For a reflexive lattice polytope4in Rn =Zn⊗ZR, we have a Fano toric manifoldX4⊃(C∗)n with a (C∗)n action. In the following, we will sometimes just writeX forX4 for simplicity.
Let (S1)n⊂(C∗)n be the standard real maximal torus. Let{zi}be the standard coordinates of the dense orbit (C∗)n, and xi = log|zi|2. We have a standard lemma about toric K¨ahler metric, which we omit the proof. See for example [WaZh].
Lemma 24. Any (S1)n invariant K¨ahler metricω onX has a potentialu=u(x)on(C∗)n, i.e.
ω=√
−1∂∂u.¯ uis a proper convex function onRn, and satisfies the momentum map condition:
Du(Rn) =4. find the appropriate potential on (C∗)n for the pull back of Fubini-Study metric. (Cf. [WaZh])
Recall that, for any sectionsofKX−1, the Fubini-Study metric as a Hermitian metric onKX−1 is defined up to the multiplication by a positive constant:
|s|2F S=e−C˜∑|s|2
β|sβ|2. (4.5)
The righthand side is well defined by using local trivializations. ˜Cis some normalizing constant which we choose now to simplify the computation later.
First, let ˜s0be the section corresponding to the origin 0∈ 4. On the open dense orbit (C∗)n, by standard toric geometry, we can assume
sα
In other words, we define
Now we can choose ˜C by the normalization condition:
∫
On the other hand,Ric(ω) is the curvature form of Hermitian line bundleKM−1 with Hermitian metric determined by the volume formωn. Note that we can take ˜s0=z1 ∂
i when acting on any (S1)ninvariant function on (C∗)n, we have
It’s easy to see from definition ofhω in (4.1) and normalization condition (4.8) that ehωωn/n!
dz1
z1 ∧d¯z¯z11· · · ∧dzznn∧ d¯¯zznn =ehωks˜0k2ωn=k˜s0k2F S=e−u˜0. (4.9) Remark 31. We only use vertex lattice points because, roughly speaking, later in Lemma 23, vertex lattice points alone helps us to determine which sections become degenerate when doing biholomorphic transformation and taking limit. See remark 33. We expect results similar to Theorem 20 hold for general toric reference K¨ahler metric.
So divide both sides of (∗)t by meromorphic volume form n!(dzz1
1 ∧ d¯¯zz11· · · ∧dzznn ∧ d¯z¯znn), We can rewrite the equations (∗)tas a family of real Monge-Amp`ere equations onRn:
det(uij) =e−(1−t)˜u0−tu (∗∗)t
4◦=Df(Rn), the strictly convex functionf is properly. For simplicity, let
wt(x) =tu(x) + (1−t)˜u0.
Then wt is also a proper convex function on Rn satisfying Dwt(Rn) = 4. So it has a unique absolute minimum at pointxt∈Rn. Let
mt= inf{wt(x) :x∈Rn}=wt(xt).
Wang-Zhu’s [WaZh] method for solving (∗∗)t consists of two steps. The first stepis to show some uniform a priori estimates forwt. Fort < R(X4), the proper convex functionwt obtains its minimum value at a unique pointxt∈Rn. Let
mt=inf{wt(x) :x∈Rn}=wt(xt)
Proposition 20 ([WaZh],See also [Don3]). [1.]
1. there exists a constantC, independent oft < R(X4), such that
|mt|< C
2. There existsκ >0 and a constantC, both independent oft < R(X4), such that
wt≥κ|x−xt| −C (4.10)
For the reader’s convenience, we record the proof here.
Proof. LetA={x∈Rn;mt≤w(x)≤mt+ 1}. Ais a convex set. By a well known lemma due to Fritz John, there is a unique ellipsoidEof minimum volume among all the ellipsoids containing A, and a constantαn depending only on dimension, such that
αnE⊂A⊂E
αnE means the αn-dilation ofE with respect to its center. Let T be an affine transformation with det(T) = 1, which leavesx0=the center ofE invariant, such that T(E) =B(x0, R), where
B(x0, R) is the Euclidean ball of radiusR. Then
B(x0, αnR)⊂T(A)⊂B(x0, R)
We first need to boundRin terms ofmt. SinceD2w=tD2u+ (1−t)D2u˜0≥tD2u, by ((∗∗)t), we see that
det(wij)≥tne−w Restrict to the subsetA, it’s easy to get
det(wij)≥C1e−mt
Let ˜w(x) =w(T−1x), since det(T) = 1, ˜wsatisfies the same inequality
det( ˜wij)≥C1e−mt
inT(A).
Construct an auxiliary function
v(x) =C
1 n
1 e−mtn 1 2
(|x−x0|2−(αnR)2)
+mt+ 1
Then inB(x0, αnR),
det(vij) =C1e−mt ≤det( ˜wij)
On the boundary ∂B(x0, αnR), v(x) =mt+ 1≥ w. By the Comparison Principle for Monge-˜ Am`ere operator, we have
˜
w(x)≤v(x) in B(x0, αnR)
In particular
mt≤w(x˜ 0)≤v(x0) =C
1 n
1 e−mtn 1 2(−R2
n2) +mt+ 1 So we get the bound forR:
R≤C2emt2n So we get the upper bound for the volume ofA:
V ol(A) =V ol(T(A))≤CRn≤Cemt2
By the convexity ofw, it’s easy to see that{x;w(x)≤mt+s} ⊂s· {x;w(x)≤mt+ 1}=s·A, wheres·Ais thes-dilation ofAwith respect to pointxt. So
V ol({x;w(x)≤mt+s})≤snV ol(A)≤Csnemt2 (4.11)
The lower bound for volume of sublevel sets is easier to get. Indeed, since |Dw(x)| ≤L, where L= maxy∈4|y|, we haveB(xt, s·L−1)⊂ {x;w(x)≤mt+s}. So
V ol({x;w(x)≤mt+s})≥Csn (4.12)
Now we can derive the estimate formt. First note the identity:
∫
Rn
e−wdx=
∫
Rn
det(uij)dx=
∫
4
dσ=V ol(4) (4.13)
Second, we use the coarea formula
∫
Rn
e−wdx =
∫
Rn
∫ +∞ w
e−sdsdx=
∫ +∞
−∞
e−sds
∫
Rn
1{w≤s}dx
=
∫ +∞ mt
e−sV ol({w≤s})ds
= e−mt
∫ +∞ 0
e−sV ol({w≤mt+s})ds (4.14)
Using the bound for the volume of sublevel sets (4.11) and (4.12) in (4.14), and compare with (4.13), it’s easy to get the bound for|mt|.
Now we prove the estimate (4.10) following the argument of [Don3]. We have seenB(xt, L−1)⊂ {w≤mt+ 1}, andV ol({w≤mt+ 1})≤Cby (4.11) and uniform bound formt. Then we must have {w ≤mt+ 1} ⊂ B(xt, R(C, L)) for some uniformly bounded radiusR(C, L). Otherwise, the convex set{w≤mt+ 1}would contain a convex subset of arbitrarily large volume. By the convexity of w, we have w(x) ≥ R(C,L)1 |x−xt|+mt−1 Since mt is uniformly bounded, the estimate (4.10) follows.
Thesecond stepis trying to bound|xt|. In Wang-Zhu’s [WaZh] paper, they proved the exis-tence of K¨ahler-Ricci soliton on toric Fano manifold by solving the real Monge-Amp`ere equation corresponding to K¨ahler-Ricci solition equation. But now we only consider the K¨ahler-Einstein equation, which in general can’t be solved because there is the obstruction of Futaki invariant.
Proposition 21 ([WaZh]). the uniform bound of |xt| for any 0 ≤t≤t0, is equivalent to that we can solve (∗∗)t, or equivalently solve(∗)t, for t up to t0. More precisely, (by the discussion in introduction,) this condition is equivalent to the uniform C0-estimates for the solution φt in (∗)t fort∈[0, t0].
Again we sketch the proof here.
Proof. If we can solve (∗∗)t(or equivalently (∗)t) for 0≤t≤t0. Then{w(t) = (1−t) ˜u0+tu; 0≤ t≤t0}is a smooth family of proper convex functions onRn. By implicit function theorem, the minimal pointxtdepends smoothly ont. So{xt}are uniformly bounded in a compact set.
Conversely, assume |xt|is bounded. First note thatφt=u−˜u0= 1t(wt(x)−u˜0).
As in Wang-Zhu [WaZh], we consider the enveloping function:
v(x) = max
pα∈Λ∩4hpα, xi
Then 0≤u˜0(x)−v(x)≤C, andDw(ξ)·x≤v(x) for allξ, x∈Rn. We can assumet≥δ >0.
Then using uniform boundedness of|xt|
φt(x) = 1
t(wt(x)−u˜0) = 1
t[(wt(x)−wt(xt))−v(x) + (v(x)−u˜0(x)) +wt(xt)]
≤ δ−1(Dwt(ξ)·x−v(x)−Dwt(ξ)·xt) +C≤C0
Thus we get the estimate for suptφt. Then one can get the bound for inftφtusing the Harnack inequality in the theory of Monge-Amp`ere equations. For details see ([WaZh], Lemma 3.5) (see also [Tia1]).
By the above proposition, we have
Lemma 25. If R(X4)<1, then there exists a subsequence{xti} of {xt}, such that
lim
ti→R(X4)|xti|= +∞ The observation now is that
Lemma 26. If R(X4) <1, then there exists a subsequence of {xti} which we still denote by {xti}, andy∞∈∂4, such that
lim
ti→R(X4)Du˜0(xti) =y∞ (4.15)
This follows easily from the properness of ˜u0 and compactness of4.
We now use the key relation (See [WaZh] Lemma 3.3, and also [Don3] page 29)
0 =
∫
Rn
Dw(x)e−wdx=
∫
Rn
((1−t)Du˜0+tDu)e−wdx
Since
∫
Rn
Du e−wdx=
∫
Rn
Dudet(uij)dx=
∫