1 Introduction
One of the main problems in Kähler geometry is the existence problem of canonical Kähler metrics on a Kähler manifold with fixed Kähler class. This problem began with the case of Kähler–Einstein metrics as a Calabi conjecture which states that any real
$(1,1)$
-form representing the first Chern class is given by the Ricci form of some Kähler metric in a given Kähler class. This problem was solved by Aubin [Reference Aubin2] and Yau [Reference Yau35] for the case of negative first Chern class, and by Yau for zero first Chern class. For the positive first Chern class case, several obstructions were found. One of them is the vanishing of the Futaki invariant [Reference Futaki20]. This obstruction was generalized to the case of constant scalar curvature Kähler metrics (cscK metrics) by Futaki [Reference Futaki21], Calabi [Reference Calabi, Chavel and Farkas9]. On the other hand, if there exists a cscK metric in the fixed Kähler class, then it must be unique up to automorphisms [Reference Berman and Berndtsson4], [Reference Chen and Tian11].
As a generalization of cscK metrics, Calabi introduced extremal Kähler metrics, and constructed an explicit extremal Kähler metric in each Kähler class on the Hirzebruch surfaces [Reference Calabi and Yau8]. This led to the speculation that extremal Kähler metrics exist in every Kähler class for any compact Kähler manifold. However, his expectation was not true in general [Reference Levine26], [Reference Wang and Zhou34]. In honor of Calabi’s original vision, X. Chen and J. Cheng introduced the notion of a Calabi dream manifold (CDM), that is a compact Kähler manifold which admits an extremal Kähler metric in each Kähler class [Reference Chen and Cheng10, Definition 1.12]. At this moment, several examples of CDMs have been discovered, for instance, [Reference Apostolov, Calderbank, Gauduchon and Tønnesen-Friedman1], [Reference Calabi and Yau8], [Reference Guan22], [Reference Hwang25], [Reference Song and Weinkove33]. In particular, Apostolov–Calderbank–Gauduchon–Tønnesen–Friedman established the admissible construction, and then found CDMs among some classes of projective bundles. In [Reference Boyer, Calderbank and Tønnesen-Friedman6], Boyer–Calderbank–Tønnesen–Friedman applied the admissible construction to Bott manifolds, which is the main focus of this paper. Also, they found many explicit examples of Bott manifolds with extremal Kähler metrics.
The existence of cscK/extremal metrics on a polarized manifold has been expected to be equivalent to K-polystability/relative K-polystability (Yau–Tian–Donaldson conjecture). In particular, the existence problem of extremal Kähler metrics on Bott Fano manifolds is relevant to classify certain class of toric Fano manifolds in terms of relative K-polystability (see, [Reference Nitta, Saito and Yotsutani29, Section 3.5]). Specifically, the third author and B. Zhou partially classified relative K-polystable toric Fano threefolds using a combinatorial criteria of relative K-stability established in [Reference Yotsutani and Zhou36]. Although it still remains an open problem to verify relative K-polystability for some of toric Fano threefolds (see, [Reference Yotsutani and Zhou37, Table 1]), we can adapt Bott structure to verify relative K-polystability for certain toric Fano manifolds. For example, toric Fano fourfold
$\mathbb P_{(\mathbb P^1)^3}(\mathcal O\oplus \mathcal O(1,1,1))$
, (which is labeled
$L_1$
in [Reference Batyrev3]), is an example of Fano Bott manifolds at stage four, and the criterion in [Reference Yotsutani and Zhou36] cannot be applied. However, it is known that
$L_1$
is a CDM, so that we conclude that
$L_1$
is relatively K-polystable with respect to the anticanonical polarization (see, [Reference Nitta, Saito and Yotsutani29, Corollary 1.9]).
As a special class of CDM, Martinez–Garcia [Reference Martinez-Garcia27] called a compact Kähler manifold X a strong Calabi dream manifold (SCDM) if X admits a cscK metric in each Kähler class. It was proved in [Reference Martinez-Garcia27, Corollary 1.7] that any smooth projective rational strong Calabi dream surface is either
$\mathbb P^2$
or
$\mathbb P^1\times \mathbb P^1$
. The main interest of this paper is to classify all SCDMs of arbitrary dimension. Our main result is the following.
Theorem 1.1. Let X be a stage n Bott manifold. Then the Futaki invariant of X vanishes for any Kähler class if and only if X is isomorphic to an n-fold product of the projective line i.e.,
$X\cong (\mathbb P^1)^n$
.
The uniqueness of cscK metrics in the fixed Kähler class implies the following splitting theorem on Bott manifolds with cscK metrics.
Corollary 1.2. Let X be a stage n Bott manifold which admits a cscK metric in each Kähler class. Then X is isomorphic to the products of the projective lines. Moreover, such a cscK metric in every Kähler class is the Riemannian product of
$c_i\omega _{\mathrm {FS}}$
for
$1\leqslant i \leqslant n$
, where
$c_i$
are some positive constants and
$\omega _{\mathrm {FS}}$
is the Fubini-Study metric of the projective line.
Finally, we explain a related work to Theorem 1.1. Following the Yau–Tian–Donaldson conjecture, we can consider an algebraic counterpart of SCDM: a polarized manifold
$(X,L)$
which is K-polystable for any polarization. By using the notion of slope semistability in [Reference Ross and Thomas31], which is a weaker notion of K-semistability, Fujita and the third author proved the following.
Theorem 1.3 [Reference Fujita and Yotsutani18].
Let X be a stage n Bott manifold. For any ample
${\mathbb Q}$
-line bundle L on X, if
$(X,L)$
is slope semistable along any smooth prime divisor D in X, then
$X\cong (\mathbb P^1)^n$
.
As a corollary of Theorem 1.3, they concluded that any strong Calabi dream Bott manifold is isomorphic to the products of projective lines. We remark that on a Bott manifold, the assumptions in Theorems 1.1 and 1.3 would be equivalent under certain condition. In fact, for any torus-invariant prime divisor D on a Bott manifold X, we assume that
$\big ((L-\varepsilon D)^{\cdot n}\big )=0$
, where
$\varepsilon $
is the Seshadari constant of D with respect to a polarization L. Under this assumption, we see that the Futaki invariant for the holomorphic vector field vanishing on D is equal to the limit of the Donaldson–Futaki invariant of the degeneration to the normal cone of D, when the parameter goes to the Seshadri constant. Thus, we conclude that the vanishing of the Futaki invariant is equivalent to slope semistability for any smooth prime divisor on a Bott manifold with a given polarization, if
$\big ((L-\varepsilon D)^{\cdot n}\big )=0$
. For the reader’s convenience, we give more details in Section A. Although the conclusions are close to each other, the approaches of this article and [Reference Fujita and Yotsutani18] are different: in the former, we go through integration over the moment polytope of X. In particular, a key ingredient of our proof is the classical Brunn–Minkowski inequality for convex polytopes (see, Section 4.2). In the latter, the proof was based on the computations of intersection numbers on Bott manifolds.
This paper is organized as follows. Section 2 gives a quick review of toric varieties and their associated convex polytopes. Section 3 collects basic facts of Bott manifolds. In Section 3.2, we recall a combinatorial characterization of Bott manifolds in terms of their associated fan (see, Proposition 3.2). We prove Theorem 1.1 in Section 4 by induction of the dimension of the Bott tower. In the final section, we give the detailed explanation of the relation between Theorems 1.1 and 1.3, and clarify the difference between the main theorems of this article and the paper by Fujita and the third author [Reference Fujita and Yotsutani18].
Through the paper, we work over a complex number field
${\mathbb C}$
, and a manifold is a smooth irreducible complex projective variety.
2 Toric geometry and convex polytopes
This section collects some fundamental results in toric geometry which will be used at a later stage. See [Reference Cox, Little and Schenck14] and [Reference Fulton19] for more details. A toric variety X is an algebraic normal variety with an effective holomorphic action of
$T_{\mathbb C}:=({\mathbb C}^\times )^n$
with
$\dim _{\mathbb C} X=n$
. Let
$T_{\mathbb R}:=(S^1)^n$
be the real torus in
$T_{\mathbb C}$
and
$\mathfrak {t}_{\mathbb R}$
be the associated Lie algebra. Let
$N_{\mathbb R}:=J\mathfrak {t}_{\mathbb R}\cong {\mathbb R}^n$
be the associated Euclidean space where J is the complex structure of
$T_{\mathbb C}$
. We denote the dual space
$\operatorname {\mathrm {Hom}} (N_{\mathbb R}, {\mathbb R})\cong {\mathbb R}^n$
of
$N_{\mathbb R}$
by
$M_{\mathbb R}$
. Setting the group of algebraic characters of
$T_{\mathbb C}$
by M, we see that
$M_{\mathbb R}=M\otimes _{\mathbb Z}{\mathbb R}$
.
Let us denote the dual lattice of M by N. Then, N consists of the algebraic one parameter subgroups of
$T_{\mathbb C}$
. A fan
$\Sigma $
in
$N_{\mathbb R}$
is defined to be a collection of strongly convex polyhedral lattice cones such that any face of a cone in
$\Sigma $
is also an element of
$\Sigma $
and the intersection of two cones in
$\Sigma $
is a face of each. The support of
$\Sigma $
is defined as the union of all cones contained in
$\Sigma $
. We denote the support of
$\Sigma $
by
$|\Sigma |$
, that is,
$|\Sigma |={\bigcup _{\sigma \in \Sigma }}\sigma $
. A fan
$\Sigma $
is said to be smooth if any cone
$\sigma \in \Sigma $
is smooth (i.e., its minimal generators form part of a
$\mathbb Z$
-basis of N). A fan
$\Sigma $
is called complete if its support is the whole space
$N_{\mathbb R}$
. A fan
$\Sigma $
is polytopal if there exists a convex lattice polytope P in
$M_{\mathbb R}$
such that
$0\in P$
and whose normal fan
$\Sigma _P$
coincides with
$\Sigma $
.
From now on
$X_\Sigma $
denotes the toric variety associated with a fan
$\Sigma $
in
$N_{\mathbb R}$
. Then, geometric properties of
$X_\Sigma $
can be characterized by properties of
$\Sigma $
as follows.
Proposition 2.1 [Reference Cox, Little and Schenck14, Theorem 3.1.19].
Let
$X_\Sigma $
be the toric variety defined by a fan
$\Sigma $
in
$N_{\mathbb R}$
. Then:
-
1.
$X_\Sigma $
is a smooth variety if and only if
$\Sigma $
is smooth. -
2.
$X_\Sigma $
is compact (in the Euclidean topology) if and only if
$\Sigma $
is complete. -
3.
$X_\Sigma $
is projective if and only if
$\Sigma $
is polytopal.
Let
$\Sigma $
be a complete fan in
$N_{\mathbb R}$
. For
$1\leqslant k \leqslant n$
, we let
$\Sigma (k)$
denote the set of all k-dimensional cones in
$\Sigma $
; 
. For any finite subset
$F\subset N_{{\mathbb R}}$
, we set
$$\begin{align*}\mathrm{Cone\,}(F):=\sum_{x\in F}{\mathbb R}_{\geqslant 0}\,x. \end{align*}$$
Then, we define
Definition 2.2. A nonempty subset
$R\subset G_\Sigma $
is a primitive collection of
$\Sigma $
, if it satisfies the conditions
-
(a)
$\mathrm {Cone\,}(R)\notin \Sigma $
, and (b) for every
$x\in R$
.
for every 
We denote by
$\mathrm {PC}(\Sigma )$
the set of primitive collections of
$\Sigma $
.
Let
$\Sigma _1$
and
$\Sigma _2$
be smooth complete fans in
$N_{\mathbb R}$
.
$\Sigma _1$
and
$\Sigma _2$
are said to be combinatorially equivalent if there exists a bijective map
$\psi :G_{\Sigma _1}\to G_{\Sigma _2}$
such that for any nonempty subset
$R\subset G_{\Sigma _1}$
, we have
$\mathrm {Cone\,}(R)\in \Sigma _1$
if and only if
$\mathrm {Cone\,}(\psi (R))\in \Sigma _2$
. Furthermore,
$\mathrm {PC}(\Sigma _1)$
and
$\mathrm {PC}(\Sigma _2)$
are isomorphic if there exists a bijective map
$\phi : G_{\Sigma _1}\to G_{\Sigma _2}$
which induces a well-defined bijective map
$$\begin{align*}\phi_*:\mathrm{PC}(\Sigma_1) \longrightarrow \mathrm{PC}(\Sigma_2), \qquad R\longmapsto \phi(R). \end{align*}$$
Let P be a full dimensional convex polytope in
$M_{\mathbb R}$
, and let
$\mathcal V(P)$
be the set of all vertices of P. Here and hereafter, we assume unless otherwise stated that all polytopes are full dimensional convex polytopes. Two convex polytopes
$P_1$
and
$P_2$
are combinatorially equivalent if there exists a bijective map
$\varphi :\mathcal V(P_1)\to \mathcal V(P_2)$
which induces a well-defined map
that preserves dimensions, intersections, and the face relation. For example, there is a seven dimensional non-symmetric reflexive Kähler–Einstein moment polytope
$P\subset M_{\mathbb R}$
with
$64$
vertices used in [Reference Ono, Sano and Yotsutani30]. This seven dimensional polytope P is combinatorially equivalent to
$\Delta _3\times (\Delta _1)^4$
, where
$\Delta _n$
denotes the standard n-simplex.
Finally, we recall that an n-dimensional convex polytope P in
$ M_{\mathbb R}$
is Delzant if it satisfies the following conditions:
-
(i) At each vertex p of P, there are exactly n edges
$l_{p,1}, \ldots , l_{p,n}$
of P emanating from p. -
(ii) Each edge
$l_{p,k}$
in (i) is of the form
$p+tu_k$
$(t\geqslant 0)$
with some primitive vector
$u_k\in M$
. -
(iii) For each vertex p, the primitive vectors
$u_1, \ldots , u_n$
corresponding to the edges
$l_{p,1}, \ldots , l_{p,n}$
generate the lattice M over
$\mathbb Z$
.
From these definitions, we readily obtain the following proposition.
Proposition 2.3 [Reference Batyrev3, Proposition 2.1.6], [Reference Sato32, Proposition 1.3.4].
Let
$P_1$
and
$P_2$
be lattice Delzant polytopes in
$M_{\mathbb R}$
. Let
$\Sigma _{P_1}$
and
$\Sigma _{P_2}$
be the associated normal fans in
$N_{{\mathbb R}}$
. Then, the followings are equivalent.
-
(a)
$P_1$
and
$P_2$
are combinatorially equivalent. -
(b)
$\Sigma _{P_1}$
and
$\Sigma _{P_2}$
are combinatorially equivalent. -
(c)
$\mathrm {PC}(\Sigma _{P_1})$
and
$\mathrm {PC}(\Sigma _{P_2})$
are isomorphic.
For simplicity, we use
$X_P$
to denote the projective toric variety associated with the normal fan
$\Sigma _P$
of a Delzant polytope
$P\subset M_{\mathbb R}$
.
3 Bott manifolds
Historically, Bott and Samelson introduced a family of complex manifolds obtained as the total spaces of iterated
$\mathbb P^1$
-bundles over
$\mathbb P^1$
[Reference Bott and Samelson5]. The notion of Bott tower was first defined by Grossberg and Karshon in their study of complex geometric structure of Bott–Samelson varieties [Reference Grossberg and Karshon23]. Each Bott tower is a (smooth) projective toric variety whose corresponding moment polytope
$P\subset M_{\mathbb R}$
is combinatorially equivalent to a cube. In this section, we recall the toric description of Bott manifolds and see their basic properties. For further details, we refer the reader to [Reference Buchstaber and Panov7, Section 7.8] and [Reference Choi, Masuda and Suh12].
3.1 Definition and basic properties
For a given
$n\in \mathbb N$
with
$n\geqslant 2$
, we inductively construct complex manifolds
$X_k$
for
$k=1,2, \ldots , n$
as follows. Let
$X_0$
be a point
$\mathrm {Spec\:} {\mathbb C}$
, and let
$X_1$
be the projective line
$\mathbb P^1$
. For
$k\geqslant 2$
, we assume that
$X_{k-1}$
is already determined and choose a holomorphic line bundle
$N_k$
on
$X_{k-1}$
. Then, we define
$X_k$
as the total space of the
$\mathbb P^1$
-bundle over
$X_{k-1}$
and denote it by
$$ \begin{align} X_k:=\mathbb P_{X_{k-1}}(\mathcal O_{X_{k-1}}\oplus N_k)\stackrel{\pi_k}{\longrightarrow} X_{k-1}. \end{align} $$
Definition 3.1. A Bott tower of height n is a sequence of
$\mathbb P^1$
-bundles
$$ \begin{align} X_n \stackrel{\pi_n}{\longrightarrow} X_{n-1} \stackrel{\pi_{n-1}}{\longrightarrow} \cdots \stackrel{\pi_2}{\longrightarrow} X_1 \stackrel{\pi_1}{\longrightarrow} \mathrm{Spec\:} {\mathbb C} \end{align} $$
over complex manifolds, where
$X_1=\mathbb P^1$
and
$X_k$
for
$2\leqslant k \leqslant n$
is the
$ \mathbb P^1$
-bundle defined in (3.1).
The last stage
$X_n$
in (3.2) is referred to as a stage n Bott manifold although it is also often called as the same name Bott tower.
A Bott tower
$X_n$
is topologically trivial if each projection
$\pi _k:X_k\to X_{k-1}$
gives a trivial
$\mathbb P^1$
-bundle. In particular, such
$X_n$
is diffeomorphic to the product of real two dimensional spheres
$(S^2)^n$
. The (topological) twist is the number of topologically nontrivial
$\mathbb P^1$
-bundles in the Bott tower (3.2) that is well-defined due to the work of Choi and Suh [Reference Choi and Suh13]. In fact, we may have several Bott tower structures for a given Bott manifold (cf. [Reference Fujita and Yotsutani18, Section 3.2]). Hence it is not obvious whether the number of nontrivial
$\mathbb P^1$
-fibration
$\pi _k: X_k\to X_{k-1}$
in the sequence is well-defined or not. The well-definedness of the twist of a Bott manifold was proved in [Reference Choi and Suh13, Corollary 3.3].
Let
$({\mathbb C}^2_*)^n$
be n copies of 
. Then, one can construct a stage n Bott manifold
$X_n$
by taking a quotient of
$({\mathbb C}^2_*)^n$
by an algebraic torus
$T_{\mathbb C}$
. Alternatively
$X_n$
is obtained as the symplectic quotient of
$(S^3)^n$
with respect to the free
$(S^1)^n$
-action as we shall see below. Remark that the former gives a complex geometric viewpoint of Bott manifolds, while the latter gives a symplectic geometric viewpoint of them.
Let A be a lower triangular unipotent matrix of the form
$$ \begin{align} A=\begin{pmatrix} 1 & 0 & \cdots & 0 & 0 \\ a_2^1 & 1 & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{n-1}^1 & a_{n-1}^2 & \cdots & 1 & 0 \\ a_n^1 & a_n^2 & \cdots & a_n^{n-1} &1 \end{pmatrix} \end{align} $$
with
$a_j^i\in \mathbb Z$
. The induced action
$\lambda $
of
$\boldsymbol t=(t_1,\ldots , t_n)\in T_{\mathbb C}$
on
$$\begin{align*}(\boldsymbol z; \boldsymbol w)=((z_1, \ldots ,z_n);(w_1, \ldots ,w_n))\in ({\mathbb C}^2_*)^n \end{align*}$$
is given by
For the unit sphere
$S^3$
in
${\mathbb C}^2_*$
, there is the induced action of
$({\mathbb R}_{>0})^n$
on
$({\mathbb C}^2_*)^n$
by (3.4), which is transverse to
$(S^3)^n$
. Hence the orbits of this action are bijectively correspond to orbits of the induced free
$(S^1)^n$
-action on
$(S^3)^n$
. Consequently, the geometric quotient
$({\mathbb C}^2_*)^n/T_{\mathbb C}$
gives a complex n-dimensional compact manifold which is denoted by
$X_n(A)$
in some literature. In this way, an n-dimensional Bott manifold can be constructed as a quotient of
$({\mathbb C}^2_*)^n$
by an algebraic torus. In particular, we see that any Bott manifold is a smooth projective toric variety.
3.2 A Bott manifold and its associated fan
For a given stage n Bott manifold X, let
$\Sigma $
be its associated polytopal fan. Recall that an n-cross polytope is a smooth polytope which is dual to an n-cube. According to [Reference Masuda and Panov28, Corollary 3.5],
$\Sigma $
is combinatorially equivalent to the fan consisting of the cones over the faces of an n-cross polytope. This means
Furthermore, for the Bott tower with a matrix A given in (3.3), we may assume that
$\boldsymbol e_1, \ldots , \boldsymbol e_n$
are the standard basis of
${\mathbb R}^n$
and each
$\boldsymbol u_i$
is written as
$$ \begin{align} \boldsymbol u_i=-\sum_{j=1}^na^i_j\boldsymbol e_j \end{align} $$
for
$i=1, \ldots , n$
. For later use, we denote 
as 
in (3.5). In particular, if a stage n Bott manifold X has twist zero, then we have
$\boldsymbol v_{2i-1}+\boldsymbol v_{2i}= 0$
for all
$1\leqslant i \leqslant n$
. As we have seen in (3.6), each column vector in the matrix (3.3) corresponds to the primitive generator
$\boldsymbol v_{2i-1}$
(i.e.,
$\boldsymbol u_i$
) for
$1\leqslant i \leqslant n$
. Summing up these arguments, we have the following proposition.
Proposition 3.2. Let X be a stage n Bott manifold having twist k with the associated fan
$\Sigma \subset N_{\mathbb R}$
. Then, the primitive collection of
$\Sigma $
has the form of
for
$1\leqslant i \leqslant 2n$
. In particular, there exist integers
satisfying
$\boldsymbol v_{i_j}+\boldsymbol v_{i_j+1}=0$
.
4 Proof of Theorem 1.1
In this section, we prove the main result. Our proof consists of the following steps:
Step 1. For a stage n Bott manifold X, we fix a Kähler class
$\Omega $
of X. Denoting by P the corresponding moment polytope of
$(X,\Omega )$
, we see that P is written in the form of (4.3). Then, the Futaki invariant of our targeted Bott manifold
$(X,\Omega )$
is given by (4.10).
We prove that if the Futaki invariant of the vector field generated by
$v=-x_n\partial /\partial x_n$
is identically zero, where
$x_n$
corresponds to the “height” of P, then the volume of the slice
is constant (Proposition 4.2).
Step 2. We prove that if the Futaki invariant of the vector field generated by
$v=-x_n\partial /\partial x_n$
is identically zero, where
$x_n$
corresponds to the “height” of P, then the volume of the slice
is constant (Proposition 4.2).
Step 3. In a more general setting, looking at the projection
$X\to \mathbb P^1$
for a smooth projective variety and considering the lift of
${\mathbb C}^{\times }$
-action on
$\mathbb P^1$
to X, we consider the vector field v generated by this
${\mathbb C}^{\times }$
-action. We can show that if the Futaki invariant of v vanishes on the entire space of Kähler cone, then X splits into
$\mathbb P^1$
and another smooth projective variety
$X'$
. In the toric setting, using the Brunn–Minkowski inequality for convex polytopes, we prove it in Corollary 4.6.
Step 4. For the splitting
$X_P=X_Q\times \mathbb P^1$
in Step
$3$
, we see that the Futaki invariant of
$X_P$
is proportional to the Futaki invariant of the base manifold
$X_Q$
in Lemma 4.7. By an inductive argument in the dimension of a Bott manifold, we verify the assertion.
Following the outline described in the above, we shall give a proof as follows. Let X be the stage n Bott manifold associated with a smooth polytopal fan
$\Sigma $
. Then,
$\Sigma (1)$
consists of
$2n$
rays such as
where
$\boldsymbol v_{i}$
is the primitive generator of each ray
$\rho _i$
. Since a Bott tower X (of height n) has the trivial
$\mathbb P^1$
-bundle at the first stage, i.e.,
$X_1=\mathbb P^1$
, we may assume
$$ \begin{align} \boldsymbol v_{2n-1} + \boldsymbol v_{2n} =\boldsymbol 0, \qquad \text{with} \quad \boldsymbol v_{2n}=\boldsymbol e_n. \end{align} $$
Let
$D_i$
be the
$T_{\mathbb C}$
-invariant Cartier divisor on X corresponding to
$\rho _i\in \Sigma (1)$
. We consider a Kähler class
$\Omega \in H^{1,1}(X,{\mathbb R})$
given by
$$\begin{align*}\Omega=\sum_{i=1}^{2n}a_i[D_i], \quad a_i\in {\mathbb R}, \end{align*}$$
where
$[D_i]$
denotes the Poincaré dual of
$D_i$
. Then, the corresponding moment polytope of
$(X,\Omega )$
is (a part of) the pillar given by
with
$\boldsymbol x=(x_1, \ldots ,x_n)$
.
4.1 Volume preserving constraints for each slice of P
Through this subsection, we fix
$a_i$
for each
$1 \leqslant i \leqslant 2n-2$
in (4.2). Let
$P_{(-\infty , +\infty )}$
be the unbounded pillar defined by
We regard
$a_{2n-1}$
and
$a_{2n}$
as real parameters s and
$-t$
, respectively.
Then, the polytope P can be written as
$$ \begin{align} P_{[t, \, s]} = P_{(-\infty, \, +\infty)} \cap \{ t\leqslant x_{n} \leqslant s\}. \end{align} $$
Let us denote the slice of
$P_{(-\infty ,\, +\infty )}$
at
$x_n=x$
by
$Q_x$
:
$$ \begin{align} Q_{x} = P_{(-\infty,\, +\infty)}\cap \{x_{n}=x\}. \end{align} $$
Taking a suitable parallel translation of
$P_{(-\infty , +\infty )}$
along the
$x_n$
-axis, we may assume
$t=0$
in (4.3). Henceforth we shall take
$t=0$
unless otherwise stated.
Let 
be the defining affine function of the facet
$F_i$
of P. Let
$dv=dx_1\wedge \cdots \wedge dx_n$
be the standard volume form of
$M_{\mathbb R}$
. On each facet 
, we define the
$(n-1)$
-dimensional Lebesgue measure
$d\sigma _i$
by
$$ \begin{align} dv=\pm d\sigma_i \wedge d\ell_i. \end{align} $$
Analogously, we consider the relative volume
$\mathrm {Vol} (\partial Q_x)=\int _{\partial Q_x}d\sigma _{Q_x}$
, where
$d\sigma _{Q_x}$
is the
$(n-2)$
-dimensional Lebesgue measure defined as follows: let
$d\sigma _i$
be the volume form of the facet
$F_i$
defined in (4.5). Then, the corresponding
$(n-2)$
-dimensional Lebesgue measure
$d\sigma _{Q_x,i}$
on
$\partial Q_x\cap F_i$
is defined to be
$$ \begin{align} d\sigma_{Q_x,i} \wedge dx_n=\pm d\sigma_i. \end{align} $$
For a fixed
$s>0$
, let us consider a convex polytope given by
which has a pair of parallel facets
$Q_s$
(the top facet) and
$Q_0$
(the bottom facet). Here and hereafter, we shall abbreviate the polytope
$P_{[0,s]}$
by P. The following lemma is an immediate consequence of the definition of
$d\sigma _{Q_x,i}$
in (4.6).
Lemma 4.1. Let s be a fixed positive constant. For any
$0\leqslant u \leqslant s$
, we consider the slice
$Q_u$
of P given in (4.4), so that each
$Q_u$
is parallel to both
$Q_0$
and
$Q_s$
. Regarding
$Q_u$
as a full-dimensional polytope in
${\mathbb R}^{n-1}$
, we denote the standard volume form of
$Q_u$
by
$dv_{Q_u}$
. Let us define the functions
$f(u)$
and
$g(u)$
by
$$\begin{align*}f(u):=\mathrm{Vol}(Q_u)=\int_{Q_u}dv_{Q_u} \qquad \text{and} \qquad g(u):=\mathrm{Vol}(\partial Q_u)=\int_{\partial Q_u}d\sigma_{Q_u} \end{align*}$$
respectively. Then, the volume of
$\partial P$
, and the barycenter of
$\partial P$
with respect to the
$x_n$
-axis are given as follows:
$$ \begin{align} \begin{aligned} \int_{\partial P} d\sigma&=\int_0^s g(u)\, du+f(0)+f(s), &\text{and} \\ \int_{\partial P} x_n \, d\sigma&=\int_0^s ug(u)\, du+sf(s). \end{aligned} \end{align} $$
Proof. Firstly, we remark that the polytope
$P=P_{[0,s]}$
is combinatorially equivalent to the product of
$Q_{0}$
and
$[0,s]$
. Thus, the boundary
$\partial P$
of P is also combinatorially equivalent to
This implies that
$$ \begin{align*} \mathrm{Vol}(\partial P)&=\int_{\partial P}d\sigma =\int_0^s\left(\int_{\partial Q_u}1\,d\sigma_{Q_u}\right)du+\left. \int_{Q_u}1\,dv_{Q_u} \right|{}_{u=0}+\left. \int_{Q_u}1\,dv_{Q_u} \right|{}_{u=s}\\ &=\int_0^s\mathrm{Vol}(\partial Q_u)\,du+\mathrm{Vol}(Q_0)+\mathrm{Vol}(Q_s)\\ &=\int_0^sg(u)\,du+f(0)+f(s). \end{align*} $$
This proves the first equality of (4.8).
For the second equality of (4.8), we compute
$$ \begin{align*} \int_{\partial P}x_n\, d\sigma&=\int_0^su\left(\int_{\partial Q_u}1\,d\sigma_{Q_u} \right)du +u\left. \int_{Q_u}1\,dv_{Q_u} \right|{}_{u=0}+u\left. \int_{Q_u}1\,dv_{Q_u} \right|{}_{u=s}\\ &=\int_0^sug(u)\,du+sf(s). \end{align*} $$
Thus, the assertions are verified.
Using the terminology defined in (4.7), we recall that the Futaki invariant of the vector field generated by
$v=-x_{n}\partial /\partial x_{n}$
on the toric Kähler manifold
$\big (X_{P_{[0,s]}}, \Omega _{P_{[0,s]}} \big )$
is equal to
$$ \begin{align} F_{X_{P_{[0, \, s]}}}(v) =\mathrm{Vol} (P_{[0, \, s]})\int_{\partial P_{[0, \, s]}} x_{n}\, d\sigma -\mathrm{Vol}(\partial P_{[0, \, s]})\int_{P_{[0, \, s]}} x_{n}\, dv. \end{align} $$
See, for example [Reference Donaldson16].
Proposition 4.2. For a fixed
$s>0$
with
$\dim Q_0=\dim Q_s=n-1$
, we assume that
$F_{X_{P_{[0, \, x]}}}(v)=0$
for any
$x\in [0, \, s]$
, as long as
$Q_x$
has dimension
$n-1$
. Then, the volume
$\mathrm {Vol}(Q_{x})$
is constant for any
$x\in [0,s]$
.
Proof. We use the same notation as in Lemma 4.1. Using the equalities in (4.8), we set
$$ \begin{align} \begin{aligned} a(x) &:= \mathrm{Vol}(P_{[0, \, x]})= \int^x_0 f(u)du, \\ b(x) &:= \mathrm{Vol}(\partial P_{[0, \, x]}) = \int^x_0 g(u)du + f(0) + f(x), \\ c(x) &:= \int_{P_{[0, \, x]}} x_n dv = \int^x_0 u f(u) du, \quad \text{and} \\ d(x) &:= \int_{\partial P_{[0, \, x]}} x_n d\sigma = \int^x_0 u g(u) du + xf(x). \end{aligned} \end{align} $$
Then, one can see that
$a(x), b(x), c(x)$
and
$d(x)$
are all polynomial functions. In particular, we have
$$\begin{align*}F_{X_{P_{[0, \, x]}}}(v) = a(x) d(x) - b(x)c(x). \end{align*}$$
Lemma 4.3. Let
$a^{(m)}(x)$
(resp.
$b^{(m)}(x), c^{(m)}(x), d^{(m)}(x)$
) denote the mth derivative of the function
$a(x)$
(resp.
$b(x), c(x), d(x)$
). Then, we have the following:
$$\begin{align*}\begin{array}{lclcl} a(0)=0, &\quad & a^{(m)}(0)= f^{(m-1)}(0), &\quad & \text{for}\quad m \geqslant 1,\\ \\ b(0)=2f(0), &\quad & b^{(m)}(0)= g^{(m-1)}(0)+ f^{(m)}(0), &\quad & \text{for}\quad m\geqslant 1,\\ \\ c(0)=c^{(1)}(0)=0, &\quad & c^{(m)}(0)= (m-1)f^{(m-2)}(0), &\quad & \text{for}\quad m\geqslant 2, \\ \\ d(0)=0, &\quad & d^{(1)}(0)=f(0)=\mathrm{Vol}(Q_0), \\ \\ &\quad & d^{(m)}(0)= (m-1)g^{(m-2)}(0)+ mf^{(m-1)}(0), &\quad & \text{for}\quad m\geqslant 2. \end{array} \end{align*}$$
Proof of Lemma 4.3
The values of
$a(0), b(0), c(0)$
and
$d(0)$
are obvious from (4.11).
All other equalities follows from the direct computations using the Leibniz rule. Firstly, we find that
$a^{(m)}(x)=f^{(m-1)}(x)$
for a positive integer m by calculating the derivatives of
$a(x)$
. This yields that
$$ \begin{align*} a^{(m)}(0)= f^{(m-1)}(0). \end{align*} $$
Next we compute
$b^{(m)}(0)$
. The straightforward computation shows that
$$ \begin{align} b^{(m)}(x)=g^{(m-1)}(x)+f^{(m)}(x), \end{align} $$
for
$m\geqslant 1$
. Plugging
$x=0$
into (4.12), for
$m\geqslant 1$
, we conclude that
$$ \begin{align*} b^{(m)}(0)=g^{(m-1)}(0)+f^{(m)}(0). \end{align*} $$
Let us calculate the derivatives of
$c(x)$
. By Leibniz rule, we find that
$$ \begin{align*} c^{(m)}(x)=(xf(x))^{(m-1)}&=\sum_{i=0}^{m-1}{m-1 \choose i} x^{(i)}f^{(m-1-i)}(x) \\ &=xf^{(m-1)}(x)+(m-1)f^{(m-2)}(x). \end{align*} $$
Thus, we have
$$ \begin{align*} c^{(1)}(0)=0 \qquad \text{and} \qquad c^{(m)}(0)=(m-1)f^{(m-2)}(0) \quad \text{for} \quad m\geqslant 2. \end{align*} $$
Finally, we compute the derivatives of
$d(x)$
. The direct computation shows that
$$ \begin{align*} d^{(m)}(x)&=(xg(x))^{(m-1)}+(xf(x))^{(m)}\\ &=\sum_{i=0}^{m-1}{m-1\choose i}x^{(i)}g^{(m-1-i)}(x)+\sum_{j=0}^{m}{m\choose j}x^{(j)}f^{(m-j)}(x)\\ &=xg^{(m-1)}(x)+(m-1)g^{(m-2)}(x)+xf^{(m)}(x)+mf^{(m-1)}(x). \end{align*} $$
Consequently, we have
$$ \begin{align*} d^{(1)}(0)&=f(0)=\mathrm{Vol}(Q_0), \quad \text{and} \qquad \\ d^{(m)}(0)&=(m-1)g^{(m-2)}(0)+mf^{(m-1)}(0), \end{align*} $$
for
$m\geqslant 2$
.
By Lemma 4.3, we find
$$ \begin{align} \begin{aligned} \frac{d^3}{d x^3}F_{X_{P_{[0, \, x]}}}(v)\bigg|_{x=0} &= a^{(3)}(0)d(0)+3a^{(2)}(0)d^{(1)}(0) +3a^{(1)}(0)d^{(2)}(0)+a(0)d^{(3)}(0) \\ &\quad -b^{(3)}(0)c(0)-3b^{(2)}(0)c^{(1)}(0) -3b^{(1)}(0)c^{(2)}(0)-b(0)c^{(3)}(0) \\ &= 2 f'(0)f(0). \end{aligned} \end{align} $$
Since
$f(0)>0$
and
$F_{X_{P_{[0, \, x]}}}(v)$
is identically
$0$
by our assumption, we have
$f'(0)=0$
. By taking an appropriate parallel translation of
$P_{(-\infty ,\, +\infty )}$
along the
$x_n$
-axis again if necessary, we conclude that
$f'(x)$
is identically
$0$
for arbitrary
$x\in [0,\, s]$
. Thus, the volume
$\mathrm {Vol}(Q_{x})$
is constant.
4.2 The Brunn–Minkowski inequality for convex polytopes
In this subsection, we prove that any two slices
$Q_x$
and
$Q_{x'}$
for
$0\leqslant x,x'\leqslant s$
are congruent. For this, we apply the Brunn–Minkowski inequality which estimates the (Euclidean) volume
$\mathrm {Vol}(C+D)$
of the Minkowski sum
for two convex bodies C and D. We refer the reader to [Reference Gruber24] for more detail.
A set C in
${\mathbb R}^n$
is said to be convex if it satisfies
$$\begin{align*}(1-\lambda)\boldsymbol x+\lambda \boldsymbol y \in C \end{align*}$$
for any
$\boldsymbol x, \boldsymbol y\in C$
and
$0\leqslant \lambda \leqslant 1$
. We call a compact convex set a convex body. A convex body C is proper if its interior is non-empty, and improper otherwise. Let
$\mathcal C({\mathbb R}^n)$
be the space of all convex bodies in
${\mathbb R}^n$
. We say that two convex bodies
$C,D \in \mathcal {C}({\mathbb R}^n)$
are homothetic to each other, if there exist
$r>0$
and
$\boldsymbol a\in {\mathbb R}^n$
such that
Considering the Minkowski sum
$$\begin{align*}(1-\lambda)C+\lambda D \qquad \text{in~~~}~~~ {\mathbb R}^n \end{align*}$$
for
$C, D\in \mathcal {C}({\mathbb R}^n)$
and
$0\leqslant \lambda \leqslant 1$
, we introduce the function
$\Psi $
defined by
In order to prove Proposition 4.5 below, we need the following result. See [Reference Gruber24, Theorem 8.3] for the proof.
Theorem 4.4 (The Brunn–Minkowski inequality).
Let
$\Psi $
be the function defined in (4.14). If
$\Psi $
is not strictly concave, then either one of the following conditions is satisfied:
-
(i) C or D is a point.
-
(ii) Both C and D are improper, and lie in parallel hyperplanes.
-
(iii) Both C and
$D $
are proper, and they are homothetic to each other.
We state the key result for proving Theorem 1.1.
Proposition 4.5. Let us take any two distinct
$u_1$
and
$u_2$
in
$[0, s]$
with
$u_1<u_2$
. For any x between
$u_1$
and
$u_2$
, let
$Q_x$
be the slice of
$P_{(-\infty ,\,+\infty )}$
defined in (4.4). If
$\mathrm {Vol}(Q_x)$
is constant for all
$x\in [u_1, u_2]$
, then
$Q_{u_1}$
and
$Q_{u_2}$
are congruent.
Proof. Denoting
$C:=Q_{u_1}$
and
$D:=Q_{u_2}$
, we remark that the Minkowski sum
$(1-\lambda )C+\lambda D$
is the slice
$Q_{(1-\lambda )u_1+\lambda u_2}$
. Let us define
$\widetilde {Q}_\lambda :=(1-\lambda )C+\lambda D$
. Then,
$$\begin{align*}\mathrm{Vol}(\widetilde{Q}_\lambda)=\int_{\widetilde{Q}_\lambda} dx_1\wedge \cdots \wedge dx_{n-1} \end{align*}$$
is a constant function in
$\lambda $
by assumption. Hence, the function
$$\begin{align*}\Psi:[0,1] \longrightarrow {\mathbb R}, \qquad \lambda \longmapsto \mathrm{Vol}(\widetilde{Q}_\lambda)^{\frac{1}{n-1}} \end{align*}$$
is not strictly concave. If we regard each
$\widetilde {Q}_\lambda $
as a convex polytope in the hyperplane 
, we note that
-
(i) neither C nor D is a point, and
-
(ii) both C and D are proper convex polytopes in
${\mathbb R}^{n-1}$
,
because
$\dim \widetilde {Q}_\lambda =n-1$
in
${\mathbb R}^{n-1}$
for any
$\lambda $
. By applying Theorem 4.4 into C and D in
${\mathbb R}^{n-1}$
, we conclude that C and D are homethetic, i.e., there exist some
$r>0$
and
$\boldsymbol a\in {\mathbb R}^{n-1}$
such that
$$\begin{align*}C=rD+\boldsymbol a \quad \Leftrightarrow \quad Q_{u_1}=rQ_{u_2}+\boldsymbol a. \end{align*}$$
From this, we find
$$ \begin{align} \mathrm{Vol}(Q_{u_1})=r^{n-1}\mathrm{Vol}(Q_{u_2}). \end{align} $$
Since
$\mathrm {Vol}(Q_{u_1})=\mathrm {Vol}(Q_{u_2})$
, we conclude
$r=1$
. Consequently,
$Q_{u_1}$
and
$Q_{u_2}$
are congruent.
Corollary 4.6. If the Futaki invariant
$F_{X_{P_{[0, \, s]}}}(v)$
in (4.10) vanishes for any
$ s>0$
, then our stage n Bott manifold X splits into
$X' \times \mathbb P^{1}$
, where
$X'$
is the
$(n-1)$
-dimensional smooth toric variety whose associated fan
$\Sigma '$
has the primitive collection
Moreover, the Futaki invariant on the base manifold
$X'$
for any Kähler class also vanishes, whenever the Futaki invariants on X for all Kähler classes vanish.
Proof. Let P be the moment polytope of a toric Kähler manifold
$(X_{P_{[0,s]}}, \Omega _{P_{[0,s]}})$
. By Proposition 4.5, the moment polytope P is a quadrangular prism with the bottom
By taking an appropriate parallel translation of P along the
$x_n$
-axis if necessary, we assume that
$a_{2n-1}=0$
and
$a_{2n}=a$
for some constant
$a>0$
. Since P is a quadrangular prism with the bottom facet Q, we have the splitting of P such that
$$ \begin{align} P=Q\times [0, a]. \end{align} $$
The first statement follows from the fact that
$\Sigma '$
is the normal fan of Q.
The second assertion is a consequence of Lemma 4.7. We remark that the splitting (4.16) implies that the associated toric manifold
$X_Q$
is the torus invariant toric divisor of
$X_P$
. Then, we have the following:
Lemma 4.7. For each
$i=1, \ldots , n$
, let
$F_{X_P}(x_i)$
be the Futaki invariant on
$(X_P, \Omega _P)$
given by
$$ \begin{align} F_{X_P}(x_i):=\mathrm{Vol}(P)\int_{\partial P}x_i\,d\sigma-\mathrm{Vol}(\partial P)\int_P x_i\,dv. \end{align} $$
Then,
$F_{X_P}(x_i)$
is proportional to the Futaki invariant
$F_{X_Q}(x_i)$
on the base manifold
$(X_Q,\Omega _Q)$
, for each
$i=1,\ldots , n-1$
.
Hence, the second assertion of Corollary 4.6 is verified.
Proof of Lemma 4.7
Let
$\boldsymbol x=(x_1,\ldots ,x_n)$
be the coordinate function of
$M_{\mathbb R}$
, and let
$\widetilde {\boldsymbol x}=(x_1, \ldots , x_{n-1})$
be its projection. We note that the splitting (4.16) yields that the boundary of P can be written as
We also remark that two measures on
$\partial Q$
, that is,
-
• the measure given by the restriction of
$d\sigma $
(on
$\partial P$
) to
$\partial Q$
, and -
• the induced measure on
$\partial Q$
when we regard Q as a full dimensional polytope in the hyperplane , where c is some constant,
, where c is some constant,coincide with each other. Thus, by (4.16) and (4.18), we have
$$\begin{align*}\mathrm{Vol}(P)=a\mathrm{Vol}(Q) \qquad \text{and} \qquad \mathrm{Vol}(\partial P)=a\mathrm{Vol}(\partial Q)+2\mathrm{Vol}(Q). \end{align*}$$
Furthermore, for
$i=1, \ldots , n-1$
, we find
$$ \begin{align*} \int_Px_i\,dv&=a\int_Q x_i\,dv_Q, \qquad \hspace{2.2cm} \int_Px_n \,dv=\frac{a^2}{2}\mathrm{Vol}(Q),\\ \int_{\partial P}x_i\,d\sigma&=a\int_{\partial Q}x_i\,d\sigma_Q+2\int_Qx_i\,dv_Q, \quad \text{and}\\ \int_{\partial P}x_n\, d\sigma&=\frac{a^2}{2}\mathrm{Vol}(\partial Q)+a\mathrm{Vol}(Q) \end{align*} $$
respectively. Plugging these values into (4.17), we see that
$$ \begin{align*} F_P(x_i)&=a^2\left(\mathrm{Vol}(Q)\int_Q x_i\,d\sigma_Q -\mathrm{Vol}(\partial Q)\int_Qx_i\, dv_Q\right)\\ &=a^2F_Q(x_i) \end{align*} $$
for
$i=1,\ldots , n-1$
.
Remark 4.8. Moreover, we can see a more general result of Corollary 4.6 as follows. Let
$X\to Y$
be a fibration for smooth projective varieties X and Y. Let
$L_X$
and
$L_Y$
be very ample line bundles on X and Y respectively. If a vector field v on Y lifts to
$\tilde v$
on X, then the Futaki invariant
$F_{L_X+jL_Y}(\tilde v)$
of
$\tilde v$
with respect to the Kähler class
$L_X+jL_Y$
for j sufficiently large has the leading order term given by the Futaki invariant
$F_{L_Y}(v)$
of v with respect to
$L_Y$
. This yields that if the Futaki invariant of
$\tilde v$
vanishes identically on the Kähler cone of X, then the Futaki invariant of
$ v$
also vanishes identically on the Kähler cone of Y.
4.3 The induction on the dimension of an SCD Bott manifold
Now we complete the proof of Theorem 1.1 by an inductive argument in n. Assume that X is an n-dimensional Bott manifold with vanishing Futaki invariants for all Kähler classes. By Corollary 4.6, we see that
$\boldsymbol u_1, \ldots , \boldsymbol u_{n-1}$
are perpendicular to
$\boldsymbol e_n$
, for primitive generators in (3.5). Hence, we have
with
$\boldsymbol u_i=-\sum _{j=1}^na_j^i \boldsymbol e_j$
. Then, (4.19) is equivalent to
$$\begin{align*}a_n^1=a_n^2=\cdots =a_n^{n-1}=0 \end{align*}$$
which implies that the matrix A given in (3.3) is written as
$$ \begin{align*} A=\begin{pmatrix} 1 & 0 & \cdots & 0 & 0 \\ a_2^1 & 1 & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{n-1}^1 & a_{n-1}^2 & \cdots & 1 & 0 \\ 0 & 0 & \cdots & 0 &1 \end{pmatrix}. \end{align*} $$
Let us denote the
$(n-1)$
-square submatrix
$$ \begin{align*} \begin{pmatrix} 1 & 0 & \cdots & 0& 0 \\ a_2^1 & 1 & \cdots & 0& 0 \\ \vdots & \vdots & \ddots & \vdots &\vdots \\ a_{n-2}^1 & a_{n-2}^2 & \cdots & 1 & 0 \\ a_{n-1}^1 & a_{n-1}^2 & \cdots & a_{n-1}^{n-2} & 1 \end{pmatrix} \end{align*} $$
by
$\widetilde {A}$
which corresponds to the stage
$n-1$
Bott manifold
$X'$
in Corollary 4.6. Inductively, we apply Corollary 4.6 into
$\widetilde A$
which yields
$$\begin{align*}a_{n-1}^1=a_{n-1}^2=\cdots =a_{n-1}^{n-2}=0. \end{align*}$$
As a result, we conclude that A is the unit matrix. Thus, Theorem 1.1 has been proved.
A Appendix
In this section, we shall see that Theorems 1.1 and 1.3 are equivalent as long as we work over a complex number field
${\mathbb C}$
.
Firstly, we recall the argument in [Reference Fujita and Yotsutani18] briefly. Let X be an n-dimensional smooth projective variety, and let L be an ample
${\mathbb Q}$
-line bundle on X. For an irreducible smooth subvariety Z in X with
$\mathrm {Codim}_X(Z)=c$
, we take the blow-up
$\sigma :\hat {X}\rightarrow X$
along Z. Denoting the ideal sheaf of Z by
$I_Z$
, we consider the Cartier divisor F in
$\hat {X}$
defined by
$\mathcal {O}_{\hat {X}}(-F)=I_Z\cdot \mathcal {O}_{\hat {X}}$
. We denote the Seshadri constant of Z with respect to L by
Setting
$$\begin{align*}\mu_L=\frac{\big(L^{\cdot n-1}\cdot(-K_X) \big)}{(L^{\cdot n})}, \end{align*}$$
we define the invariant
$\xi _L(Z)$
by
$$ \begin{align} \xi_L(Z)=n\int_0^\varepsilon\big((\sigma^*L-xF)^{\cdot n-1}\cdot(\sigma^*(K_X+\mu_L L)+(c-\mu_Lx)F) \big)dx. \end{align} $$
The invariant
$\xi _L(Z)$
was originally introduced in [Reference Fujita17] for the case that
$L=-K_X$
, i.e.,
$(X,-K_X)$
is a Fano variety. Later, this invariant was extended in [Reference Dervan and Legendre15, Definition 2.8] for the case of any polarized variety
$(X,L)$
, which is called the
$\beta $
-invariant in the literature of K-stability.
Now, we shall consider the relation between
$\xi _L(Z)$
and the Futaki invariant. For Z, let us consider the test configuration
$(\mathcal {X},\mathcal {L}_t)$
consisting of the blow up
$\eta :\mathcal {X}\to X\times \mathbb {P}^1$
along
$Z\times \{0\}$
and
$\mathcal {L}_t:=\eta ^* p^*L-t E$
for
$t\in (0, \varepsilon _L(Z))\cap {\mathbb Q}$
. The space
$\mathcal {X}$
is called the degeneration of X to the normal cone of Z in [Reference Ross and Thomas31]. Here E is the exceptional divisor of the blow-up
$\eta $
. See [Reference Ross and Thomas31] for the details of the notions that appeared here. As seen in [Reference Fujita and Yotsutani18, Remark 2.6], the invariant
$\xi _L(Z)$
coincides with the limit of the Donaldson–Futaki invariant
$\mathrm {DF}(\mathcal {X}, \mathcal {L}_t)$
as t goes to
$\varepsilon _L(Z)$
.
For the relation among the
$\beta $
-invariant, the Donaldson–Futaki invariant and
$\xi _L(Z)$
, they are equal to each other up to the multiplicative positive constant. Precisely, we have
$$ \begin{align} \xi_L(Z)=\lim_{t\to \varepsilon_L(Z)} {\mathrm{DF}}(\mathcal X, \mathcal L_t)=\frac{1}{2(n-1)!}\beta_L(Z), \end{align} $$
where the first equality follows from [Reference Fujita and Yotsutani18, Remark 2.6] and the second equality follows from [Reference Dervan and Legendre15, Proposition 3.9].
Remark A.1. In order to explain about geometric motivation of the invariant
$\xi _L(D)$
, we assume that an n-dimensional polarized variety
$(X,L)$
is slope semistable along a smooth prime divisor D. As it was observed in the proof of [Reference Fujita and Yotsutani18, Proposition 2.2], slope semistability implies that
$$ \begin{align*} \frac{1}{2(n-1)!}&\big( (L^{\cdot n-1})\cdot(-K_X) \big)\int_0^{\varepsilon}\frac{1}{n!}\big((L-x D)^{\cdot n}\big)dx \\ \geqslant&\frac{1}{n!}\left( \int_0^\varepsilon\frac{1}{2(n-1)!}\big((L-x D)^{\cdot n-1}\cdot(-K_X)\big)dx+\frac{1}{2} \left[ \frac{1}{n!}\big((L-x D)^{\cdot n}\big) \right]_0^\varepsilon \right) \\ \Leftrightarrow \quad \big((L^{\cdot n-1})&\cdot(-K_X)\big)\int_0^\varepsilon \big((L-x D)^{\cdot n}\big)dx -(L^{\cdot n})\int_0^\varepsilon \big((L-x D)^{\cdot n-1}\cdot(-K_X)\big)dx \\ \hspace{3cm} & \geqslant \frac{(L^{\cdot n})}{n}\big((L-\varepsilon D)^{\cdot n} -(L^{\cdot n})\big). \end{align*} $$
Plugging the equality
$$ \begin{align*} \int_0^\varepsilon \big((L-x D)^{\cdot n-1}\cdot D\big)dx &= \left[ -\frac{1}{n}\big((L-x D)^{\cdot n} \big) \right]_0^\varepsilon\\ &= \frac{1}{n}\big((L^{\cdot n})-(L-\varepsilon D)^{\cdot n} \big) \end{align*} $$
into the above last inequality, we have
$$ \begin{align} &\int_0^\varepsilon\Bigg( \big(L^{\cdot n-1}\cdot(-K_X)\big)\cdot(L-xD)^{\cdot n} -(L^{\cdot n})\cdot\big( (L-xD)^{\cdot n-1}\cdot (-K_X)\big) \notag \\ & \hspace{7.6cm} +(L^{\cdot n})\cdot\big( (L-xD)^{\cdot n-1}\cdot D\big) \Bigg) dx \geqslant 0 \notag \\ \Leftrightarrow \quad & \int_0^\varepsilon \big((L-xD)^{\cdot n-1} \big)\cdot \big( \mu_L(L-xD)+K_X+D \big)dx \geqslant 0. \end{align} $$
We remark that our assumption implies that
$c=1$
and
$\sigma $
is the identity map in (A.1). Thus, we see that the left hand side of (A.3) equals
$\xi _L(D)/n$
.
From Remark A.1 and [Reference Ross and Thomas31], we can replace the slope semistability along D with the semipositivity of
$\xi _L(D)$
. Then, Fujita and the third author proved that if
$\xi _L(D) \geqslant 0$
for any ample
${\mathbb Q}$
-line bundle L and any smooth prime divisor D on an n-dimensional Bott manifold X, then
$X\cong (\mathbb P^1)^n$
.
For our purpose, we assume that
$(X,L)$
is an n-dimensional smooth polarized toric variety and D is a smooth toric prime divisor on X. Let P be the moment polytope of
$(X,L)$
, and
$\boldsymbol v_D \in N_{\mathbb R}$
the ray generator corresponding to D. Let
$$\begin{align*}\widetilde{\ell}_D(\boldsymbol x):= \langle \boldsymbol x, \boldsymbol v_D\rangle + \widetilde{a}_D \end{align*}$$
be the affine function induced by
$\boldsymbol v_D$
where the constant
$\widetilde {a}_D$
is determined so that
Let
$\boldsymbol v_D^\sharp $
be the holomorphic vector field induced by
$\boldsymbol v_D$
. Then, the Futaki invariant of
$\boldsymbol v_D^\sharp $
is equal to the Donaldson–Futaki invariant
$$\begin{align*}\mathrm{Vol}(P) \int_{\partial P} \widetilde{\ell}_D(\boldsymbol x) d\sigma -\mathrm{Vol}(\partial P)\int_{P} \widetilde{\ell}_D(\boldsymbol x) dv \end{align*}$$
of the product test configuration induced by
$\boldsymbol v_D^\sharp $
.
Lemma A.2. Let
$(X,L)$
be a smooth polarized toric variety. Let D be a smooth toric prime divisor on X. If the pseudo-effective threshold
is equal to the Seshadri constant
$\varepsilon _L(D)$
, then the Futaki invariant of
$\boldsymbol v_D^\sharp $
is equal to
$\xi _L(D)$
.
Proof. Let
$\ell _D(\boldsymbol x)$
be the defining affine function of the facet of P corresponding to D:
$$\begin{align*}\ell_D(\boldsymbol x):=\langle \boldsymbol x, \boldsymbol v_D \rangle + a_D. \end{align*}$$
Let
$\mathcal {P}$
be the
$(n+1)$
-dimensional moment polytope of (the compactification of) the test configuration
$(\mathcal {X},\mathcal {L}_t)$
as above. Then, the polytope
$\mathcal {P}$
coincides with
Remark that the level set
is the moment polytope of
$(X, L-tD)$
as long as
$t\in (0,\varepsilon _L(D))\cap {\mathbb Q}$
.
On the other hand, the moment polytope
$\mathcal P'$
of (the compactification of) the product test configuration induced by
$\boldsymbol v_D^\sharp $
is given by
The assumption that
$\varepsilon _L(D)=\tau _L(D)$
implies
$\ell _D=\widetilde {\ell }_D$
. In particular,
$\mathcal {P}_{\varepsilon _L(D)}$
coincides
$\mathcal {P}'$
. Hence, we have
$$\begin{align*}F_{X_P}(\boldsymbol v_D^\sharp)= \lim_{t\to \varepsilon_L(D)}\mathrm{DF}(\mathcal{X},\mathcal{L}_t)=\xi_L(D). \end{align*}$$
The proof is completed.
Firstly, we show that Theorem 1.1 implies Theorem 1.3. Assume that X is an n-dimensional slope semistable Bott manifold whose moment polytope is given by (4.2). Let
$D_{2n-1}$
and
$D_{2n}$
be the smooth toric divisors corresponding to
$\boldsymbol v_{2n-1}$
and
$\boldsymbol v_{2n}$
respectively. Then, the moment polytope of
$(X, L-tD_{2n})$
is equal to
Since we have
$$\begin{align*}\varepsilon_L(D_{2n})=\tau_L(D_{2n})=a_{2n-1}+a_{2n}, \end{align*}$$
Lemma A.2 implies that
$$\begin{align*}F_{X}(\boldsymbol v_{D_{2n}}^{\sharp}) = F_{X}(x_{n}) = \xi_L(D_{2n}) \geqslant 0, \end{align*}$$
where the last inequality follows from the assumption of slope semistability. Similarly, we have
$$ \begin{align*}F_{X}(\boldsymbol v_{D_{2n-1}}^{\sharp}) = F_{X_P}(-x_{n}) = \xi_L(D_{2n-1}) \geqslant 0, \end{align*} $$
because the moment polytope of
$(X, L-tD_{2n-1})$
is equal to
Therefore, we have
$$ \begin{align*}F_{X_P}(x_n)=0. \end{align*} $$
Now, Corollary 4.6 implies the splitting
$X=X'\times \mathbb P^1$
where
$X'$
is the Bott manifold of dimension
$n-1$
. Applying the above argument to
$X'$
inductively as in Section 4.3, we find that
$X\cong (\mathbb P^1)^n$
.
Finally, we will prove the converse: Theorem 1.3 implies Theorem 1.1. Let us take the divisor D as in [Reference Fujita and Yotsutani18, Lemma 3.1]. Then, [Reference Fujita and Yotsutani18, Lemma 3.3(2)] assures that
$$\begin{align*}\varepsilon_L(D)=\tau_L(D)=1. \end{align*}$$
Under the above condition, we have
$$ \begin{align} \xi_L(D) = \frac{1}{2(n-1)!}\beta_L(D) \end{align} $$
by (A.2). From [Reference Dervan and Legendre15, (5.7)], we have the formula of
$\beta _L(D)$
such that
where
$A_X(D)$
is the log discrepancy of D,
$b_P$
and
$b_{\partial P}$
are the barycenters of P and
$\partial P$
respectively. Then (A.4) and (A.5) imply that
$$\begin{align*}\xi_L(D)=\frac{1}{2(n-1)!} \beta_L(D)=\frac{1}{2(n-1)!}(A_X(D)-1)\mathrm{Vol}(P)\geqslant 0. \end{align*}$$
Hence, we can apply [Reference Fujita and Yotsutani18, Proposition 4.2] to
$(X,D)$
. Repeating the above argument inductively, we get the conclusion.
Acknowledgements
The authors thank the anonymous referees for their helpful comments that improved the quality of the manuscript. In particular, we are grateful to the referee for his/her fruitful suggestion in Remark 4.8. They also thank Professor Kento Fujita for allowing us to introduce his computation in Remark A.1.
Funding
This work was partially supported by the first author’s JSPS KAKENHI Grant Number JP17K05218, JP22K03281, the second author’s JP22K03325 and the third author’s JP18K13406, JP22K03316. The second author is also supported by research funds from Fukuoka University (Grant Number 225001-000).
Data availability statement
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
