0 such that from (3.10) we get a uniform upper bound for u ′′ v ′′ . The rest of the proof follows that of Section 2.

Flows on P(O
In this section, we consider the general inverse σ k -flow on a family of projective bundles over P n and its convergence behavior under the assumption both ω, χ satisfy the Calabi Ansatz. A new geometric limit phenomenon occurs. Let E = O P n ⊕ O P n (−1) ⊕(m+1) be a vector bundle over a projective space P n , where O P n is the trivial line bundle and O P n (−1) is the tautological line bundle. Let X m,n = P(O P n ⊕ O P n (−1) ⊕(m+1) ) be the projectivization of E. X m,n is a P m+1 bundle over P n with π : X m,n → P n being the bundle map. In particular, X 0,n is P n+1 blown up at one point. Let D ∞ be the divisor in X n,m given by P(O P n (−1) ⊕(m+1) ) and D 0 be the divisor in X m,n given by P(O P n ⊕ O P n (−1) ⊕m ). In fact, the additive divisor group N 1 (X m,n ) is spanned by [D 0 ] and [D ∞ ]. We also define the divisor D H by the pullback of the divisor on P n associated to O P n (1). Then To consider the Calabi Ansatz (See [C, SY]), let ω F S be the Fubini-Study metric on P n . Let h be the hermitian metric on O P n (−1) such that Ric(h) = −ω F S . Under local trivialization of E, we write e ρ = h(z)|ξ| 2 , ξ = (ξ 1 , ξ 2 , · · · , ξ m+1 ), where h(z) is a local representation of h. In particular, if we choose an inhomogeneous coordinate z = (z 1 , z 2 , · · · , z n ) on P n , we have h(z) = 1 + |z| 2 .
If ω, χ are given as above, then the evolution of f (x, t) defined via (4.7) is Proof. From (4.4), we may calculate the eigenvalues of χ with respect to ω to be: Taking time derivative of (4.7), we get Taking spacial derivative of (1.1), we get We also have Similarly, We also have Using (4.11), (4.12), (4.13) and (4.14) and (4.10), we get Following our method developed in Section 2, we consider the ODE  Here c k is the topological constant n+m+1 k Xm,n χ m+n+1−k ∧ ω k Xm,n χ m+n+1 . with equality holds only at x = 0. By definition, (4.21) from which c k ≥ n k follows. It is clear that f ′ (x) ≥ 0. We claim that the function f is strictly increasing. If not, there exists a Both f x and 1+f 1+x are strictly decreasing near x 0 . Note that Hence the points where f ′ = 0 are discrete. It follows that f is strictly increasing.
Finally, to calculate f ′ (0), we expand (4.19) near x = 0 and compare the lowest order terms of both sides. Assume f (x) = Ax + higher order terms, We may derive that (1 + t) n m + 1 = n k m + 1 x m+1 + higher order terms. (4.23) Hence A = f ′ (0) > 0 if and only if α > n k when k ≤ n and α > 0 when k > n.
If c k < n k , the solution of (4.16) with the boundary condition f (0) = 0 and f (b ′ ) = b is not positive near x = 0. We define and let f be the corresponding solution of (4.16) with f (λ) = 0 and f (b ′ ) = b. By the definition of λ, f ′ (λ) = 0.  It is easy to see there is a unique solution (α, β, λ) to (4.26).
Then 1+ft 1+x ≤ 1 as well. Take x = x 0 in σ k ( , by (4.29) and lower bound in (4.28), we get a uniform lower bound C on ft(x) x depending only on C 1 , k, m, n, b, b ′ not on t. We have finished the proof of Claim 4.6.
We continue our proof of Claim 4.5, Case 2. By Claim 4.5, both ft(x) x and 1+ft(x) 1+x are bounded uniformly from below, using the upper bound in (4.28) again, we get a uniform upper bound for f ′ t . We have thus finished the proof of Claim 4.5.
One more thing to mention is the handle of nonlinearity in this situation. As a matter of fact we already have seen that Finally, let us discuss the geometric behavior. First of all, if f ′ ∞ (0) > 0, which is the case for k > n and c k > n k when k ≤ n, then as previously discussed, we get smooth convergence.
If f ′ ∞ (0) = 0, we have smooth convergence away from {ρ = −∞} which corresponds to P 0 . Then the corresponding Kähler metric dd c v ∞ has a conical singularity of cone angle π transverse to P 0 . P 0 can be also regarded as the intersection of m+1 effective divisors P(O P n ⊕ O P n (−1) ⊕m ), which is of codimension m + 1. Therefore the convergence of general inverse σ k -flow can produces Kähler metrics which are singular on subvarieties of higher codimension.
We consider Kähler metrics of the form with u satisfies proper asymptotic behavior (2.1) near ρ = −∞ and ρ = ∞. The convergence behavior is very similar to that on P n #P n so we will omit the detail here.