Malek Abid
email: abid@irphe.univ-mrs.fr

Christian Kharif

Effect of water vorticity on wind-generated gravity-waves in finite depth

Keywords: Shear instability, Rayleigh equation, wind-wave generation, vorticity 1 Introduction

], in the shallow Lake George (Australia), and the least squares fit of these data by [Young (1997)]. It is shown that without vorticity in water there is a deviation between theory and experimental data. However, a good agreement between the theory and the fit of experimental data is obtained when negative vorticity is taken into account. Furthermore, it is shown that the amplitude growth rate increases with vorticity and depth. A limit to the wave energy growth, corresponding to the vanishing of the growth rate, is obtained. The corresponding limiting wave age is derived in a closed form showing its explicit dependence on vorticity and depth. The limiting wave age is found to increase with both vorticity and depth.

parallel shear flow described by a boundary layer in the air above a flat surface of irrotational water of infinite depth, at rest. Miles found that the transfer of energy occurs at the critical layer level where the wind velocity equals the phase velocity of the surface waves. [Hristov, Miller & Friehe (2003)] confirmed experimentally the importance of the critical layer theory on the structure of the wave-induced air flow in open ocean. Many theoretical and numerical investigations in deep water have been devoted to this problem, among them one can cite [Valenzuela(1976)], [Kawai(1979)], [Beji & Nadaoka(2004)], [ Stiassnie & al. (2007)] and more recently [Young & Wolfe(2014)]. For a detailed review on wind wave generation one can refer to the book by [Janssen(2004)]. Very recently, within the framework of deep water, [Kharif & Abid (2020)] found in the case of water flows of negative vorticity a limit to wave energy growth depending on the magnitude of the vorticity. Until now there is no theoretical study on the effect of finite depth on wind-wave generation, except the paper by [Montalvo & al. (2013)] who considered an irrotational water flow. They considered the Miles' theory of wave amplification by wind in the case of finite depth and found that the wave growth of young waves is comparable to that of deep water whereas for old waves a finite-depth limited growth is reached. [START_REF] Young | [END_REF]] conducted experiments in a shallow lake which is 20 km long by 10 km wide with an approximately uniform water depth of 2 m, and collected data set of approximately 1000 observations. From these experiments they showed that at large fetch the evolution of both the total energy and peak frequency ceases, and both parameters become depth limited. They confirmed the existence of an asymptotic limit to growth depending on the depth. However, they have not considered the possibility of the presence of vorticity in water that is likely to occur in a shallow lake due to wind at the water surface and friction at the bottom. This feature will be addressed in the sequel.

In this study we revisit the Miles theory of wind wave generation at the surface of a pre-existing underlying water flow of constant vorticity in finite depth. Constant vorticity is the first approximation that one may consider to simplify the mathematical computation to obtain preliminary results on the effect of the vorticity on wind wave generation in shallow water, and the first step toward a more general situation with depth-dependent vorticity. We consider the case of a logarithmic wind profile in the air. In section 2 the mathematical formulation is presented. Section 3 is devoted to the validation of our numerical results. The finite depth effect in the presence of vorticity is presented in section 4, as well as the comparison of our theoretical results with the experimental data of [START_REF] Young | [END_REF]] and the least squares fit of these data by [Young (1997)]. Our conclusions are summarized in section 5.

Mathematical formulation

The approach developed in what follows is similar to those of [Janssen(2004)] and [Thomas(2012)] except that we now consider a water flow of constant vorticity in finite depth. For a detailed description of the method without water vorticity in deep water one can refer to [Thomas(2012)].

The inviscid governing equations of the flow in air and water are the following

∇ • u = 0, (1) du dt = - ∇p ρ + g, (2)
dρ dt = 0, (3)
with d dt = ∂ ∂t + u • ∇,
and where u is the fluid velocity, ρ is the fluid density, p is the pressure and g is the acceleration due to gravity. Equation (1) corresponds to mass conservation, equation ( 2) is the Euler equation and equation ( 3) means incompressible fluids.

We consider the linear stability of the following solution of the system of equations (1)-( 3) which corresponds to a flat air-water interface

u = U 0 e x , g = -ge z , ρ = ρ 0 (z), p 0 (z) = g ρ 0 (z)dz, (4)
where U 0 corresponds to the velocity in the air and in the water, ρ(z) corresponds to atmospheric density and water density and e x and e z are unit vectors in the x-direction and z-direction, respectively.

U 0 (z) = U a (z) , z > 0 U w (z) , -h < z < 0 , ρ 0 (z) = ρ a , z > 0 ρ w , -h < z < 0 ,
where U a is the wind velocity and U w the flow velocity in the water, h is the water depth, ρ a and ρ w are the atmospheric density and water density, respectively. We assume the conservation of the momentum flux in the atmospheric boundary layer and the water flow is assumed to be vertically sheared with constant vorticity. Consequently The following velocity profiles will be used:

U a (z) = u * κ ln(1 + z z 0 ), U w (z) = U s + Ωz,
where u * is the friction velocity, κ is the von Karman constant and z 0 is the roughness length of the air-water interface given by the Charnock relation z 0 = α ch u 2 * /g. For the flow velocity in the water the shear Ω and U s are constant.

Let us perturb the equilibrium given by equations ( 4) with an infinitesimal perturbation

U = U 0 + u , p = p 0 + p , ρ = ρ 0 + ρ . (5)
Note that the fluid is incompressible and nonhomogenous. Therefore, the fluid density is constant on streamlines and this is ensured by equation (3). Consequently, this conservation leads to an equation for ρ .

Substituting the expressions (5) into equations ( 1)-( 3) and linearising gives

∂u ∂x + ∂w ∂z = 0, (6) ∂u ∂t + U 0 ∂u ∂x + w U 0 (z) = - 1 ρ 0 ∂p ∂x , (7)
∂w ∂t + U 0 ∂w ∂x = - 1 ρ 0 ∂p ∂z + ρ ρ 2 0 dp 0 dz , (8)
∂ρ ∂t + U 0 ∂ρ ∂x + w dρ 0 dz = 0, ( 9
)
where w is the vertical component of the velocity perturbation.

The solutions of the linearized problem are sought in the following form

(normal modes) [u , w , p , ρ ] = [u 1 (z), w 1 (z), p 1 (z), ρ 1 (z)] exp[i(kx -ωt)], (10)
where k and ω are the wavenumber and frequency of the perturbation, respectively.

Substituting the expressions (10) into the linearized equations and using g = -p 0 /ρ 0 gives the following Sturm-Liouville problem

d dz ρ 0 W 2 dψ dz -k 2 ρ 0 W 2 + g dρ 0 dz ψ = 0, (11)
where

W = U 0 -c, c = ω/k and ψ = w 1 /W .
As stated above, the water flow is assumed to be vertically sheared with constant vorticity: U 0 = U s + Ωz, where the shear Ω and U s are constant. Note that the vorticity is -Ω. Without loss of generality we consider a frame of reference in which U s = 0. Hence, equation ( 11) in water reads

d dz ρ w (Ωz -c) 2 dψ w dz -k 2 ρ w (Ωz -c) 2 + g dρ w dz ψ w = 0.
Let us assume dρ w /dz = 0, then

(Ωz -c) d 2 ψ w dz 2 + 2Ω dψ w dz -k 2 (Ωz -c)ψ w = 0. ( 12
)
A critical layer in water corresponding to Ωz -c = 0 exists for negative values of Ω (positive vorticity). Consequently, (12) reads

d 2 ψ w dz 2 + 2Ω Ωz -c dψ w dz -k 2 ψ w = 0. ( 13
)
The equation ( 13) can be transformed to a reduced form

d 2 θ dz 2 (z) + q 1 (z)θ(z) = 0, ( 14
)
with the following change of variables

ψ w (z) = θ(z) exp - 1 2 z 0 2Ω Ωz -c dz , q 1 (z) = -k 2 - 1 2 d dz 2Ω Ωz -c - 1 4 2Ω Ωz -c 2 , that is ψ w = c c -Ωz θ(z), q 1 (z) = -k 2 .
The reduced form of ( 13) is

d 2 θ dz 2 (z) -k 2 θ(z) = 0, (15)
Hence, the solution of equation ( 13) satisfying the boundary condition

ψ w (z = -h) = 0 is ψ w (z) = Ac c -Ωz sinh k(z + h) cosh kh .
A critical layer occurs in the water flow if the following relation is satisfied

c -Ωz = 0.
Let z c = c/Ω be the critical layer level. The critical layer depth in water, h c , only occurs for Ω < 0. Using the leading order of the phase velocity, c 0 , given by equation ( 22), the critical layer depth is

h c = tanh kh 2k + tanh kh 2k 1 + 4gk Ω 2 tanh 2 kh .
Note that this critical layer depth has been derived using the linear phase velocity, c 0 , in the presence of constant vorticity and without wind. In the presence of wind, c = c 0 + O(ρ a /ρ w ). To avoid the presence of a critical layer in water we assume that h < h c + O(ρ a /ρ w ). Due to the quite small values of Ω used, this assumption does not imply that our study is confined to extremely shallow water waves.

Equation ( 11) is integrated between two points below (z = 0-) and above (z = 0+) the air-water interface

ρ 0 W 2 dψ dz 0+ 0- = 0+ 0- k 2 ρ 0 W 2 + g dρ 0 dz ψ dz, with dρ 0 dz = (ρ w -ρ a )δ(z),
where δ is the Dirac delta function. Note that the formal relation

+∞ -∞ δ(z)dz = 1,
indicates that δ has the physical dimension of m -1 . It follows

ρ a W 2 (0+)ψ a (0+) -ρ w W 2 (0-)ψ w (0-) = g(ρ a -ρ w )ψ(0)
where

ψ(0) = ψ a (0) = ψ w (0) due to continuity of ψ. It results ρ a (U a (0+) -c) 2 ψ a (0+) -ρ w (Ωz -c) 2 (0-)ψ w (0-) = g(ρ a -ρ w )ψ(0), c 2 (ρ a ψ a (0+) -ρ w ψ w (0-)) = g(ρ a -ρ w )ψ(0), c 2 ρ a ψ a (0+) -ρ w (Ak + AΩ c tanh kh) = g(ρ a -ρ w )ψ(0).
Because we consider linear waves, without loss of generality we can set A = 1

c 2 ρ a ψ a (0+) -ρ w (k + Ω c tanh kh) = g(ρ a -ρ w ) tanh kh. (16)
If we ignore wind effect in equation ( 16), we obtain the linear dispersion relation of gravity waves propagating at the free surface of a flow of finite depth of constant vorticity kc 2 + (Ω tanh kh)c -g tanh kh = 0.

Let = ρ a /ρ w and c = c 0 + c 1 + O( 2 ) the Taylor series in in the presence of wind (note that without wind = 0). Substituting the expansion of c into equation ( 16) gives

At 0 kc 2 0 + (Ω tanh kh)c 0 -g tanh kh = 0. At 1 c 1 = c 2 0 ψ a (0+) -g tanh kh 2kc 0 + Ω tanh kh .
Following [Janssen(2004)] and [Thomas(2012)], the equation ( 11), in the atmospheric medium, is reduced to the following form

d dz W 0 dψ a dz -k 2 W 2 0 ψ a = 0, ( 17
)
ψ a (0) = 1, limψ a (z) = 0 as z → +∞,
where

W 0 = U 0 -c 0 .
The growth rate γ a of wave amplitude is

γ a = Im(kc 0 + kc 1 ) = k Im(c 1 ), γ a ω 0 = c 0 2kc 0 + Ω tanh kh Im(ψ a (0+)),
where ω 0 = kc 0 and Im denotes imaginary part.

[ Thomas(2012)] has shown that

Im(ψ a (0+)) = i 2 W(ψ a , ψ * a )(0+),
where W is the Wronskian given by

W(ψ a , ψ * a )(0+) = ψ a (0+)ψ * a (0+) -ψ a (0+)ψ * a (0+) = -2iIm(ψ a (0+)),
and ψ * a denotes the complex conjugate. Then

γ a ω 0 = i c 0 2(2kc 0 + Ω tanh kh) W(ψ a , ψ * a )(0+). ( 18
)
Let χ = w/w(0) be the normalised vertical component of air velocity. Then, equation (17) becomes the following Rayleigh equation

W 0 d 2 dz 2 -k 2 χ = W 0 χ, (19)
with χ(0) = 1, and limχ(z) = 0 as z → +∞.

The Rayleigh equation has a singular point where the phase velocity, c 0 , of the waves equals the mean wind velocity U 0 . Consequently, the height, z c , of the critical layer in the atmosphere satisfies U 0 (z c ) = c 0 .

The growth rate can be rewritten as a function of the Wronskian of the solutions of the Rayleigh equation

γ a ω 0 = i c 0 2(2kc 0 + Ω tanh kh) W(χ, χ * )(0+). ( 20
)
One can show that W (χ, χ * ) = 0. Consequently, the Wronskian is constant for z > z c and z < z c as well and may show a jump W(z c + ε) -W(z c -ε), with ε > 0, at the critical height. Due to the boundary condition at infinity, lim W = 0 as z → +∞, W(z) = 0, ∀z > z c . Finaly, the jump is equals to -W(z c -ε) and is given by the following expression

-W(z c -ε) = lim I(∆, ε) as ∆ → 0, ∆ > 0, with I(∆, ε) = -4i W 0c W 0c |χ c | 2 arctan ε W 0c ∆ .
The result is

-W(z c -ε) = -2iπ W 0c |W 0c | |χ c | 2 ,
where

W 0c = W 0 (z = z c ), W 0c = W 0 (z = z c ) and χ c = χ(z = z c ).
The expression of the Wronskien is

W = 2iπ W 0c |W 0c | |χ c | 2 , z < z c .
The normalised growth rate of surface wave amplitude is

γ a ω 0 = - π c 0 2kc 0 + Ω tanh kh W 0c |W 0c | |χ c | 2 , (21)
with

c 0 = - Ω tanh kh 2k + g tanh kh k + Ω 2 tanh 2 kh 4k 2 . ( 22
)
Equation ( 21) can be written differently

γ a ω 0 = - π c 0 4gk tanh kh + Ω 2 tanh 2 kh W 0c |W 0c | |χ c | 2 . ( 23
)
As mentioned previously, we assume the conservation of the momentum flux in the atmospheric boundary layer. Consequently, the wind profile is given by the following logarithmic law

U a (z) = u * κ ln(1 + z z 0 ), (24)
where u * is the friction velocity, κ is the von Karman constant and z 0 is the roughness length of the air-water interface given by the Charnock relation z 0 = α ch u 2 * /g. Within the framework of a logarithmic law we obtain

W 0c |W 0c | = - 1 z 0 exp(-κ c 0 u * ).
To derive the expression of the growth rate of the wave amplitude as a function of the wave age c 0 /u * we use as reference velocity u * and reference length u 2 * /g.

Let c * = c 0 /u * , k * = u 2 * k/g, Ω * = u * Ω/g, h * = gh/u 2
* and z 0 * = gz 0 /u 2 * be the dimensionless variables and parameters. Note that z 0 * = α ch . In dimensionless form, the growth rate of the wave amplitude is

γ a ω 0 = ρ a ρ w π z 0 * c 2 * 2 -c * Ω * tanh(kh) exp(-κc * )|χ c | 2 , (25)
with

c * = - Ω * tanh(k * h * ) 2k * + tanh(k * h * ) k * + Ω 2 * tanh 2 (k * h * ) 4k 2 * . Note that 2 -c * Ω * tanh(k * h * ) = 2 -tanh(k * h * ) + k * c 2 * > 1. Using z 0 * = α ch , equation (25) reads γ a ω 0 = ρ a ρ w π α ch c 2 * 2 -c * Ω * tanh(kh) exp(-κc * )|χ c | 2 . ( 26
)
The dimensionless amplitude growth rate depends only on the wave age and vorticity.

The Rayleigh equation ( 19) is written in dimensionless form

(U a * -c * )( d 2 dz 2 * -k 2 * )χ * = U a * χ * , (27)
where

U a * = 1 κ ln(1 + z * z 0 * ), χ * = w * w * (0) , w * = w u * .
The dimensionless unknown χ * is computed numerically by solving equation ( 27) with the method of [Conte & Miles(1959)]. The dimensionless growth rate of the wave amplitude γ a /ω 0 is calculated once the critical value of χ * is known. Rayleigh equation could be written for water. However, U w (z) = 0 cancels the singularity at z = h c + O(ρ a /ρ w ) and consequently any instability mechanism in the water flow. [START_REF] Young | [END_REF]], * , and the least squares fit by [Young (1997)] -•. [Montalvo & al. (2013)] dot-dashed. The solid line corresponds to present results. The depth used corresponds to δ = gh/U 2 10 = 0.2, using [Young (1997)] notation.

Validation

To validate our approach in finite depth we have compared our results, in the absence of vorticity (Ω = 0), with numerical results of [Montalvo & al. (2013)], the experimental data of [START_REF] Young | [END_REF]], and the least squares fit by [Young (1997)]. The empirical relationship derived by [Young (1997)] to fit the experimental data reads

γ E = A( U 10 c 0 -0.83) tanh n ( U 10 c 0 -B), (28)
where γ E is the energy growth rate of water waves, A = 6.8×10 -5 , B = 1.25δ -0.45 and U 10 11.6u * /κ (see [Montalvo & al. (2013)]). The value n = -0.45 was obtained by Young from a least squares fit to the data. Note that this relationship should depend on vorticity through c 0 while Young did not consider vorticity effect. Figure 1 displays the dimensionless growth rate of wave energy as a function of the inverse of wave age in the absence of vorticity. A good agreement is found between our results and those of [Montalvo & al. (2013)] despite the occurrence of a small deviation when the slope of the curve becomes very steep. One can observe a deviation between the result of the least squares fit of experimental data and the theoretical ones. A physical ingredient seems to be absent in the theoretical approach and will be addressed in the next section. Note that in the limiting case of deep water (kh → ∞) our results are in full agreement with those of [Beji & Nadaoka(2004)] and [ Stiassnie & al. (2007)].

Vorticity effects in finite depth

Very recently, [Kharif & Abid (2020)] have shown for the infinite depth case that the amplitude growth rate, of wind generated waves, increases with vorticity except for quite old waves. They also found a limit to the wave energy growth, in the case of negative vorticity, corresponding to the vanishing of the growth rate. The limiting wave age they found is equal to 1/Ω * . We consider here the case of finite depth and find that the limit of the growth rate of waves depends not only on vorticity, Ω * , but also on the depth h * . Figure 2 shows the dimensionless growth rate of the wave amplitude as a function of the wave age for different values of the dimensionless depth and different values of the dimensionless vorticity. We can see that the growth rate of waves generated at the surface of a vertically sheared flow of constant vorticity decreases as the intensity Ω * increases and vanishes when a limit to the wave amplitude growth is reached corresponding to a limiting wave age. The opposite of this behavior, observed in the infinite depth case for old waves when the vorticity is positive, does not occur in finite depth. There is no wave age for which the amplitude growth rate curve with Ω * < 0 crosses the curve with Ω * = 0, in finite depth, at least for h * ≤ 300. Furthermore, for a given finite depth, figure 2 also shows that the limiting wave age is greater when Ω * < 0, and that the limiting wave age, for a given vorticity, increases with depth.

The wave age corresponding to γ a = 0 can be determined easily as follows. In the limit k → 0, the limiting phase speed, corresponding to the wave age with γ a = 0, is given by (for a finite depth h)

lim c 0 = - hΩ 2 + gh + h 2 Ω 2 4 , as k → 0.
Hence,

lim c * = - h * Ω * 2 + h * + h 2 * Ω 2 * 4 , as k * → 0, (29)
which is the dimensionless phase speed of linear long waves in the presence of constant water vorticity. Contrarily to the case of infinite depth this expression of the limiting wave age is valid for both positive and negative vorticity. Note that without vorticity the limiting wave age is given by lim c * = √ h * . Hence, the growth rate of wind generated waves is depth limited in this case. When Ω = 0, for a given finite depth, the limiting phase speed increases with positive vorticity (Ω < 0) and decreases otherwise. For a given vorticity, the limiting phase speed increases with h. These features are in agreement with the results presented in figure 2 where dimensionless amplitude growth rate as a function of the wave age, for different values of the dimensionless depth and vorticity, is depicted.

The dimensionless limiting wave-age as a function of dimensionless depth, for different values of vorticity, is presented in figure 3, as well as the AUSWEX experimental data from [Donelan & al. (2006) ]. It is clearly seen that taking the vorticity in water into account is important in determining the limiting wave age, namely in the case of negative vorticity (Ω * > 0). Indeed, the AUXEWEX data are obtained from experiments on wind waves conducted in the Lake George which is a shallow lake (Australia). The combined action of wind and the friction at the bottom leads to the formation of a shear flow in the water of negative vorticity. To go further in this direction, let us revisit the experimental data of [START_REF] Young | [END_REF]] and the fit of [Young (1997)], presented in figure 1, using the vorticity in water.

Knowing the limiting wave age and the depth we can compute the corresponding vorticity using equation ( 29):

Ω * = (-(lim c * ) 2 + h * )/(h * lim c * ). (30)
From this expression, it is interesting to note that lim c * = 1/Ω * when h * → ∞.

From both equations ( 28) and (30) we have obtained the value Ω * = 0.0226. The corresponding dimensionless energy growth-rate curve, obtained with our approach, is presented in figure 4. As it can be seen in this figure the agreement between our vortical flow results and the least squares fit of the experimental data is very good, stressing the importance of taking the vorticity in water into account, namely in finite depth.

Conclusion

The Miles theory of wind wave generation has been revisited by considering a preexisting underlying water flow of constant vorticity in arbitrary depth. We found that the wave energy growth rate is an increasing function of the vorticity and depth and showed that this growth rate is limited. This limit depends both on vorticity and depth. The corresponding limiting wave age is related to vorticity and depth according to the following relation

Ω = -(lim c 0 ) 2 + gh h lim c 0 ,
and it increases with both vorticity and depth. We have compared our theoretical results with the experimental data of [START_REF] Young | [END_REF]] and the empirical expression derived by [Young (1997)] in the light of the dependence of the limiting wave age on vorticity. We found that water vorticity could play an important role in the growth of wind waves generated in the presence of an underlying vortical flow, whatever the depth.

To go beyond the assumption of constant vorticity, it would be interesting to extend this study to depth-dependent water vorticity and this is the object of a future work.

Fig. 1

 1 Fig.1Dimensionless energy growth rate as a function of the inverse of wave age without vorticity (Ω = 0). Experimental data of[START_REF] Young | [END_REF]], * , and the least squares fit by[Young (1997)] -•.[Montalvo & al. (2013)] dot-dashed. The solid line corresponds to present results. The depth used corresponds to δ = gh/U 2 10 = 0.2, using[Young (1997)] notation.

Fig. 2 (

 2 Fig. 2 (Color online) Dimensionless amplitude growth rate as a function of the wave age for different values of the dimensionless depth h * : 100 (•), 200 ( * ), 300 (squares), ∞ (dashed) and different values of the dimensionless vorticity Ω * : a) -0.02, b) 0, c) 0.02.

Fig. 3 (

 3 Fig. 3 (Color online) Limit phase speed as a function of depth for different values of vorticity: blue Ω * = -0.03, black Ω * = 0, red Ω * = 0.0226, magenta Ω * = 0.06 and green Ω * = 0.1.The squares correspond to AUSWEX experimental data from[Donelan & al. (2006) ].

Fig. 4 (

 4 Fig.4(Color online) Dimensionless energy growth rate as a function of the inverse of wave age. Without vorticity (Ω * = 0): experimental data of[START_REF] Young | [END_REF]], * , and the least squares fit by[Young (1997)] -•.[Montalvo & al. (2013)] dot-dashed. The solid line corresponds to present results. With vorticity: red, Ω * = 0.0226, blue, Ω * = -0.0226. The depth used corresponds to δ = gh/U 2 10 = 0.2, using[Young (1997)] notation.

Acknowledgements

The project leading to this publication has received funding from the Excellence Initiative of Aix-Marseille University -A*Midex, a French "Investissements d'Avenir programme" AMX-19-IET-010.

Data availability

Data will be made available on reasonable request.

Conflict of interest

The authors declare that they have no conflict of interest.


Thibaut Delcroix

Fano Manifolds

EXAMPLES OF K-UNSTABLE

HAL is

Introduction

When does a Fano manifold admits a Kähler-Einstein metric? If it does not, how can we measure this failure? What alternative canonical metric can we obtain on a Fano manifold? The goal of this article is not to answer these questions but to give them more visibility and provide a detailed study of a few examples. By studying the blow-ups of projective spaces at two disjoint linear subspaces, we provide an infinite family of examples of manifolds with no Kähler-Einstein metrics but coupled Kähler-Einstein metrics. By studying the blow-ups of quadrics at a linear subquadric, we provide an infinite family of examples of manifolds that are not K-semistable and admit no Kähler-Ricci solitons.

Let us be a bit more precise. Consider a Fano manifold X. A Kähler form (abusively, metric) ω ∈ c 1 (X) is Kähler-Einstein if it satisfies the equation

Ric(ω) = ω
where Ric(ω) denotes the Ricci form of ω. We say a Fano manifold is Kähler-Einstein if it admits a Kähler-Einstein form. A convincing numerical measure of how a Fano manifold fails to be Kähler-Einstein is given by the greatest Ricci lower bound : R(X) = sup {t ∈ [0, 1] | ∃ω ∈ c 1 (X), Ric(ω) ≥ tω} .

If X is Kähler-Einstein then R(X) = 1 while the converse direction does not hold. In K-stability terms, R(X) = 1 is equivalent to K-semistability of X.

Date: 2019.

Several candidate alternative metrics on Fano manifolds with no Kähler-Einstein metrics have been proposed over the years. The best known are the Kähler-Ricci solitons, which satisfy the equation

Ric(ω) -L ξ ω = ω
where ξ is a holomorphic vector field and L ξ denotes the Lie derivative with respect to ξ. They actually belong to a large family of metrics introduced by Mabuchi as multiplier Hermitian structures: X is equipped with a multiplier Hermitian structure if there exists a Kähler form ω ∈ c 1 (X), a holomorphic vector field ξ and a smooth real valued concave function h such that ( 1)

Ric(ω) - √ -1∂ ∂h • θ = ω
where θ is defined by L ξ ω = √ -1∂ ∂θ. Kähler-Einstein metrics correspond to the case g = 0 and Kähler-Ricci solitons to the case g = id. When g is the logarithm of an affine function, such metrics are called Mabuchi metrics. The advantage of Mabuchi metrics over Kähler-Ricci solitons is that such metrics have a more algebraic nature. For example, the holomorphic vector field is essentially well determined by the manifold alone in the case of Kähler-Ricci solitons or of Mabuchi metrics. In the case of solitons, the vector field may very well generate a non-closed subgroup whereas this is impossible for Mabuchi metrics. This can make an important difference in the study of moduli spaces of Fano manifolds with canonical metrics. Motivated by this, we want to advertise the following open question. Question 1.1. Is it true that a Fano manifold X admits a Kähler-Ricci soliton if and only if there exists a multiplier Hermitian structure as defined by Equation [START_REF] Blanchard | Sur les variétés analytiques complexes[END_REF], where h is the logarithm of a polynomial?

Another candidate for canonical metrics is given by coupled Kähler-Einstein metrics. These were introduced by Hultgren and Witt Nyström in [12] and in the simplest case of pairs of metrics, may be defined as follows. A Fano manifold X admits a pair of coupled Kähler-Einstein metrics if there exists a pair (ω 1 , ω 2 ) of Kähler forms such that

Ric(ω 1 ) = Ric(ω 2 ) = ω 1 + ω 2 .
The advantage here is that no holomorphic vector field is involved, and the drawback is that several Kähler classes have to be considered. Until the present paper, only two examples of Fano manifolds with no Kähler-Einstein metrics but a pair of coupled Kähler-Einstein metrics were known [11,[START_REF] Delcroix | Coupled complex monge-ampère equations on fano horosymmetric manifolds[END_REF].

The first example of a Fano manifold X with no Kähler-Ricci solitons and R(X) < 1 was obtained by the author in [START_REF] Delcroix | Métriques de Kähler-Einstein sur les compactifications de groupes[END_REF][START_REF] Delcroix | Kähler-Einstein metrics on group compactifications[END_REF]. The example is a biequivariant compactification of the group Sp 4 , obtained by blowing up the unique closed orbit in the wonderful compactification of this group. The author later provided two additional examples in [START_REF] Delcroix | K-Stability of Fano spherical varieties[END_REF], as compactifications of the group SO 4 , however the examples were still sporadic at that point. Such examples provide a useful illustration of the variety of situations regarding Kstability, and are the only known illustrations of type 2 singularity formation along the Kähler-Ricci flow on Fano manifolds (see [START_REF] Li | Singular limits of Kähler-Ricci flow on Fano G-manifolds[END_REF]). It should be noted that Fujita obtained in [START_REF] Fujita | Examples of K-unstable Fano manifolds with the Picard number 1[END_REF] examples of Fano manifolds of Picard number one which are K-unstable. To the author's knowledge, it is not known if these examples admit Kähler-Ricci solitons.

The infinite family of examples presented here share a strong similarity with the previous examples: they are equivariant compactifications of symmetric spaces (actually one of the above mentioned compactifications of SO 4 is part of this family). This allows us to use the K-stability criterions proved by the author in [START_REF] Delcroix | K-Stability of Fano spherical varieties[END_REF], which reduces the problem to determining the sign of a finite number of integrals of polynomials on polytopes. While such integrals can be computed, it yields in general complicated expressions. In particular, the sign is not easily determined without an explicit computation, making the determination of K-stability on infinite families a challenging task.

The family of examples we consider here is formed as we said by blow-ups of quadrics along lower dimensional linear subquadrics. An infinity of these are Kähler-Einstein, and an infinity are as stated (not K-semistable and with no Kähler-Ricci solitons).

Theorem 1.2. Let Q n denote the quadric of dimension n, and

Q k denote a linear subquadric of Q n of dimension k. Then if 2 ≤ k ≤ n -3, the blow-up Bl Q k (Q n ) of Q n along Q k is K-unstable
and does not admit any Kähler-Ricci soliton. In the other cases, In the process of introducing the symmetric structure on the family of manifolds that provide our examples, a natural detour is to start by considering the blow-ups of projective spaces along linear subspaces. The blow-up of a projective space along a linear subspace has a non-reductive automorphism group, satisfying Matsushima's obstruction [START_REF] Matsushima | Sur la structure du groupe d'homéomorphismes analytiques d'une certaine variété kählérienne[END_REF]. They provided as such the first example of a Fano manifold with no Kähler-Einstein metrics: the blow-up of P 2 at a point.

Bl Q k (Q n ) is Kähler-Einstein. The lowest dimensional example is Bl Q 2 (Q 5 ).
Furthermore, the blow-up of P 4 along a line and a disjoint plane provided the first example of a Fano manifold with reductive automorphism group and non-vanishing Futaki invariant as highlighted by Futaki himself [10]. As a toric manifold, Bl P 1 ,P 2 P 4 admits a Kähler-Ricci solitons: Wang and Zhu proved in [START_REF] Wang | Kähler-Ricci solitons on toric manifolds with positive first Chern class[END_REF] that any toric Fano manifold admits a Kähler-Ricci soliton. It actually admits better metrics in two directions: it admits Mabuchi metrics and coupled Kähler-Einstein metrics.

More generally, the blow-up of a projective space along two complementary linear subspaces also has reductive automorphism group and non-vanishing Futaki invariant. We show that an infinity of these manifolds are manifolds with no Kähler-Einstein metrics but pairs of coupled Kähler-Einstein metrics.

Theorem 1.4. For k large enough, Bl P k-1 ,P k P 2k+1 admits pairs of coupled Kähler-Einstein metrics but no Kähler-Einstein metrics.

We believe this holds more generally and our proof could easily be adapted to prove other particular cases but we found no general proof. Question 1.5. Does there exist a pair of coupled Kähler-Einstein metrics on all blow-ups of the projective space along complementary subspaces ? Is it true that there exist no coupled Kähler-Einstein metrics on

Bl Q k Q n , for 2 ≤ k ≤ n -3?
Similarly, we are convinced that Mabuchi metrics exist on all blow-ups of projective space we consider. Finding no elegant general proof of this fact, we instead provide a positive answer to Question 1.1 in a particular case.

Theorem 1.6. Let X be a homogeneous P 1 -bundle over a rational homogeneous manifold G/P , where G is a reductive group and P a parabolic subgroup of G. Assume X is Fano, then X admits Kähler-Ricci solitons, and X admits multiplier Hermitian structures as in Equation [START_REF] Blanchard | Sur les variétés analytiques complexes[END_REF], where h is the logarithm of a polynomial of degree equal to the dimension of G/P .

In the body of the paper we will prove the results stated in this introduction. The results to prove are translated into combinatorial conditions by using [START_REF] Delcroix | Coupled complex monge-ampère equations on fano horosymmetric manifolds[END_REF]. More precisely, Theorem 1.2 is proved using [8, Corollary 1.3], Theorem 1.4 is proved using [8, Corollary 1.5], and Theorem 1.6 is proved using [8, Theorem 1.2]. As already mentioned, it does not mean that checking the conditions is trivial, as soon as we consider infinite families. Nonetheless the arguments to be used pertain to elementary mathematics. An additional goal of the paper is the description of all the combinatorial data associated to the families of examples considered (in particular a full description of the Picard group and moment polytopes associated to ample line bundles). Our hope is that these detailed examples will provide an easier entry point to the theory of horosymmetric and spherical manifolds, and can more readily be used to check properties of other canonical metrics or other geometric properties.

In section 2 we consider the blow-ups of projective spaces along complementary linear subspaces as horospherical manifolds (the combinatorial data can be used as well to study blow-ups at a single linear subspace as horospherical manifolds), and prove Theorem 1.4 and Theorem 1.6. In sections 3 and 4, we study the blow-up of a quadric along a linear subquadric as a symmetric variety and prove Theorem 1.2. In section 3 we treat the case when the symmetric space is semisimple, which contains the K-unstable cases. In section 4 we treat the remaining case. This final section also contains an interesting family: the blow-up of a quadric at a point admits no Mabuchi metrics, starting from dimension three. Each one of the three sections follows the same structure. We begin by a geometric description of the manifolds and of the orbits under the natural group action to consider. We then describe the full combinatorial data associated to these manifolds, as horosymmetric varieties. First we describe the combinatorial data encoding the manifold itself, then we describe the combinatorial data associated to each ample R-divisor. Finally we examine the combinatorial conditions encoding the existence of canonical metrics to prove our main results.

Acknowledgements. It is my pleasure to thank Jakob Hultgren for our joint work which allowed to prove half of the results in this article, and which provided motivation to study these particular families of examples. Part of this work was initially written in preparation for talks at two conferences during the summer of 2019. I thank Ivan Cheltsov and Simone Calamai for the invitations to these conferences. This research was accomplished in part during my postdoc in Strasbourg funded by the Labex IRMIA, and received partial funding from ANR Project FIBALGA ANR-18-CE40-0003-01.

Blow-ups of projective space along complementary linear subspaces

2.1. Description of the manifolds. Let p and n be two integers such that

2 ≤ p ≤ n -2,
Consider the complex reductive group G := S(GL p × GL n-p ). We consider G as embedded in GL n by block-diagonal matrices: if (A, B) ∈ G, we identify it with A 0 0 B .

Write C n = C p ⊕ C q and consider the induced action of the group G. There are four orbits under this action: the fixed point {0}, the pointed linear subspaces (C p \ {0}) × {0} and {0} × (C q \ {0}), and the open dense orbit (C p \ {0}) × (C q \ {0}). It induces as well an action of G on P n-1 with three corresponding G-orbits: an open dense orbit, and two disjoint linear subspaces (which we denote by P p-1 and P q-1 in an abuse of notations).

The blow-ups of the projective space along one or both of these linear subspaces are Fano manifolds. We will focus on the manifold obtained by blowing up both orbits, and we will denote the manifold by X n,p := Bl P p-1 ,P q-1 (P n-1 ).

By the description above, this manifold is equipped with an action of G.

It is a standard fact, though not obvious, that the neutral components of the automorphism group Aut(Bl Y (X)) of a blow-up and of the stabilizer Stab Aut(X) (Y ) of the blown-up submanifold in the automorphism group of the initial manifold are isomorphic. The non-obvious direction may be seen as a consequence of Blanchard's Lemma (see [2,[START_REF] Blanchard | Sur les variétés analytiques complexes[END_REF]).

The stabilizer of P p-1 in the automorphism group Aut(P n-1 ) = PGL n of the projective space is the image of the group of upper block-triangular invertible matrices with two square diagonal blocks of size p and q. The neutral component Aut 0 (Bl P p-1 (P n-1 )) of Aut(Bl P p-1 (P n-1 )) is thus exactly the stabilizer described above. In particular, this group is not reductive and the image of G in it is a Levi subgroup. For Bl P p-1 ,P q-1 (P n-1 ), the same argument shows that Aut 0 (Bl P p-1 ,P q-1 (P n-1 )) is the Levi subgroup above, hence the image of G is the full connected automorphism group. Note in particular that the automorphism group of X n,p is reductive.

Under the action of G, the manifold X n,p is a horospherical manifold, as we will explain in the next section. This structure, taking into account the full group of automorphisms, is the most natural to consider. However, we should note that X n,p is:

• a toric manifold if we consider only the action of a maximal torus, • a cohomogeneity one manifold if we consider only the action of a maximal compact subgroup • a P 1 -bundle over the product of projective spaces P p-1 × P q-1 .

2.2.

Combinatorial data associated to the manifold.

2.2.1.

The open orbit. Let T ⊂ SL n denote the maximal torus (both in G and SL n ) formed by diagonal matrices, and denote the roots of SL n with respect to T by

α i,j : T → C * t → d i d j where t = diag(d 1 , . . . , d n ), for 1 ≤ i = j ≤ n. The roots of G are the α i,j with 1 ≤ i = j ≤ p or p + 1 ≤ i = j ≤ n.
The set of characters of T is denoted by X(T ) and the set of one-parameter subgroups is denoted by Y(T ). The Killing form of G is denoted by {x, y} for x, y ∈ g.

It is immediate to determine the stabilizer H of the point

(1 : 0 • • • : 0 : 1 : 0 : • • • : 0) in P n-1 ,
which is in the open orbit (the 1 are at position 1 and (p + 1)). Let P denote the parabolic subgroup of G containing T such that the roots of its unipotent radical are

Φ P u = {α 1,2 , . . . , α 1,p , α p+1,p+2 , . . . , α p+1,n }.
Then H is the kernel in P of the character of P defined by α 1,p+1 .

As a consequence, H contains the unipotent radical of P , and G/H is a rank one horospherical homogeneous space. To be consistent with the conventions in [START_REF] Delcroix | Coupled complex monge-ampère equations on fano horosymmetric manifolds[END_REF] and [7], we fix an involution σ of t such that t σ = ker(α 1,p+1 ). It amounts to choosing the fixed point set of σ, a complement to Rα 1,p+1 in X(T ) ⊗ R. Here, we choose the orthogonal of α 1,p+1 with respect to the Killing form to define σ.

Let us describe the combinatorial invariants corresponding to the spherical homogeneous space G/H. We do this with respect to the Borel subgroup

B Figure 1. M ⊗ R • 0 • e = α * 1,p+1 = α ∨ p+2,p+1 | M⊗R • -e = α ∨ 2,1 | M⊗R of G containing T whose corresponding simple roots are S = {α 2,1 , . . . , α p,p-1 , α p+2,p+1 , . . . α n,n-1 },
to ensure again consistency with the conventions in [START_REF] Delcroix | Coupled complex monge-ampère equations on fano horosymmetric manifolds[END_REF] and

[7]. The spherical lattice M is then X(T /T ∩ H) = Zα 1,p+1
. The valuation cone is the vector space generated by α 1,p+1 : V = M ⊗ R. The colors are in bijection with the set -Φ P u ∩ S = {α 2,1 , α p+2,p+1 }, we denote these by D α 2,1 and D α p+2,p+1 respectively. Recall that given a root α ∈ X(T

) of G, the coroot α ∨ ∈ Y(T ) is defined by α(x) = 2 {x,α ∨ } {α ∨ ,α ∨ } for all x ∈ Y(T ) ⊗ R.
The color map sends D α to the restriction α ∨ | M⊗R for α ∈ {α 2,1 , α p+2,p+1 }. The images are thus respectively -e and e where e = α * 1,p+1 denotes the element dual to α 1,p+1 in (M ⊗ R) * , as illustrated in Figure 1.

Note that if we considered the case p = 1 or np = 1, only one of the two roots α 2,1 or α p+2,p+1 would be a well defined roots of G. There would correspondingly be only one projective subspace giving rise to a non-trivial blow-up. We will not give the details but the combinatorial description is a minor variation of the one presented here. where the first cone corresponds to the orbit P p-1 , the second to P q-1 and the last to the open orbit. The colored fan of Bl P q-1 P n-1 is

{(R + e, D α p+2,p+1 ), (-R + e, ∅), ({0}, ∅)}.

The colored fan of Bl

P p-1 P n-1 is {(R + e, ∅), (-R + e, D α 2,1 ), ({0}, ∅)}.
Finally, the colored fan of Bl P p-1 ,P q-1 P n-1 is the colorless fan

{(R + e, ∅), (-R + e, ∅), ({0}, ∅)}.
2.3. Combinatorial data associated to divisors.

2.3.1.

Picard group, isotropy characters and polytopes: Recall that the Picard group of a general spherical variety is described by Brion in [3]. We recalled this description and introduced a few convenient notions in [7,[START_REF] Delcroix | Coupled complex monge-ampère equations on fano horosymmetric manifolds[END_REF] to deal with moment polytopes in the special case of horosymmetric manifolds. We will use the language of [7] here, notably we will use the terms special divisor and isotropy character as defined in that paper.

Figure 2. Moment polytope for an ample divisor on Bl

P p-1 ,P q-1 P n-1 • • -a + α 1,p+1 • a -α 1,p+1 • •∆ + (a c , a + , a -) • 2a c χ 2,1
The Picard group of Bl P p-1 ,P q-1 P n-1 is of rank 3. Let D ± denote the Ginvariant divisor associated to the ray ±R + e. The Picard group is generated by D α 2,1 , D α p+2,p+1 , D + and D -with the relation

D + + D α p+2,p+1 = D -+ D α 2,1 .
For D ± , they already are special divisors, and the isotropy characters are trivial. Up to renormalizing the action by a character of G, the linearized line bundle associated to D α 2,1 admits a B-semi-invariant section whose Bweight ̟ 2,1 is given by diag

(d 1 , • • • , d n ) → d -1
1 (this is, up to a character of the reductive group G the fundamental weight of α 2,1 ). Then the linearly equivalent special Q-divisor is

D α 2,1 - {̟ 2,1 , α 1,p+1 } {α 1,p+1 , α 1,p+1 } (D + + D α p+2,p+1 -D --D α 2,1 ) = 1 2 (D + + D α p+2,p+1 + D α 2,1 -D -)
and its isotropy character is

χ 2,1 := ̟ 2,1 - {̟ 2,1 , α 1,p+1 } {α 1,p+1 , α 1,p+1 } α 1,p+1 = ̟ 2,1 + 1 2 α 1,p+1 .
As a consequence, special R-divisors are all the

D(a c , a + , a -) := a c (D α 2,1 + D α p+2,p+1 ) + a + D + + a -D - for a c , a + , a -∈ R.
The isotropy character of such a divisor is 2a c χ 2,1 , and the associated special polytope is defined by

∆(a c , a + , a -) = {tα 1,p+1 | max(-a + , -a c ) ≤ t ≤ min(a -, a c )}.
The moment polytope, provided the divisor is ample, is

∆ + (a c , a + , a -) = 2a c χ 2,1 + ∆(a c , a + , a -).
Note that Brion's ampleness criterion [3, Théorème 3.3] states, in our case, that D(a c , a + , a -) is ample if and only if a + < a c , a -< a c and a + + a -> 0.

The special divisor D n 2 , p + 1 -n 2 , n 2p + 1 represents the anticanonical divisor, its isotropy character is nχ 2,1 , thus the corresponding moment polytope is

∆ + n 2 , p + 1 - n 2 , n 2 -p + 1 = nχ 2,1 + ∆ n 2 , p + 1 - n 2 , n 2 -p + 1 .
2.3.2. Duistermaat-Heckman polynomial and barycenters. Key ingredients in the combinatorial criterions for existence of canonical metrics are the Duistermaat-Heckman polynomial and barycenters of moment polytopes with respect to this polynomial. We give here a first expression of these in our special case.

It is enough to express the Duistermaat-Heckman polynomial P DH in the plane generated by α 1,p+1 and 2χ 2,1 since every moment polytope lives in this plane. We have

P DH (xα 1,p+1 + 2yχ 2,1 ) = α∈-Φ P u {α, xα 1,p+1 + 2yχ 2,1 } = (-x + y) p-1 (x + y) n-p-1
The Duistermaat-Heckman barycenter corresponding to the ample divisor D(a c , a + , a -) is, for any choice of Lebesgue measure dp,

Bar(a c , a + , a -) = ∆ + (ac,a + ,a -) pP DH (p)dp ∆ + (ac,a + ,a -) P DH (p)dp = 2a c χ 2,1 + a - -a + t(t + a c ) n-p-1 (a c -t) p-1 dt a - -a + (t + a c ) n-p-1 (a c -t) p-1 dt α 1,p+1
2.4. Kähler-Einstein criterion. Since X is a toroidal horospherical manifold, the combinatorial criterion ruling the existence of Kähler-Einstein metrics on X was originally proved by Podesta and Spiro [START_REF] Podestà | Kähler-Ricci solitons on homogeneous toric bundles[END_REF]. It was reproved on several occasions by the author, in [START_REF] Delcroix | K-Stability of Fano spherical varieties[END_REF] and [7]. The criterion states that X is Kähler-Einstein if and only if the barycenter of the moment polytope associated to the anticanonical divisor with respect to the Duistermaat-Heckman polynomial P DH coincides with the opposite of the sum of positive roots of the parabolic subgroup P . In formulas:

Proposition 2.1 (application of Theorem B and Proposition 4.7 in [START_REF] Podestà | Kähler-Ricci solitons on homogeneous toric bundles[END_REF]).

The manifold Bl P p-1 ,P q-1 P n-1 is Kähler-Einstein if and only if

Bar n 2 , p + 1 - n 2 , n 2 -p + 1 = α∈-Φ P u α.
Note that

α∈-Φ P u α = α 2,1 + • • • + α p,1 + α p+2,p+1 + • • • + α n,p+1 = nχ 2,1 + n 2 -p α 1,p+1
Hence the manifold Bl P p-1 ,P q-1 P n-1 is Kähler-Einstein if and only if

I(n, p) := n 2 -p+1 -(p+1-n 2 ) t + p - n 2 t + n 2 n-p-1 n 2 -t p-1 dt = 0.
Lemma 2.2. The manifold Bl P p-1 ,P q-1 P n-1 is Kähler-Einstein if and only if n is even and p = n 2 . Else we have I(n, p) > 0 if p < n 2 and I(n, p) < 0 if p > n 2 . Proof. By using the variable x = t + p -n 2 , we get

I(n, p) = 1 -1 x(x + n -p) n-p-1 (p -x) p-1 dx.
The integrand has a simple primitive, providing the explicit formula

I(n, p) = -1 n (p -x) p (n -p + x) n-p 1 -1 = -1 n (p -1) p (n -p + 1) n-p -(p + 1) p (n -p -1) n-p
We claim that the function f :

x → x-1 x+1
x is strictly increasing on [1, ∞[. Once the claim is proved, the theorem is as well in view of the last expression of I(n, p). A direct computation of the derivative yields

f ′ (x) = ln x -1 x + 1 + 2x x 2 -1 x -1 x + 1 x .
By a power series expansion, for 0 < y < 1 we have

ln (1 -y) + y(2 -y) 2 1 1 -y = k∈N k + 1 2(k + 3) y k+3 > 0.
Setting y = 2 x+1 , we obtain f ′ (x) > 0 for x > 1, thus the claim. 2.5. Mabuchi-type metrics. The combinatorial criterion, proved by the author and Jakob Hultgren [8, Corollary 1.5], ruling the existence of Mabuchi metrics on Bl P p-1 ,P q-1 P n-1 reads as follows. We place ourselves in the situation where the manifold does not admit any Kähler-Eintein metrics, that is p = n 2 . Set

J := 1 -1 x 2 (x + n -p) n-p-1 (p -x) p-1 dx > 0.
Choose A and B ∈ R such that

1 -1 x(Ax + B)(x + n -p) n-p-1 (p -x) p-1 dx = AJ + BI = 0.
There does not exist a Mabuchi metric if and only if for all such choices, the affine function x → Ax + B vanishes on [-1, 1], that is, if and only if

-B A = J I ∈ [-1, 1]. If p < n 2 then I > 0 and the latter condition is equivalent to J -I ≤ 0. If p > n 2 then it is equivalent to J + I ≤ 0.
From checking on examples, it appears that the manifolds Bl P p-1 ,P q-1 P n-1 admit Mabuchi metrics in all cases. However we did not find an elegant way to prove this. On the other hand, we can easily show that there always exists a Mabuchi-type metric (or multiplier Hermitian structure) on any rank one toroidal horospherical Fano manifold.

Recall that the notion of multiplier Hermitian structure was introduced by Mabuchi in [START_REF] Mabuchi | Multiplier Hermitian structures on Kähler manifolds[END_REF] and consists of a Kähler metric ω ∈ c 1 (X) solving the complex Monge-Ampère equation

Ric(ω) - √ -1∂ ∂h(θ) = ω
for some smooth real valued concave function h defined on the range of θ, where θ is the Hamiltonian function of ω with respect to some holomorphic vector field (equivalently, some action of C * ).

In the remaining of this subsection, we consider a Fano manifold X, which is an arbitrary rank one toroidal horospherical manifold under the action of a reductive group G with a one-dimensional center containing a C * subgroup. Equivalently, the manifold X is a homogeneous P 1 -bundle over a rational homogeneous space G/P . The action of the central subgroup C * is the natural action on the fibers.

Proof of Theorem 1.6. Let G/H be the open orbit in X. Let P denote the normalizer of H in G and choose a Borel subgroup B opposite to P , and T a maximal torus in B ∩ P . Then the moment polytope ∆ + of X is of the form

∆ + =    t̟ + α∈-Φ P u α | -1 ≤ t ≤ 1   
where ̟ is a generator of the one dimensional lattice M (see [16, p. 207]). Then by [8, Theorem 1.2], there exists a multiplier Hermitian structure on X if and only if we can find a smooth, real valued, concave function h on

[-1, 1] such that 1 -1 te h(t) P DH   t̟ + α∈-Φ P u α   dt = 0
where, as usual, P DH denotes the Duistermaat-Heckman polynomial for X: P DH (x) = α∈-Φ P u {α, x} for x ∈ X(T ). Thanks to the symmetry of the polytope, an obvious choice is given by

e h(t) = P DH   -t̟ + α∈-Φ P u α   .
The corresponding function h is obviously smooth, real valued, and concave: α, -t̟ + α∈-Φ P u α > 0 for all -1 ≤ t ≤ 1 and α ∈ -Φ P u since the moment polytopes are strictly contained in the positive Weyl chamber for toroidal horospherical manifolds. 2.6. Coupled Kähler-Einstein metrics. In this section, we prove that there exists infinitely many couples (n, p) such that the manifold X n,p admits no Kähler-Einstein metrics but admits a couple of coupled Kähler-Einstein metrics, by proving Theorem 1.4 as stated in the introduction.

Proof of Theorem 1.4. We place ourselves in the situation where n = 2k + 1 is odd, and p = k.

Using Theorem 1.2 of [START_REF] Delcroix | Coupled complex monge-ampère equations on fano horosymmetric manifolds[END_REF], we want to find a c , a + and a -such that

• Bar(a c , a + , a -) + Bar(k + 1 2 -a c , 1 2 -a + , 3 2 -a -) = nχ 2,1 + 1 2 α 1,k+1 • a + < a c < k + a + • a -< a c < k -1 + a - • 0 < a + + a -< 2. Set a ′ c = k + 1 2 -a c , a ′ + = 1 2 -a + and a ′ -= 3 2 -a -.
The three last conditions ensure that the divisors D(a c , a + , a -) and D(a ′ c , a ′ + , a ′ -) are ample. Their sum is obviously the anticanonical divisor, and the first condition means that the coupled Kähler-Einstein equation is satisfied for this decomposition.

Let C(a c , a + , a -) denote the quantity

a - -a + t(t + a c ) k (a c -t) k-1 dt a - -a + (t + a c ) k (a c -t) k-1 dt + a ′ - -a ′ + t(t + a ′ c ) k (a ′ c -t) k-1 dt a ′ - -a ′ + (t + a ′ c ) k (a ′ c -t) k-1 dt - 1 2
then the first condition is equivalent to C(a c , a + , a -) = 0. Note that the Kähler cone is connected and that (a c , a + , a -) → C(a c , a + , a -) is continuous. Since 2p = 2k < 2k + 1 = n, Theorem 2.2 implies that C( n 4 , 1 4 , 3 4 ) > 0, hence it is enough to find a triple (a c , a + , a -) (satisfying the ampleness conditions) with C(a c , a + , a -) < 0 to prove the existence of coupled Kähler-Einstein metrics.

Set a c = k -2, a + = 1 2 and a -= 0. Then we have a c = 5 2 , a + = 0 and a -= 3 2 . Such a data satisfies the ampleness conditions. We will show C(a c , a + , a -) < 0 for k large enough by showing that the first quotient of integrals in C(a c , a + , a -) converges to -1 4 and that the second quotient of integrals converges to 0. In fact, if a c is linear in k, a + and a -are constant, then the quotient of integrals converges to the middle of [-a + , a -], and if a + = 0, a ′ c and a ′ -are constant, then the quotient of integrals converges to 0.

Using the equation

(t + k -2) k (k -2 -t) k-1 (k -1) 2k-1 = 1 + t -1 k -1 1 + t -1 k -1 k-1 1 - t + 1 k -1 k-1 we see that, uniformly in t ∈ [-1 2 , 0], lim k→∞ (t + k -2) k (k -2 -t) k-1 (k -1) 2k-1 = e -2 hence lim k→∞ 0 -1 2 t(t + k -2) k (k -2 -t) k-1 dt 0 -1 2 (t + k -2) k (k -2 -t) k-1 dt = - 1 4 .
The other limit is a bit less immediate. Fix 0 < ǫ < 3 2 and consider

(2)

3 2 0 (t -ǫ) t + 5 2 k 5 2 -t k-1
dt.

If we prove that the latter integral is negative for k large enough, then

0 < 3 2 0 t t + 5 2 k 5 2 -t k-1 dt 3 2 0 t + 5 2 k 5 2 -t k-1 dt < ǫ
for k large enough. Actually, proving this for just one ǫ < 3 4 is enough to get the conclusion. In the integral (2), the negative contribution is the integral on [0, ǫ], while the positive contribution is the integral on [ǫ, 3 2 ]. For ǫ ≤ t ≤ 3 2 , we have

t + 5 2 k 5 2 -t k-1 = 5 2 2k-1 1 + 2t 5 1 - 2t 5 2 k-1 ≤ 5 2 2(k-1) 1 + 3 5 1 - 2ǫ 5 2 k-1 hence 3 2 ǫ (t-ǫ) t + 5 2 k 5 2 -t k-1 dt ≤ 9 8 - 3ǫ 2 + ǫ 2 2 5 2 2(k-1) 1 + 3 5 1 - 2ǫ 5 2 k-1
For the negative contribution, it suffices to restrict to 0, 1 2(k-1) . Set t = s k-1 where s ∈ 0, 1 2 . Then

(t + 5 2 ) k ( 5 2 -t) k-1 ( 5 2 ) 2k-1 = 1 + 2s 5(k -1) 1 + 2s 5(k -1) k-1 1 - 2s 5(k -1) k-1 Hence, uniformly in s ∈ [0, 1 2 ] (or in t ∈ 0, 1 2(k-1) ), lim k→∞ (t + 5 2 ) k ( 5 2 -t) k-1 ( 5 2 ) 2k-1 = 1.
We deduce that

1 2(k-1) 0 (t -ǫ) t + 5 2 k 5 2 -t k-1 dt ∼ k→∞ -ǫ 5 2 2k-1 2(k -1) Since 9 8 - 3ǫ 2 + ǫ 2 2 5 2 2(k-1) 1 + 3 5 1 - 2ǫ 5 2 k-1 = o -ǫ( 5 2 ) 2k-1 2(k -1)
, the negative contribution in integral (2) is dominant over the positive contribution. We deduce that integral (2) is negative for k large enough.

Unstable blow-ups of quadrics along linear subquadrics

3.1. Description of the manifolds. Choose as before two positive integers p and q such that p + q = n. For this first section on blow-ups of quadrics, we assume 3 ≤ p ≤ n -3. We will deal with the case p = n -2 in the next section.

Consider the standard quadratic form n j=1 z 2 j on C n . Let Q n-2 denote the associated n -2-dimensional projective quadric. It is equipped with an action of G := SO p × SO q ⊂ SO n with four orbits, easily described in terms of the rational projections from P n-1 to P p-1 and P q-1 . There is an open dense orbit, formed by the points at which both projections are defined and no projection lie in the corresponding quadrics, the codimension one orbit formed by the points at which both projections are defined and lie in the corresponding quadrics, and two closed orbits formed by the points where one of the projections is undefined.

We will be interested in the blow-up Y n,p of Q n-2 along one of the closed orbits, say the quadric Q p-2 in P p-1 .

The group of automorphisms of Q n-2 is up to isogeny the group SO n . The connected group of automorphisms of X is thus isogenous to the neutral component of the stabilizer of the quadric in P p-1 inside SO n . It is isogenous to SO p × SO q .

3.2. Combinatorial data associated to the manifold.

3.2.1.

The open orbit. The quadric and its blow-up, equipped with the action of G, are symmetric varieties. The open dense orbit is the symmetric space SO p × SO q /H where H is the stabilizer of say

[1 : 0 • • • : 0 : 1 : 0 • • • : 0]. It is the unique proper subgroup sandwiched between {1} × SO p-1 ×{1} × SO q-1 and S(O 1 × O p-1 )×S(O 1 × O q-1 ).
In particular, it is not, properly speaking a product of symmetric spaces even though its restricted root system does decompose as A 1 × A 1 . Let α and β denote the simple restricted roots corresponding the first and second factor of G. We denote by α ∨ and β ∨ the corresponding simple restricted coroots. Note that the multiplicity of the restricted root α is p -2 and the multiplicity of β is q -2.

Given the restricted root system, it is easy to determine the combinatorial data associated to the spherical homogeneous space G/H. The spherical lattice M is the unique proper lattice sandwiched between the restricted weight lattice and the restricted root lattice for the restricted root system of type A 1 ×A 1 . It is thus generated by α and 1 2 (α+β). The valuation cone V is

• • β ∨ • α ∨ • • β ∨ • α ∨
the negative restricted Weyl chamber. The colors are in bijection with simple restricted roots and the image of the color map is the set of simple restricted coroots. The description of colors in the last sentence is not immediate as it is not immediate that the color map is injective. However, it follows for example from the description of the Picard group which will follow in the next section, as the quadric is a Picard rank one manifold.

3.2.2.

Colored fans and embeddings. The colored fan of the quadric consists of the following colored cones:

• the point ({0}, ∅), for the open orbit,

• the ray (-R + (α ∨ + β ∨ ), ∅) for the codimension one orbit,

• (Cone(α ∨ , -(α ∨ + β ∨ )), {α ∨ }) for the subquadric Q q-2 , • and (Cone(β ∨ , -(α ∨ + β ∨ )), {β ∨ }) for the subquadric Q p-2 .
To obtain the colored fan of the blow up Y n,p of the quadric along Q p-2 , one needs only replace the last cone by (Cone(-α ∨ , -(α ∨ + β ∨ )), ∅) and add the ray (-R + α ∨ , ∅) (see Figure 3). Knowing the restricted root system (with multiplicities), it is easy to determine the anticanonical divisor, which is here D( p+q 2 -1, p -1). Note also for later use that the half-sum of positive restricted roots with multiplicities is ( p 2 -1)α + ( q 2 -1)β. 3.3.2. Duistermaat-Heckman polynomial and barycenters. Up to a multiplicative constant, the Duistermaat-Heckman polynomial is

P DH (xα + yβ) = {α, xα + yβ} p-2 {β, xα + yβ} q-2 = (2x) p-2 (2y) q-2
we set the following notation for the barycenter of moment polytopes with respect to P DH and an arbitrary Lebesgue measure dp. Bar(a d , a e ) = ∆(a d ,ae) pP DH (p)dp ∆(a d ,ae) P DH (p)dp 3.4. Existence of Kähler-Einstein metrics. The notation k = p -1, l = q -1 used in the following are introduced for convenience in the proofs. With these notations, p -2 = k -1 is the multiplicity of one of the restricted roots and q -2 = l -1 is the multiplicity of the other. The Duistermaat-Heckman polynomial is thus, up to a constant,

P DH (xα + yβ) = x k-1 y l-1 .
3.4.1. Non-existence of Kähler-Einstein metrics. Set, for k, l in N ≥2 ,

I(k, l) := k x=0 k+l-x y=0 (x -(k -1))x k-1 y l-1 dydx. Lemma 3.1. The expression I(k, l) is negative for k ∈ N >2 and l ∈ N≥ 2.
The direct translation in geometrical terms, by [8, Corollary 1.3], justifies the K-unstable members of the family, a major step in proving Theorem 1.2:

Corollary 3.2. The Fano manifold Y n,p = Bl Q p-2 (Q n-2 ) does not admit a Kähler-Einstein metric if 3 < p ≤ n -3.
It is a direct consequence from this and the fact that G is semisimple that there cannot exist any Kähler-Ricci soliton or Mabuchi metrics on these manifolds (the soliton or Mabuchi vector field must commute with the action of G hence is zero here). Furthermore, the above Theorem has a more precise interpretation: from [5, p. 653] we deduce that Y n,p is not K-semistable, or similarly from [8, Corollary 1.3] we deduce that the greatest Ricci lower bound R(Y n,p ) is strictly less than 1.

Proof. We first derive an explicit formula for the integral. Integration in the y variable yields

I(k, l) = k 0 (x k -(k -1)x k-1 ) (k + l -x) l l dx
Using the change of variables t = x k , we have

I(k, l) = k k+l+1 l 1 0 t k - k -1 k t k-1 1 -t + l k l dt
By the binomial formula applied to 1t + l k l , we write this as a sum

k k+l+1 l l j=0 l j l l-j k l-j B(k + 1, j + 1) - k -1 k B(k, j + 1)
where B denotes the beta function, defined by

B(a, b) = 1 0 t a-1 (1 -t) b-1 dt = (a -1)!(b -1)! (a + b -1)! at positive integers a, b
Let A j denote the j-th summand, so that lk -(k+l+1) I(k, l) = l j=0 A j . We simplify a bit the expression of the summands:

A j = l j l l-j k l-j k!j! (k + j + 1)! - k -1 k (k -1)!j! (k + j)! = l j l l-j k l-j+1 (k -1)!j! (k + j + 1)! k 2 -(k -1)(k + j + 1) = l j l l-j k l-j+1 (k -1)!j! (k + j + 1)! (j(1 -k) + 1)
The major advantage of this expression (over the numerous different ways to compute the integral) is that for k ≥ 3, all but the first term are negative. It thus suffices to compensate the positive first term with some of the negative terms to conclude.

For most cases, it is enough to consider the first two summands:

A 0 + A 1 = l l k l+1 1 k(k + 1) + l l l-1 k l 2 -k k(k + 1)(k + 2) = l l k l+2 (k + 1)(k + 2) -k 2 + 3k + 2
The polynomial -

x 2 + 3x + 2 is negative for x > 3+ √ 17 2 , hence A 0 + A 1 is negative for k ≥ 4.
For the case k = 3, we consider the additional summand A 2 . We compute

A 0 + A 1 + A 2 = l l-1 36 • 3 l (-5l + 9
) and this is negative for l > 9 5 hence in particular for l ≥ 2. 3.4.2. Existence of Kähler-Einstein metrics.

Lemma 3.3. We have, for all l ∈ N ≥2 ,

J(l) := 2 x=0 2+l-x y=0 x(y -(l -1))y l-1 dydx > 0 I(l) := 2 x=0 2+l-x y=0 x(x -1)y l-1 dydx > 0
The geometrical translation of the above statement finishes the proof of Theorem 1.2 in the cases 3 ≤ p ≤ n -3 by dealing with the case p = 3.

Proof. Let us start with the first inequality. We compute

J(l) = 2 0 x (2 + l -x) l+1 l + 1 -(l -1) (2 + l -x) l l dx = l+2 l t l l(l + 1) -lt 2 + (2l 2 + 2l -1)t -(l + 2)(l 2 -1) dt = (l + 2) l+1 (4l 2 + 7l -2) -l l+1 (-4l 2 + l + 12) l(l + 1)(l + 2)(l + 3)
Note that, for l ≥ 2, 4l 2 + 7l -2 is positive, and (l + 2) l+1 > l l+1 , so we have

l -l (l + 1)(l + 2)(l + 3)J(l) > (4l 2 + 7l -2) -(-4l 2 + l + 12) > 8l 2 + 6l -14.
It is easy to check that the latter polynomial is positive for l > 1 hence a fortiori for l ≥ 2.

For the second inequality, we compute using the same method

I(l) = (l + 2) l+2 -(7l + 12)l l+1 (l + 2)(l + 3) = (l + 2) l+2 (l + 2)(l + 3) 1 -1 - 2 l + 2 l+2 7l + 12 l By a classical inequality, 1 -2 l+2 l+2 < e -2 , hence it is enough to have 7l + 12 < le 2 .
The inequality above is satisfied as long as l ≥ 31 and the statement is proved in these cases. It remains to check a finite number of cases (for 2 ≤ l ≤ 30). We omit the details as it is tedious and without difficulty.

4.

Remaining cases of blow-ups of quadrics along a linear subquadric 4.1. Description of the manifolds. In this last section, we will be interested in the blow-ups of Q n-2 along Q n-4 , the blow-up along Q 0 , and the blow up along a single point (which is one half of Q 0 ). In particular, in the notations of the previous section, we deal with the cases p = 2 and p = n -2.

We hence consider the case when the group acting on the quadric is G = SO n-2 × SO 2 (and we assume 5 ≤ n, the case of the blow up of P 1 × P 1 at one or two points being well-known). Since SO 2 = C * , the group G is not semisimple. Furthermore, there are more orbits in the quadric: the zerodimensional quadric Q 0 consists of two orbits of G, there is still the closed orbit Q n-4 , there are two codimension one orbits, formed by points whose projections belong to Q n-4 and one point in Q 0 , and finally the complement of all these lower dimensional orbits is the open dense orbit. The valuation cone V is still the negative restricted Weyl chamber, defined here by α ≤ 0 in (M ⊗ R) * . There is a single color D α which is sent to α ∨ by the color map. Let f be the element of (M ⊗ R) * defined by α(f ) = 0 and ̟(f ) = 2, so that f and f +α ∨ 2 generate the lattice dual to M.

4.2.2.

Colored fans and embeddings. The colored fan of the quadric is formed by the following cones:

• ({0}, ∅) corresponding to the open dense orbit,

• the ray (R + (fα ∨ ), ∅) corresponding to a codimension one orbit whose closure we denote by D + , • the ray (-R + (f + α ∨ ), ∅) corresponding to a codimension one orbit whose closure we denote by D -,

• the colorless cone (Cone(f -α ∨ , -f -α ∨ ), ∅) corresponding to Q n-4 • the colored cone (Cone(α ∨ , f -α ∨ ), {α ∨ }) corresponding to one point in Q 0 , • and the colored cone (Cone(α ∨ , -f -α ∨ ), {α ∨ }) corresponding to the other point in Q 0 . Figure 5. Images of divisors in (M ⊗ R) * α ∨ f • E + • E - • D + • D - • D α • E
To obtain the colored fans of the blow-ups, it suffices to make the obvious modifications while adding:

• the ray (±R + f, ∅) for the blow-up at one point of Q 0 , or both for the blow-up along the full Q 0 , we denote by E ± the corresponding divisors, • the ray (-R + α ∨ , ∅) for the blow-up at Q n-4 , we denote by E the corresponding divisor.

The images of the above divisors in (M⊗R) * are as illustrated in Figure 5.

∆ e (a, a e ) = {xα + y̟ | 0 ≤ x ≤ a e 2 , x -a ≤ y ≤ a -x}.
The divisor D e (a, a e ) is ample if and only if 0 < a e < 2a. The anticanonical divisor is D e ( n 2 -1, n -3). We now consider the blow-up of Q n-2 along one point of Q 0 , say the one whose exceptional divisor we denoted by E + . The relations in the Picard group are 2E + + D + -D -= 0 and 2D α 1 -D + -D -= 0. We may then write any R-divisor as combinations

D + (a, a + ) = a(D + + D -) + a + E + Figure 6. The polytope ∆ e (a, a e ) • 0 • ae 2 α • a̟ Figure 7. The polytope ∆ + (a, a + ) • 0 • aα • -a + 2 ̟
with a, a + ∈ R. They are special divisors, with trivial isotropy characters. The corresponding special (or moment) polytope is

∆ + (a, a + ) = xα + y̟ | 0 ≤ x, -a + 2 ≤ y, x -a ≤ y ≤ a -x .
The divisor D + (a, a + ) is ample if and only if 0 < a + < 2a. The anticanonical divisor is D + ( n 2 -1, 1). We now consider the blow-up of Q n-2 along both points of Q 0 . The relations are now 2(E + -E -) + D + -D -= 0 and 2D α 1 -(D + + D -) = 0. We may then write any R-divisor as combinations

D ± (a, a + , a -) = a(D + + D -) + a + E + + a -E - Figure 8. The polytope ∆ ± (a, a + , a -) • 0 • aα • -a + 2 ̟ • a - 2 ̟
with a, a + , a -∈ R. They are special divisors, with trivial isotropy characters. The corresponding special (or moment) polytope is

∆ ± (a, a + , a -) = xα + y̟ | 0 ≤ x, -a + 2 ≤ y ≤ a - 2 , x -a ≤ y ≤ a -x .
The divisor D ± (a, a + , a -) is ample if and only if 0 < a + < 2a and 0 < a -< 2a. The anticanonical divisor is D ± ( n 2 -1, 1, 1). The half sum of restricted roots is ( n 2 -2)α 1 We set the following notations for the barycenters of the several blow-ups of Q n-2 : x n-4 dydx

I α = n-3 x=0 n-2-x y=x-n+2
x n-3 dydx

I ̟ = n-3 x=0 n-2-x y=x-n+2 yx n-4 dydx then Bl Q n-4 (Q n-2
) is Kähler-Einstein if and only if

I ̟ = 0 and I α I V > n -4.
The vanishing I ̟ = 0 is obvious by symmetry with respect to the reflection leaving R̟ invariant, while a straightforward computation yields

I α I V = 2(n -3) 2 (n -2) (n -1)(2n -5) .
Then Iα I V > n -4 is equivalent to 7n 2 -11n + 2 > 0 which is true for n > 11+ √ 65 14

hence certainly for all n ≥ 5. So we have proved that Bl Q n-4 Q n-2 is Kähler-Einstein for all n ≥ 5.

Case of Bl

Q 0 Q n-2 .
Here there is an obvious symmetry of the polytope yielding the vanishing of the Futaki character, hence no obstruction in the direction of ̟. It remains to check the inequality

I α I V -(n -4) > 0
where

I V = 2 1 y=0 n-2-y x=0
x n-4 dxdy

I α = 2 1 y=0 n-2-y x=0
x n-3 dxdy.

We have, by computation of the integrals,

I α I V = (n -3) (n -2) n-1 -(n -3) n-1 (n -1) ((n -2) n-2 -(n -3) n-2 )
Using the fact that

(n -2) n-1 -(n -3) n-1 > (n -2) (n -2) n-2 -(n -3) n-2
the previous inequality is implied by (n -3)(n -2) -(n -4)(n -1) > 0 which is true (recall that we can assume n ≥ 5 throughout). We have proved that Bl Q 0 Q n-2 is Kähler-Einstein.

4.5.

There are no Mabuchi metrics on Bl pt Q n-2 . We finally consider the blow-up of Q n-2 at a point. There is an obvious lack of symmetry in the polytope (even with respect to the Duistermaat-Heckman measure) which ensures the the Futaki character will not vanish. Hence there are no Kähler-Einstein metrics on Bl pt Q n-2 . An even shorter proof is that the automorphism group is not reductive. We prove in fact a stronger result: We will show that when the first condition is satisfied, the third condition is violated. Assume A and B are such that the first condition is satisfied. Then -B A is equal to The numerator is obviously positive, and the denominator is positive thanks to the lack of symmetry, and comparison with the Kähler-Einstein case of Bl Q 0 Q n-2 . In order to show that there are no Mabuchi metrics, it thus suffices to show that Integrating, then multiplying by 2 n n(n -1)(n -3), this is equivalent to 4(n -2) n-1 -(n -3) n-2 (2n 2 + n -9) ≤ 0.

Since n -2 n -3

n-3 = 1 - 1 n -3 n-3 ≤ e -1 ,
the inequality is implied by 0 ≤ 2n 3 -9n 2 + 4n + 11 which certainly holds for n ≥ 5. We have shown that Bl pt Q n-2 admits no Mabuchi metrics (for n ≥ 5).

  It has dimension five, so the following question remains open. Question 1.3. Does there exists examples of Fano threefolds with no Kähler-Ricci solitons that are K-unstable? What about Fano fourfolds?

  2.2.2.Colored fans and embeddings. The colored fan of P n-1 is the fully colored fan {(R + e, D α p+2,p+1 ), (-R + e, D α 2,1 ), ({0}, ∅)},

Figure 3 .

 3 Figure 3. Colored fans of the quadric and its blow-up

3. 3 .

 3 Combinatorial data associated to divisors. 3.3.1. Picard group, isotropy characters and polytopes. The Picard group of Y n,p is generated by: the two colors D α and D β whose image under the color map are respectively α ∨ and β ∨ , and the G-invariant divisors D d and D e , whose image in (M ⊗ R) * are respectively -1 2 (α ∨ + β ∨ ) and -α ∨ . There are two relations, given by 2(D α -D e )-D d = 0 and D α +D β -D e -D d . Since we are not interested in integrality issues, we write R-divisors as combinations of the two G-invariant divisors: D(a d , a e ) = a d D d + a e D e , and note that any such divisor is special and has trivial isotropy character. The special polytope and moment polytope thus coincide and live in M ⊗ R, where they are given by ∆(a d , a e ) = xα + yβ | 0 ≤ x ≤ a e 2 , 0 ≤ y, x + y ≤ a d .

Figure 4 .

 4 Figure 4. Moment polytope for Y n,p

4. 2 .

 2 Combinatorial data associated to the manifolds. 4.2.1. The open orbit. The open dense orbit is a (reductive) symmetric space of type A 1 and rank two. The simple root is denoted, consistently with the previous section, by α. Its multiplicity is p -2 = n -4. The spherical lattice M is generated by some element ̟ ∈ X(T /(T ∩ [G, G])) and ̟+α 2

4. 3 .

 3 Combinatorial data associated to divisors. 4.3.1. Picard group, isotropy characters and polytopes: The Picard group of the blow-up of Q n-2 along Q n-4 is generated by D + , D -, D α , and E, with the relations D -= D + and D + + D -+ 2E -2D α = 0. It is convenient to represent the R-divisors as combinations D e (a, a e ) = a(D + + D -) + a e E with a, a e ∈ R. They are special divisors, with trivial isotropy characters. The corresponding special (or moment) polytope is

4.3. 2 .

 2 Duistermaat-Heckman polynomial and barycenters. Up to a multiplicative constant, the Duistermaat-Heckman polynomial is P DH (xα + y̟) = {α, xα + y̟} n-4 = (2x) n-4

4 . 4 . 1 .

 441 Bar e (a, a e ) = ∆e(a,ae) pP DH (p)dp ∆e(a,ae) P DH (p)dpBar + (a, a + ) = ∆ + (a,a + ) pP DH (p)dp ∆ + (a,a + ) P DH (p)dp Bar ± (a, a + , a -) = ∆ ± (a,a + ,a -) pP DH (p)dp∆ ± (a,a + ,a -) P DH (p)dp 4.4. Existence of Kähler-Einstein metrics for blow-ups along a full quadric. We will now finish the proof of Theorem 1.2. Case of Bl Q n-4 Q n-2 . Set

Theorem 4 . 1 .

 41 For n ≥ 5, the blow-up of Q n-2 at one point does not admit Mabuchi metrics.Proof. By [8, Corollary 1.4] and the combinatorial description above, there exists a Mabuchi metric on Bl pt Q n-2 if and only if we can find a linear function y → Ay + B with 0 = A, B ∈ R, such that the following three conditions are satisfied: