Type I Almost-Homogeneous Manifolds of Cohomogeneity One — IV

This paper is one of a series in which we generalize our earlier results on the equivalence of existence of Calabi extremal metrics to the geodesic stability for any type I compact complex almost homogeneous manifolds of cohomogeneity one. In this paper, we actually carry all the earlier results to the type I cases. In Part II, we obtained a substantial amount of new Kähler–Einstein manifolds as well as Fano manifolds without Kähler–Einstein metrics. In particular, by applying Theorem 15 therein, we obtained complete results in the Theorems 3 and 4 in that paper. However, we only have partial results in Theorem 5. In this note, we provide a report of recent progress on the Fano manifolds Nn,m when n > 15 and N′ n,m when n > 4. We provide two pictures for these two classes of manifolds. See Theorems 1 and 2 in the last section. Moreover, we present two conjectures. Once we solve these two conjectures, the question for these two classes of manifolds will be completely solved. By applying our results to the canonical circle bundles, we also obtain Sasakian manifolds with or without Sasakian–Einstein metrics. These also provide open Calabi–Yau manifolds.


Introduction
This paper is the fourth part of [1].In [1], we prove the following: Proposition 1.For any simply connected Type I, compact, Kähler, complex, almost-homogeneous manifold of cohomogeneity one with a hypersurface end, there is an extremal metric in a given Kähler class if and only if Condition (7) in [1] holds.Condition (7) therein was represented as a sign of a topological integral, as was shown in Section 6.1 of Part III [2].See also our integral at the beginning of the next section.
We obtained similar results for the higher codimensional end case and the general case, in Sections 3 and 4 of Part II [3].See also Theorem 15 in the last section for the Kähler-Einstein case therein.
As an application, considering the canonical circle bundle, we also obtained Sasakian manifolds with and without Sasaki-Einstein metrics (with the same Reeb vector field and CR structure, see [4] Theorem 2.4 (iv), also [5,6]).
We shall prove the converse for the Type II cases in [7].We finished all cases in which the existence of the extremal metrics could be reduced to an ordinary differential equation problem.We also provided many examples for both the stable and unstable cases, as promised in [8].It was difficult for us to find any example that is semistable but not stable.We note that, since the automorphism group was semisimple, the original Futaki invariants were zero for all manifolds considered in this paper.Therefore, we provide more classical examples than in [9].
The authors in [10] studied a few of the first cases in [1], i.e., the cases with F = F(OP n ) in which S = SO(4, C), SO(6, C), SO(8, C), SO(10, C), where S is the induced group action on the fibers.
We classify the manifolds into three types: Types I, II, and III.This classification can be found in [8] Section 12.
Basically, Type III manifolds are completions of a C * bundle over a homogeneous manifold.They were dealt with in [11,12].See also [13].Once the automorphism group has a nontrivial solvable radical, it has a C * action.It then has a Type III structure.See [8,14,15].
In the case in which the automorphism group is semisimple, let K be the isometric group, which is compact, and let L be the isotropic subgroup of K at a point of a generic real hypersurface orbit.Let N = Norn K (L) be the normalizer.Then on the Lie algebra level N is a direct product of the Lie algebra of L and an algebra A. A is either of real dimension one or three.When dim A = 1, its Lie group is a torus.When dim A = 3, it is so(3).We call the corresponding manifolds Types I and II.One can look at [2] for more information.See [16][17][18][19] for examples.
In [1], we used very explicit and elementary calculations to avoid the Cauchy-Riemann structure and other very abstract tools in [10,20] that also cumulating results from other papers of Spiro.
We dealt with the uniqueness in the second section therein, where we proved the existence of smooth geodesics in the Mabuchi moduli spaces of the Kähler metrics.Proposition 2. For any two smooth Kähler metrics, which are equivariant under the maximal compact subgroup in a given Kähler class on a Type I compact complex almost-homogeneous manifold of cohomogeneity one, there is a smooth geodesic connecting them in the Mabuchi moduli space of Kähler metrics.
The same result was proven for the toric manifolds and the Type III manifolds in [21], for some Type II manifolds in [8], and for all Type II manifolds in [7].
We also obtained many Kähler-Einstein manifolds as well as Fano manifolds without the Kähler-Einstein metric of Type I in the fifth section of Part II [3].The Futaki invariants are zero in our case since the automorphism groups are semisimple.It turns out that our method is also easier than that in [22] for finding Kähler-Einstein metrics since they depended on the zeros of Futaki invariants, which might be very rare.Therefore, their method is more suitable for finding manifolds without any Kähler-Einstein metric.In our case, both the Kähler-Einstein manifolds and manifolds without Kähler-Einstein metric might be dense in a certain Zariski sense.It seems to us that it is hard to see any example with a vanishing generalized Futaki invariant.Furthermore, most of our manifolds are Fano, but it is not that easy in [22].Moreover, it is very easy to check that [22] can be a Corollary of [23] and does not involve much stability.
In Part II [3], Theorems 3 and 4, we completely solved the existence of the Kähler-Einstein metric for the two classes of manifolds M n,m and M n.m therein.
To understand better the M n in our Section 5 in Part II [3], that is, the M n,1 in our Theorem 3 there, it is not difficult to see that M n is a Q n bundle over Then it is our manifold with F = CP n .Therefore, N n is never Fano (see [24,25]).To construct M n , we notice that by [26] (pp.590-593) there is a holomorphic conformal structure on Q n+1 with respect to the line bundle The exceptional divisor comes from the zero set of the corresponding holomorphic symmetric 2-tensor in N n .Therefore, the vector bundle O(1) ⊕ T Q n+1 also has a conformal structure with respect to N. M n is just the zero set of the corresponding holomorphic symmetric 2-tensor in As in the early parts, we shall deal with Lie algebras.A general reader might use [27] as a reference.In Part II [3], Theorem 5, we completely solved the integral problem for N n,m .They are all homogeneous and hence Kähler-Einstein.However, for N n,m and N n,m therein, we only had partial results.
From Part II [3], the picture in the case of the C k = Sp(k, C) fiber structure (the paragraph after Theorem 9, compared with Theorems 3-5 and the results of this note, for example) is quite different from that in the case of SO(n + 1, C) or Spin(7, C) structures.The results of Theorems 6-12 are quite complete compared with Theorem 5 in that paper.
In the next two sections, we shall provide more details.In Section 3, we provide two new theorems and some conjectures, which might lead to a complete solution of the two classes of manifolds N n,m and N n,m .

Preliminaries
Now, we apply our arguments in [1][2][3]8].In this note, we only consider the case in which S = SO(n + 1, C).In Part II [3], we first consider the case in which G = S = SO(n + 1, C).Let m be the codimension.Then as before we have the integral in Theorem 15 of Part II [3]: Here, K(F) is related to the canonical line bundle.We know that K(F) = −n − 1 if F = CP n and −4k if F = Gr(2k, 2), and K(F) = −n if F = Q n .See the paragraph right before Theorem 15 in Part II [3].In the case of F = CP n , D = Q n−1 and therefore D(F) = 2.In the case of F = Q n , D = Q n−1 , which is the intersection of Q n ⊂ CP n+1 with a hyperplane.Therefore, D(Q n ) = 1.We also have D(Gr(2k, 2)) = 2. See the paragraph after Theorem 1 in Part II [3].In both cases of N n,m and N n,m , the divisors are of codimension 1.That is, m − 1 = 0.The upper limits of our integrals are −K(F).In both cases, we have From the proof of Lemma 6 in [1], we see that the eigenvalues of the Ricci curvature of the exceptional divisor in the direction other than the 1-strings of roots with zero eigenvalues are represented as where s are indices, similar to i.In the same way, the eigenvalues of the restriction of the Ricci curvature of the whole manifold in those direction are represented by Therefore, when F = CP n , we have When F = Q n , Ric s are the same and Therefore, according to the calculation in Section 6.1 of Part III [2] or Theorem 2 in Part I [1], we have Another way to get the a ρ,i is to look at the fiberation We see that a ρ,i are just the eigenvalues of the Ricci curvatures of G/P.However, a ρ,i only occurs in our volume if e 1 acts non-trivially on the corresponding root.The concrete a ρ,i can be calculated by our Theorem 2 in Part I [1].There, we have the Ricci curvature formula for G/P: There, the sum is over all the positive roots which contribute to the tangent space of G/P , i.e., the positive roots that are not in P.
For example, when S = SO(8, C) and G = SO(10, C) as in Item 4 in [10], the Ricci curvature of G/P is decided by H α .Here we assume that S is generated by e i , i > 1.Therefore, the parabolic subgroup P is determined, which is why we put the first index 2 to imply e 2 as our e 1 earlier and the second index 4 as S = D 4 .We see that q G/P = 8e 1 , and the only positive roots which are not in P with nontrivial e 2 actions are e 1 ± e 2 .Therefore, a ρ,1 = (q G/P , e 1 ± e 2 ) = 8.Therefore, it is not Fano for the CP 7 case, and the coefficient in Item 4 of [10], was wrong (it was 16 instead of 8).Moreover, G/P has a complex dimension of 8.This comes from e 1 ± e i .However, only e 1 ± e 2 are nontrivial by e 2 .This is why we have a power of 1 in Part II [3] right before Theorem 3; however, in [10], they had a power of 4. The one with a F = Q n fiber is our M 7,1 in our Part II [3].
One series of examples that we already know are the product P n of two copies of CP n and M n , N n in [8,14,15].Since M n are Kähler-Einstein, so are P n and N n , whenever it is Fano.

New Kähler-Einstein Metrics
If the Kähler class is the Ricci class, we have, as in Part II [3], Section 5, α = u c , l = l ρ , and In this section, we shall check case by case on the type of the groups (S, G).We say that a manifold is nef if the anti-canonical line bundle is nef.We say that a manifold is Fano if the anti-canonical line bundle is positive.
for the case in which F = CP n and for the case in which F = Q n .Therefore, there is a Kähler-Einstein metric.Again, this is known since the manifolds are homogeneous.The same formula is true for G = S = D k , n = 2k − 1.Now, we consider the situation in which G = B k+1 and S = B k .We have The manifolds are nef but not Fano (or are Fano).The same is true for G = D k+1 and S = D k .We only need to consider the case in which F = Q n .The integral is Otherwise, I n < 0. Therefore, the corresponding manifolds M n are Kähler-Einstein for n ≤ 4.
Others are non-Kähler-Einstein Fano manifolds.Each M n is a Q n bundle over Q n+1 .Now, we consider the general situation in which S = SO(n + 1, C) and G = SO(2m + n + 1, C), and P is the smallest parabolic subgroup of G containing S as a simple factor.In this case, They are nef but not Fano (or are Fano).We have the integrals: For m = 2, we can use Mathematica to check I n,2 with and have I 4,2 > 0 but I 3,2 < 0. In the same way, we can use Mathematica and check that I 3,m < 0 for m ≤ 7 but I 3,8 > 0.
We denote the corresponding manifolds by M n,m .Therefore, we have in Part II [3] the following: Proposition 3. The manifolds M n,0 are homogeneous Kähler-Einstein manifolds.
are nonhomogeneous Kähler-Einstein manifolds.Other M n,m are Fano manifolds without any Kähler-Einstein metric.
Next, we consider the case in which S = SO(n + 1, C), G = SO(2m + n + 1, C), and S 1 in Section 3 of [1] is maximal.In this case, we have positive roots in ∆ + − ∆ P m+1,k , e.g., when n = 2k − 1, e i ± e j , i ≤ m < j, e i + e l , i < l ≤ m.For each i, there are only e i ± e m+1 , which are nontrivial by e m+1 .
(q G/P , e i ± e m+1 ) = 2k with m > 1.When m > 1, they are all Fano.The integral is Therefore, J n.m < 0 (or J n,m < 0) when m is high enough.Additionally, we can compare the change rate of the factor for different m and n values.We let Therefore, if J n,m ≤ 0 with m ≥ n, then J n,m+1 < 0. The same thing is also true for J n,m .Now, we can use Mathematica to check J n,n with Integrate[(n-1-v) v^(n-1) ((2n)^2 -v^2 )^(n) , {v, 0, n}] and obtain J n,n < 0 when n = 3, 4 but J 5,5 > 0.
Similarly, we use Mathematica to check J 4,m for m = 2, 3 and J 3,m for m = 2.We find that all of them < 0. Therefore, J 3,k , J 4,k < 0 if 2 ≤ k, as are J 3,k , J 4,k .
In the same way, we find J 6,k > 0 for 2 ≤ k ≤ 5. Therefore, J 6,k > 0 if k ≤ 26; otherwise, J 6,k < 0. We can also check that J 6,k < 0 for all k > 1.One might expect that, in all cases, J n,m < 0. However, we have J n,n > 0 for n = 101, 51, 26, 25, J n,n < 0 for n = 11, 21, 24.Thus, one might expect that J n,m < 0 for n ≤ 24.However, we have J n,m > 0 for (n, m) = (24, 12), (18,9), (16,8), etc.One can check that J n,m < 0 for n ≤ 15.Moreover, J 16,m > 0 if 5 ≤ m ≤ 8; otherwise, J 16,m < 0. We can also check that J n,2 < 0. Furthermore,  We obtain Theorem 1 by a direct calculation with the help of a computer.Using a different method, we are able to extend Item 8 in Theorem 1 to n = 6000.It seems to us that it is true at least up to n = 200,000.This is an undergraduate project taken by the second author in the University of California at Riverside.We believe that it is worth publishing at this stage so far.
One might think that the same would be true for N (n).Although N (n) will not be asymptotic when n is large enough, if b(n) has a limit, then again, because everything are integers, we would obtain N (n) when n is large enough.
We should get further calculation later on.However, we could have the following conjecture now: Conjecture 2. There is a high N and an integer B such that whenever n > N, we have b(n) = B.
by the fourth paragraph after Theorem 6 in Part I [1], we have m 2 = 0.By the paragraph after Theorem 4 in Part I [1], we have c = 1.By the second paragraph of Section 3 in Part II [3], we have l ρ = −(n

Conjecture 1 .
We conjecture that N(n) has an asymptotic line 2n − a.Since all function points and values are integers, if there is truly an asymptotic line 2n − a, then we have N(n) = 2n − a eventually.More precisely, one might have the following: The manifolds M m,n are Kähler-Einstein if and only if m > N(n) or m is that in the last part of Theorem 1.Moreover, N(n) = 2n − 31 whenever n > 286.