# Generalized Kähler-Ricci flow on toric Fano varieties

Vestislav Apostolov  Vestislav Apostolov
Départment de matheématiques
Université du Québec à Montréal
Case postale 8888, succursale centre-ville Mongtréal (Québec) H3C 3P8
<EMAIL_ADDRESS>,  Jeffrey Streets  Jeffrey Streets
Rowland Hall
University of California
Irvine, CA 92617<EMAIL_ADDRESS>and  Yury Ustinovskiy  Yury Ustinovskiy
Courant Institute of Mathematical Sciences
New York University
251 Mercer Street
New York, NY, 10012-1185<EMAIL_ADDRESS>
###### Abstract.

We study the generalized Kähler-Ricci flow with initial data of symplectic
type, and show that this condition is preserved. In the case of a Fano
background with toric symmetry, we establish global existence of the
normalized flow. We derive an extension of Perelman’s entropy functional to
this setting, which yields convergence of nonsingular solutions at infinity.
Furthermore, we derive an extension of Mabuchi’s $K$-energy to this setting,
which yields weak convergence of the flow.

JS was supported by the NSF via DMS-1454854. VA was supported in part by an
NSERC Discovery Grant. The authors are grateful to Tamas Darvas, Xiaohua Zhu,
and an anonymous referee for their comments on the manuscript.

##  1\. Introduction

Generalized Kähler (GK) structures first appeared through investigations into
supersymmetric sigma models [29], and were rediscovered in a purely
mathematical context in the work of Gualtieri [35] and Hitchin [44]. They have
recently attracted interest in both the physics and mathematical communities
as natural generalizations of Kähler structures. We will focus here entirely
on the so called biHermitian description of generalized Kähler geometry (cf.
[5, 29]). Thus, a generalized Kähler manifold is a smooth manifold $M$ with a
triple $(g,I,J)$ consisting of two integrable almost-complex structures, $I$
and $J$, together with a Riemannian metric $g$ which is Hermitian with respect
to both, such that the Kähler forms $\omega_{I}$ and $\omega_{J}$ satisfy

(1.1) $\displaystyle{\mathrmsl d}^{c}_{I}\omega_{I}=H=-{\mathrmsl
d}^{c}_{J}\omega_{J},\qquad{\mathrmsl d}H=0,$

where the first equation defines $H$, and ${\mathrmsl
d}^{c}_{I}=\sqrt{-1}(\overline{\partial}_{I}-\partial_{I}),\,{\mathrmsl
d}^{c}_{J}=\sqrt{-1}(\overline{\partial}_{J}-\partial_{J})$.

The generalized Kähler-Ricci flow (GKRF) was introduced by the second named
author and Tian in [61] as a tool for constructing canonical metrics and
understanding existence and moduli problems in generalized Kähler geometry,
extending the deeply developed theory in Kähler geometry. Prior global
existence and convergence results for GKRF have appeared in for instance [6,
54, 56]. In this paper we consider normalized generalized Kähler Ricci flow
(NGKRF), expressed as a flow of ${\mathrmsl d}{\mathrmsl d}_{J}^{c}$-closed
positive definite $(1,1)$-forms $\omega_{J}$ on the complex manifold $(M,J)$,
together with a flow of $I$, as (cf. [61])

(1.2) $\displaystyle\begin{split}\frac{\partial}{\partial t}\omega_{J}=&\
-2\Big{(}\rho^{B}_{J}\Big{)}^{1,1}+2\lambda\omega_{J},\qquad\frac{\partial}{\partial
t}I={\mathcal{L}}_{\theta_{I}^{\sharp}-\theta_{J}^{\sharp}}I,\end{split}$

where $\lambda$ is a fixed real constant and
$\Big{(}\rho_{J}^{B}\Big{)}^{1,1}$ denotes the $(1,1)$-part with respect to
$J$ of the Bismut-Ricci form $\rho_{J}^{B}\in 2\pi c_{1}(M,J)$ of
$(\omega_{J},J)$. By [61], as far as a solution to (1.2) exists, the triple
$(g_{t},I_{t},J)$ defines a GK structure in the sense of (1.1). Notice that if
the initial GK structure $(g,I,J)$ was Kähler (i.e. ${\mathrmsl
d}^{c}_{J}\omega_{J}=-H=0$ and $I=J$ in (1.1)), the Bismut-Ricci form
$\rho^{B}$ is just the usual Kähler-Ricci form of the Kähler manifold $(g,J)$,
and (1.2) reduces to the normalized Kähler Ricci flow. In this setting Cao
showed [12] smooth global existence. The convergence at infinity is a very
subtle issue, and many results have appeared in recent years [14, 20, 51, 65,
66, 67, 68].

In this paper, we will study NGKRF in the case when $(M,J)$ is a smooth toric
Fano variety of complex dimension $m$, assuming that the initial GK structure
$(g,I,J)$ is compatible with a symplectic form $F\in 2\pi c_{1}(M,J)$, and is
invariant under the action of a maximal compact torus ${\mathbb{T}}$ in the
automorphism group $\mathrmsl{Aut}(M,J)$ of $(M,J)$. More precisely, suppose
that $F$ is a symplectic $2$-form on $M$ which tames $J$, i.e. whose
$(1,1)$-part $\omega_{J}:=\big{(}F\big{)}^{1,1}_{J}$ is positive definite. We
then define a Riemannian metric $g$, a $2$-form $b$ and an almost complex
structure $I$ by

(1.3) $\displaystyle g:=-(FJ)^{\rm sym},\qquad b:=-(FJ)^{\rm skew},\qquad
I:=-F^{-1}J^{*}F.$

One can show (cf. §2) that when $I$ is also integrable, $(g,I,J)$ is a GK
structure with $H=db$. Furthermore, in this case we say that $(g,I,J)$ is an
$F$-compatible GK structure, or a GK structure of symplectic type, referring
to the fact [37] that one of the corresponding generalized complex structures
on $TM\oplus T^{*}M$ is determined completely by the symplectic $2$-form $F$.

To better understand these structures, first recall that a key feature of GK
geometry, observed by Hitchin [45] (cf. [5, 52] for the $4$-dimensional case)
is that there are naturally associated holomorphic Poisson structures. More
precisely, the tensors

(1.4)
$\displaystyle\sigma=[I,J]g^{-1},\qquad\sigma_{I}=\sigma-\sqrt{-1}I\sigma,\qquad\sigma_{J}=\sigma-\sqrt{-1}J\sigma$

define respectively a real Poisson structure and holomorphic Poisson
structures with respect to $I$ and $J$. Conversely, given a compact Kähler
manifold $(M,J,{\overline{g}},\overline{\omega})$ and a non-trivial
holomorphic Poisson structure $\mu\in H^{0}(M,{}^{2}(T^{1,0}M))$, the are
deformation results [46, 32, 34, 38] producing non-Kähler GK structures
$(g_{t},I_{t},J)$ with holomorphic Poisson structure $\sigma_{J_{t}}=t\mu$ for
$|t|<\varepsilon$. This takes a particularly nice and explicit form on a
Kähler toric variety $(M,J,{\mathbb{T}})$. Recent work of Boulanger [11]
constructs invariant toric GK structures by deforming the complex structure
$J$ using $A\in{}^{2}{\mathfrak{t}}$, referred to here as deformations of type
$A$. More recently, Y. Wang [70, 71] constructed toric GK structures by
deforming $\overline{\omega}$ to a symplectic form $F$ using
$B\in{}^{2}{\mathfrak{t}}$, referred to here as deformations of type $B$. It
follows from the results in [71] (cf. Proposition 3.8 below) that, up to pull-
back by ${\mathbb{T}}$-equivariant diffeomorphism, any
${\mathbb{T}}$-invariant GK structure $(g,I,J)$ of symplectic type on $(M,J)$
with corresponding symplectic $2$-form $F$, is obtained from a
${\mathbb{T}}$-invariant Kähler structure $({\overline{g}},{\overline{J}})$ by
deformations of type $A$ and $B$. We furthermore observe in Proposition 3.9 a
geometric interpretation of $A$ and $B$-type deformations in terms of the
holomorphic Poisson structures.

Building upon this, we show in §2 that NGKRF interacts with the associated
Poisson structures in a predictable way, and thus is a natural tool for
investigating questions on the global moduli of generalized Kähler structures
on, as well as the underlying Poisson geometry of, Fano varieties. We
formulate certain precise conjectures coming from the formal picture of NGKRF
in §2. As a fundamental first step in this direction, we establish the sharp
global existence of the flow in the case of toric symmetry.

######  Theorem 1.1.

Let $(M,J,{\mathbb{T}})$ be a smooth toric Fano variety and $F\in 2\pi
c_{1}(M,J)$ a ${\mathbb{T}}$-invariant symplectic $2$-form which defines a
${\mathbb{T}}$-invariant generalized Kähler structure $(g,I,J)$ of symplectic
type on $M$. Then, the solution of normalized generalized Kähler Ricci flow
(1.2) with this initial data exists on $[0,\infty)$.

The key geometric observation behind Theorem 1.1 is that in the case of a Fano
background with toric symmetry, the normalized generalized Kähler Ricci flow
(1.2) reduces to a flow of ${\mathbb{T}}$-invariant _Kähler_ metrics
$\overline{\omega}_{t}\in 2\pi c_{1}(M,J)$. Using the Abreu–Guillemin
reduction [2, 41], these are then determined by a family of strictly convex
smooth functions $\phi_{t}$ on ${\mathbb{R}}^{m}$, satisfying

(1.5) $\displaystyle\frac{\partial}{\partial
t}\phi=\log{\det}\Big{(}\big{(}{\rm Hess}\
\phi\big{)}^{-1}+\sqrt{-1}e^{-2t}B\Big{)}^{-1}+2\phi,$

where $B$ is a skew-symmetric matrix corresponding to the $B$-type deformation
described above. Interestingly, this reduced scalar equation is independent of
$A$-type deformations. Also, when $B=0$, it coincides with the PDE describing
the reduction of the Kähler Ricci flow on a smooth toric Fano variety, studied
in [74]. We emphasize here that the scalar reduction in the Kähler-Ricci flow
case is an adaptation to the toric setting of the general scalar reduction,
itself a consequence of the transgression formula for the Ricci curvature. In
our setting the derivation of this scalar reduction is more subtle. In GK
geometry, the local Ricci potential relies on delicate constructions in
generalized geometry (cf. [28] Definition 8.15), and is not always easy to
make explicit. In §3, after a careful buildup of the differential geometry of
toric generalized Kähler structure, we derive the Bismut-Ricci potential.
However, the relevant formula involves the evolving complex structure $I$, and
therefore a scalar reduction of the flow is not immediate. Again subtle
structure of toric GK geometry enters to give the scalar reduction to (1.5) in
Proposition 4.2. Given this scalar reduction, through a series of estimates
based on the maximum principle we derive a priori $L^{\infty}$ estimates for
the associated metric in terms of a $C^{0}$ bound for $\phi$. Using the
generalization of Evans-Krylov $C^{2,\alpha}$/Calabi-Yau $C^{3}$ estimates for
pluriclosed flow ([48, 55]), we obtain $C^{\infty}$ estimates for $\phi$ in
terms of a $C^{0}$ estimate for $\phi$. Since $\phi$ grows at worst
exponentially by the maximum principle, we thus conclude the long time
existence.

One expects the convergence at infinity to be a delicate issue, and we prove
two partial results in this direction. In the toric KRF setting the
convergence was established by Zhu [74] using delicate estimates derived using
Perelman’s entropy, Mabuchi’s $K$-energy and estimates from convex geometry.
Outside of the toric setting, convergence assuming the existence of a Kähler-
Ricci soliton was shown by Tian-Zhu [67, 68]. First we prove convergence of
nonsingular solutions without the toric symmetry hypothesis, where so far
there is not a scalar reduction of the flow. The key point is a Perelman-type
entropy monotonicity, which yields that the only smooth self-similar limits of
NGKRF are in fact Kähler-Ricci solitons, and also shows a uniform
$\kappa$-noncollapsing result for NGKRF on Fano manifolds.

######  Theorem 1.2.

Let $(M,g,I,J)$ be a generalized Kähler structure of symplectic type with
respect to $F$. Let $(g_{t},I_{t},J)$ denote the solution to NGKRF with this
initial data. Then

  1. (1)

The metric $g_{t}$ is uniformly $\kappa$-noncollapsed for all times the flow
exists.

  2. (2)

Suppose $(M,J)$ is Fano, $F\in 2\pi c_{1}(M,J)$. Suppose the solution exists
on $[0,\infty)$ with uniformly bounded curvature. Then any sequence of times
$\\{t_{j}\\}\to\infty$ admits a subsequence such that
$(g_{t_{j}},I_{t_{j}},J)$ converges in the $C^{\infty}$-Cheeger-Gromov
topology to $(g_{KRS},J_{\infty},J_{\infty})$, where $g_{KRS}$ is a Kähler-
Ricci soliton.

In the convergence statement above, the complex structure $J_{\infty}$ may not
be in general biholomorphic to $J$, and as is the case in Kähler-Ricci flow
one expects this subtle issue to be related to $K$-stability of $(M,J)$. In
the case when the initial structure $(g,I,J)$ has a _toric symmetry_ , the
scalar reduction (1.5) of the GKRF and the results in [74] motivate us to
conjecture that $J_{\infty}\cong J$ (see Conjecture 2.12 below).

Returning to the setting of toric symmetry, we are able to prove a certain
kind of weak, but unconditional, convergence. First, adapting ideas from
general GIT theory, we are able to define an extension of the Mabuchi
$K$-energy to toric generalized Kähler structures, which we show is monotone
along NGKRF. Using this monotonicity and exploiting the variational approach
of [7], we obtain a weak convergence.

######  Theorem 1.3.

Let $(M,J,{\mathbb{T}})$ be a smooth toric Fano variety which admits a Kähler-
Einstein metric, and $F\in 2\pi c_{1}(M,J)$ a ${\mathbb{T}}$-invariant
symplectic $2$-form which defines a ${\mathbb{T}}$-invariant generalized
Kähler structure $(g,I,J)$ of symplectic type on $M$. Let
$(g_{t},I_{t},J),t\in[0,\infty)$ be the global solution of the normalized
generalized Kähler Ricci flow (1.2) with this initial data guaranteed by
Theorem 1.1, and $\overline{\omega}_{t}$ the corresponding family of toric
Kähler metrics defined by (1.5). Then, there exists a sequence of times
$j\to\infty$ and automorphisms $\tau_{j}\in{\mathbb{T}}_{{\mathbb{C}}}$, such
that $\tau_{j}^{*}(\overline{\omega}_{j})$ converges, with respect to the
distance $d_{1}$, to a positive $(1,1)$-current $\overline{\omega}_{\infty}$
of maximal Monge-Ampère mass and finite energy on $(M,J)$.

The precise definitions for the convergence are given in Section 5.4 below,
but in the toric case we consider it can be equivalently characterized [15,
39] in terms of the $L^{1}$ convergence over the Delzant polytope of
$(M,F,{\mathbb{T}})$ of the Legendre transforms $u_{j}$ of the convex
functions $\phi_{j}$ satisfying (1.5), up to the addition to $u_{j}$ of
affine-linear functions.

Here is an outline of the rest of the paper. In §2 we recall fundamental
properties of GK structures and GKRF, focusing on the symplectic-type case. In
§3 we discuss toric GK structures, unify the discussions of [11] and [70],
[71], and explicitly describe the relevant associated Poisson tensors. This
leads to a derivation of the underlying Kähler metric and the associated
metric potential and Bismut-Ricci potential. With these geometric results in
place, we give the proof of Theorem 1.1 in §4, relying principally on the
scalar reduction of the flow described above. In the final §5, we obtain the
monotonicity of the Perelman-type and Mabuchi-type functionals, and give the
proofs of Theorems 1.2 and 1.3.

##  2\. The generalized Kähler-Ricci flow in the symplectic-type case

###  2.1. Curvature identities for Hermitian metrics

In this subsection we recall the definitions of the Bismut-Ricci and the
Chern-Ricci forms on a Hermitian manifold $(M,g,J)$, extending the notion of
the Kähler-Ricci form of a Kähler manifold.

######  Definition 2.1.

Let $(M,g,J)$ be a Hermitian manifold with fundamental $2$-form
$\omega_{J}=gJ$. The _Bismut connection_ is the Hermitian connection defined
by

(2.1) $\displaystyle g(\nabla^{B}_{X}Y,Z)=$ $\displaystyle\
g(\nabla_{X}Y,Z)-\tfrac{1}{2}{\mathrmsl d}^{c}\omega_{J}(X,Y,Z),$

where $\nabla$ stands for the Riemannian connection of $g$. The _Chern
connection_ is the Hermitian connection defined by

(2.2) $\displaystyle g(\nabla^{C}_{X}Y,Z)=$ $\displaystyle\
g(\nabla_{X}Y,Z)+\tfrac{1}{2}{\mathrmsl d}^{c}\omega_{J}(X,JY,JZ).$

######  Definition 2.2.

Let $(M,g,J)$ be a complex manifold with a Hermitian metric $g$. The _Bismut-
Ricci curvature_ and _Chern-Ricci curvature_ are defined by

$\displaystyle\rho^{B}(X,Y)=$ $\displaystyle\
\tfrac{1}{2}\sum_{i=1}^{2m}R^{B}(X,Y,e_{i},Je_{i}),\qquad\rho^{C}(X,Y)=\
\tfrac{1}{2}\sum_{i=1}^{2m}R^{C}(X,Y,e_{i},Je_{i}),$

where $R^{B}$ and $R^{C}$ are the curvature tensors associated to the Bismut
[9] and Chern connections respectively, and $\\{e_{i}\\}$ is an orthonormal
basis for $(TM,g)$. Furthermore we define the _Bismut scalar curvature_ as

$\displaystyle s^{B}=\sum_{i=1}^{2m}\rho^{B}(e_{i},Je_{i}).$

Using the formulas (2.1) and (2.2), one can check (see e.g. [30, Rem. 5]) that
the induced unitary connections $\nabla^{B}$ and $\nabla^{C}$ on the anti-
canonical line bundle $K^{-1}(M,J)={}^{m}(T^{1,0}M)$, with respect to the
induced Hermitian metric by $g$, satisfy

(2.3) $\nabla^{B}_{X}=\nabla^{C}_{X}+i\theta_{J}(JX),$

where

$\theta_{J}:=\left<\omega_{J},{\mathrmsl
d}\omega_{J}\right>=-\omega_{J}^{-1}\righthalfcup{\mathrmsl d}\omega_{J}$

is the Lee form of $(g,J)$. It follows from (2.3) that the corresponding Ricci
curvatures are related by

(2.4) $\rho^{B}=\rho^{C}+{\mathrmsl d}(J^{\sharp}\theta_{J}),$

where for a $1$-form we have set
$(J^{\sharp}\alpha)(X):=-(J^{*}\alpha)(X)=-\alpha(JX)$ for the induced action
of $J$ on $T^{*}M$, compatible with the $g$-duality between $TM$ and $T^{*}M$.
Furthermore, as the $\nabla^{C}$ induces the Cauchy-Riemann operator
$\overline{\partial}$ on the canonical bundle $\Omega^{(m,0)}(M,J)$, the
Chern-Ricci form can be computed from any (local) holomorphic section $\Theta$
of $\Omega^{(m,0)}(M,J)$, via the formula

(2.5) $\rho^{C}=-\tfrac{1}{2}{\mathrmsl d}{\mathrmsl
d}^{c}_{J}\log\left(\frac{\omega_{J}^{[m]}}{v_{\Theta}}\right),$

where $v_{\Theta}:=(\sqrt{-1})^{m}(\Theta\wedge\overline{\Theta})$. Lastly we
record an identity relating the Bismut-Ricci form and the Riemannian Ricci
curvature. This follows from ([47], Proposition 3.1), taking in mind the sign
difference for the torsion $H$.

(2.6) $\begin{split}\rho^{B}(X,JY)=&\ {\rm
Ric}^{g}(X,Y)-\tfrac{1}{2}\langle(\imath_{X}H),(\imath_{Y}H)\rangle_{g}+\tfrac{1}{2}({\mathcal{L}}_{\theta^{\sharp}}g)(X,Y)\\\
&\ -\tfrac{1}{2}(\delta^{g}H)(X,Y)+\tfrac{1}{2}({\mathrmsl
d}\theta)(X,Y)-\tfrac{1}{2}(\imath_{\theta^{\sharp}}H)_{X,Y},\end{split}$

###  2.2. Generalized Kähler structures of symplectic type

######  Definition 2.3.

Given a smooth manifold $M$, a _generalized Kähler structure_ is a triple
$(g,I,J)$ consisting of a Riemannian metric $g$, and two integrable complex
structures $I$ and $J$ compatible with $g$ such that

$\displaystyle{\mathrmsl d}^{c}_{I}\omega_{I}=H=-{\mathrmsl
d}^{c}_{J}\omega_{J},\qquad{\mathrmsl d}H=0.$

######  Definition 2.4.

Given a smooth manifold $M$, a generalized Kähler structure $(g,I,J)$ is of
_symplectic type_ if

(2.7) $\displaystyle\det(I+J)\neq 0$

everywhere on $M$. In this setting we define

$\displaystyle F=-2g(J+I)^{-1},\qquad b=-g(J+I)^{-1}(I-J),\qquad
F^{\sharp}=Fg^{-1}=-2(J+I)^{-1}.$

It follows [34] that $F$ is non-degenerate and tames both $J$ and $I$, and $b$
is a skew-symmetric tensor which further satisfies
$b\in{}^{2,0+0,2}_{I}\cap{}^{2,0+0,2}_{J}$. Furthermore, direct computations
yield the basic relationship

(2.8) $\displaystyle-FJ=g+b,\qquad-FI=g-b.$

This decomposition yields the further useful inequality

(2.9) $\displaystyle F^{m}\geq\omega_{J}^{m}=m!dV_{g}.$

Several differential operators are relevant for these structures. In
particular, we set

$\displaystyle d^{c}_{I}=IdI,\qquad d^{c}_{J}=JdJ,\qquad
d_{F}=F^{\sharp}dF^{\sharp}.$

Using the integrability conditions for generalized Kähler structures it
furthermore follows that (cf. [6, Lemma 2.7])

$\displaystyle{\mathrmsl d}F=0,\qquad H={\mathrmsl d}b.$

######  Remark 2.5.

It turns out (cf. [26] Proposition 1) that the data above is equivalently
described by fixing first a symplectic form $F$, then asking for an integrable
complex structure $J$ which is tamed by $F$, and furthermore the almost
complex structure

$\displaystyle I=-F^{-1}J^{*}F$

is integrable. Given these conditions, the formulas in Definition 2.4 can be
inverted to derive $g$ and $b$. This in turn allows for an explicit derivation
of the associated _generalized complex structures_ (cf. [35, 36]), one of
which is of ‘symplectic type,’ determined explicitly by the global closed pure
spinor $F$.

Kähler metrics are naturally interpreted as symplectic-type generalized
Kähler, setting $I=J$. Furthermore, given a hyperKähler structure $(g,I,J,K)$,
the triple $(g,I,J)$ is GK of symplectic type, in fact is of _nondegenerate_
type (cf. [54], [6]). Hamiltonian deformations of these structures are
examples of non-Kähler symplectic-type GK structures ([5]), as are Hitchin’s
Poisson deformations of Kähler metrics on Fano surfaces [46]. We further
record here a basic further identity:

######  Lemma 2.6.

Let $(g,I,J)$ be a generalized Kähler structure of symplectic type and denote
by $\theta_{J}=J\delta^{g}\omega_{J}$ and $\theta_{I}:=I\delta^{g}\omega_{I}$
the Lee forms. Then we have

$\frac{1}{2}d\log\det(I+J)=-(b\righthalfcup
db)=\theta_{J}+\imath_{\theta_{J}^{\sharp}}b+\delta^{g}b=\theta_{I}-\imath_{\theta_{I}^{\sharp}}b-\delta^{g}b.$

###### Proof.

This follows directly from [6] Lemmas 2.10 and 2.13. ∎

###  2.3. The Bismut-Ricci forms for symplectic-type GK structures

Observe that in the context of generalized Kähler geometry we have two Bismut
connections, defined via

$\displaystyle g(\nabla^{B,I}_{X}Y,Z)=$ $\displaystyle\
g(\nabla_{X}Y,Z)-\tfrac{1}{2}{\mathrmsl
d}^{c}_{I}\omega_{I}(X,Y,Z)=g(\nabla_{X}Y,Z)-\tfrac{1}{2}H(X,Y,Z),$
$\displaystyle g(\nabla^{B,J}_{X}Y,Z)=$ $\displaystyle\
g(\nabla_{X}Y,Z)-\tfrac{1}{2}{\mathrmsl
d}^{c}_{J}\omega_{J}(X,Y,Z)=g(\nabla_{X}Y,Z)+\tfrac{1}{2}H(X,Y,Z).$

For a generalized Kähler structure $(g,I,J)$, we denote by
$\rho^{C}_{J},\rho^{C}_{I}$ and $\rho^{B}_{J},\rho^{B}_{I}$ the Chern and
Bismut-Ricci forms of the Hermitian structure $(g,J)$ and $(g,I)$,
respectively. We next derive a relationship between the two associated Bismut-
Ricci forms for GK structures of symplectic type.

######  Proposition 2.7.

Let $(g,I,J)$ be a generalized Kähler structure of symplectic type. Then

$\rho_{J}^{B}(X,Y)-\rho_{I}^{B}(X,Y)=\tfrac{1}{2}(\mathcal{L}_{\theta_{J}^{\sharp}-\theta_{I}^{\sharp}}F)(X,Y).$

###### Proof.

An important identity [28, Lemma 9.27] reads as

(2.10)
$g((\mathcal{L}_{\theta_{I}^{\sharp}-\theta_{J}^{\sharp}}I)X,Y)=\rho_{J}^{B}(X,[J,I]Y),$

where the upper script $\sharp$ stands for the bundle map $g^{-1}:T^{*}M\to
TM$. Using this together with the analogous identity involving
$\mathcal{L}_{\theta_{I}^{\sharp}-\theta_{J}^{\sharp}}J$ we have

$\begin{split}g((\mathcal{L}_{\theta_{I}^{\sharp}-\theta_{J}^{\sharp}}I)X,Y)=\rho_{J}^{B}(X,[J,I]Y),\qquad
g((\mathcal{L}_{\theta_{I}^{\sharp}-\theta_{J}^{\sharp}}J)X,Y)=\rho_{I}^{B}(X,[J,I]Y).\end{split}$

Next, using (2.6), we find

$\left(\mathcal{L}_{\theta_{I}^{\sharp}-\theta_{J}^{\sharp}}g\right)(X,Y)=\rho_{I}^{B}(X,IY)+\rho_{I}^{B}(Y,IX)-\rho_{J}^{B}(X,JY)-\rho_{J}^{B}(Y,JX).$

Combining the above for $F=-2g(I+J)^{-1}$, we find:

$\begin{split}\tfrac{1}{2}(&\mathcal{L}_{\theta_{I}^{\sharp}-\theta_{J}^{\sharp}}F)(X,Y)\\\
=&\
-\mathcal{L}_{\theta_{I}^{\sharp}-\theta_{J}^{\sharp}}g((I+J)^{-1}X,Y)+g((I+J)^{-1}\mathcal{L}_{\theta_{I}^{\sharp}-\theta_{J}^{\sharp}}(I+J)(I+J)^{-1}X,Y)\\\
=&\
\rho_{J}^{B}((I+J)^{-1}X,JY)+\rho_{J}^{B}(Y,J(I+J)^{-1}X)-\rho_{I}^{B}((I+J)^{-1}X,IY)-\rho_{J}^{B}(Y,I(I+J)^{-1}X)\\\
&\
\qquad-\rho_{J}^{B}((I+J)^{-1}X,[J,I](I+J)^{-1}Y)-\rho_{I}^{B}((I+J)^{-1}X,[J,I](I+J)^{-1}Y)\\\
=&\
\rho_{J}^{B}((I+J)^{-1}X,IY)+\rho_{J}^{B}(Y,J(I+J)^{-1}X)-\rho_{I}^{B}((I+J)^{-1}X,JY)-\rho_{J}^{B}(Y,I(I+J)^{-1}X)\\\
=&\ -\rho_{J}^{B}(X,Y)+\rho_{I}^{B}(X,Y),\end{split}$

where we used that $[J,I](I+J)^{-1}=(J-I)$ and that $\rho_{J}^{B}$ (resp.
$\rho_{I}^{B}$) is $I$ invariant (resp. $J$-invariant), see [28, Lemma 9.26].
∎

######  Proposition 2.8.

Let $(g,I,J)$ be a generalized Kähler structure of symplectic type and
$v_{\Theta}=(\sqrt{-1})^{m}\Theta\wedge\overline{\Theta}$ a local holomorphic
volume form of $(M,J)$, associated to a local non-vanishing section $\Theta$
of the canonical line bundle $\Omega^{m,0}(M,J)$. Then the smooth function

$\displaystyle\Phi_{J,\Theta}:=-\frac{1}{2}\left[\log\left(\frac{F^{[m]}}{v_{\Theta}}\right)+\log\left(\frac{F^{[m]}}{\omega_{J}^{[m]}}\right)\right]$

is a potential for $\rho^{B}_{I}$, i.e. $\rho^{B}_{I}={\mathrmsl d}{\mathrmsl
d}^{c}_{J}\Phi_{J,\Theta}$.

###### Proof.

Using (2.5) and $F^{[m]}/\omega_{J}^{[m]}=(2^{m})\det(I+J)^{-\frac{1}{2}}$, we
get

$\begin{split}{\mathrmsl d}{\mathrmsl d}^{c}_{J}\Phi_{J,\Theta}=&\
\rho^{C}_{J}+\tfrac{1}{2}{\mathrmsl d}{\mathrmsl
d}_{J}^{c}\log\det(I+J)=\rho^{C}_{J}+\tfrac{1}{2}{\mathrmsl
d}\left(J(\theta_{J}+\theta_{I})+Jb(\theta_{J}^{\sharp}-\theta_{I}^{\sharp})\right)\\\
=&\ \rho^{C}_{J}+{\mathrmsl d}(J\theta_{J})+\tfrac{1}{2}{\mathrmsl
d}\big{(}F(\theta_{I}^{\sharp}-\theta_{J}^{\sharp})\big{)}=\
\rho^{B}_{J}+\tfrac{1}{2}{\mathcal{L}}_{\theta_{I}^{\sharp}-\theta_{J}^{\sharp}}F=\
\rho^{B}_{I},\end{split}$

where for passing from the first to the second line we have used Lemma 2.6,
for passing from the second to the third line we have used (2.8), and for
passing from the third to the fourth line we have used Proposition 2.7 and
(2.4). ∎

###  2.4. The generalized Kähler-Ricci flow

The generalized Kähler-Ricci flow (GKRF) is a solution $(g_{t},I_{t},J)$ of
the equation

(2.11) $\begin{split}\frac{\partial}{\partial t}\omega_{J}=&\
-2\Big{(}\rho^{B}_{J}\Big{)}_{J}^{1,1},\qquad\frac{\partial}{\partial t}I=\
{\mathcal{L}}_{\theta_{I}^{\sharp}-\theta_{J}^{\sharp}}I.\end{split}$

We refer the reader to [28, 58, 59, 61] for the basic properties of the above
equation. The expression given here is the equivalent expresion for GKRF in
the “$J$-fixed gauge,” (cf. [28] Definition 9.22). Our first goal is to show
that the condition of being symplectic type is preserved, which follows by
recasting the GKRF in terms of $F$ and $b$.

######  Lemma 2.9.

Let $(g_{t},I_{t},J)$ denote a solution to generalized Kähler-Ricci flow
(2.11) which is symplectic-type for all times. Then the associated one-
parameter families of two-forms satisfy

(2.12) $\frac{\partial}{\partial
t}F_{t}=-2\rho^{B}_{J}(\omega_{t}),\qquad\frac{\partial}{\partial
t}b_{t}=\Delta_{g_{t}}b_{t}-\mathcal{L}_{\theta_{t}^{\sharp}}b_{t}.$

###### Proof.

By (2.6), using that $H={\mathrmsl d}b$ and Lemma 2.6, we obtain

(2.13) $\begin{split}(-2J\rho^{B}_{J})^{\rm skew}=&\ -\delta^{g}H+{\mathrmsl
d}\theta-\imath_{\theta^{\sharp}}H=\Delta_{g}b-\mathcal{L}_{\theta^{\sharp}}b.\end{split}$

As $\pi_{{}^{1,1}_{J}}F=\omega_{J}$ and $(-FJ)^{\rm skew}=b$, equation (2.13)
shows that the two equalities in the lemma are equivalent. Using the identity
(2.10) and equations (2.11) we compute below the evolution of
$F=-2g(I+J)^{-1}$:

$\begin{split}&\frac{\partial}{\partial t}F(X,Y)=\
2\left(\frac{\partial}{\partial
t}g\right)(X,(I+J)^{-1}Y)+2g\left(X,\frac{\partial}{\partial
t}(I+J)^{-1}Y\right)\\\ &\
=2\left(\rho^{B}_{J}(JX,(I+J)^{-1}Y)-\rho^{B}_{J}(X,J(I+J)^{-1}Y)\right)+2g\left((I+J)^{-1}X,(\mathcal{L}_{\theta_{I}^{\sharp}-\theta_{J}^{\sharp}}I)(I+J)^{-1}Y\right)\\\
&\
=2\left(\rho^{B}_{J}(JX,(I+J)^{-1}Y)-\rho^{B}_{J}(X,J(I+J)^{-1}Y)\right)-2\rho^{B}\Bigl{(}[J,I](I+J)^{-1}X,(I+J)^{-1}Y\Bigr{)}\\\
&\ =-2\left(\rho^{B}_{J}(X,J(I+J)^{-1}Y)+\rho^{B}_{J}(X,I(I+J)^{-1}Y)\right)\
=-2\rho^{B}_{J}(X,Y),\end{split}$

where for passing to the last line we have used that $\rho^{B}_{J}$ is
$I$-invariant, see [28, Lemma 9.26]. ∎

######  Proposition 2.10.

Let $(g_{t},I_{t},J)$ denote a solution to generalized Kähler-Ricci flow such
that $(g_{0},I_{0},J)$ is symplectic-type. Then $(g_{t},I_{t},J)$ is
symplectic-type for all $t$ for which the flow is defined.

###### Proof.

If the maximal interval on which the data is symplectic type is
$[0,T),T<\infty$, but the solution exists smoothly on $[0,T]$. By integrating
the evolution equations over $[0,T]$ it follows that the family of
endomorphisms $J+I_{t}$ extends smoothly across time $T$. Using the evolution
equation for $F$ from Lemma 2.9 and integrating in time we also conclude that

$\displaystyle(J+I_{t})^{-1}=-\tfrac{1}{2}g_{t}^{-1}F_{t}$

extends smoothly across time $T$. It follows directly that $J+I_{T}$ is well-
defined and invertible, contradicting maximality of $T$. ∎

###  2.5. The normalized generalized Kähler Ricci flow in the Fano case

As explained in the introduction, in the special case when the symplectic form
$F$ of a symplectic type generalized Kähler structure $(g,I,J)$ belongs to
$2\pi c_{1}(M,J)$, we can consider the _normalized_ GKRF

(2.14) $\begin{split}\frac{\partial}{\partial t}\omega_{J}=&\
-2\Big{(}\rho^{B}_{J}\Big{)}_{J}^{1,1}+2\omega_{J},\qquad\frac{\partial}{\partial
t}I=\ {\mathcal{L}}_{\theta_{I}^{\sharp}-\theta_{J}^{\sharp}}I.\end{split}$

To get a sense of the possible global existence behaviour of this in the Fano
case, we exhibit the behavior of some basic quantities.

######  Proposition 2.11.

Let $(M^{2n},J)$ be a Fano manifold, and suppose $(g_{t},I_{t},J)$ is a
solution of (2.14) of symplectic type such that $[F_{0}]=2\pi c_{1}(M,J)$.
Then for all $t$, the cohomology class $[F_{t}]$ and the Poisson tensor
$\sigma_{t}=[I_{t},J]g_{t}^{-1}$ (see 1.4) satisfy

  1. (1)

$[F_{t}]=2\pi c_{1}(M,J)$,

  2. (2)

$\sigma_{t}=e^{-2t}\sigma_{0}$.

###### Proof.

The solutions $(g_{t},I_{t})$ and $(\tilde{g}_{t},\tilde{I}_{t})$ of (2.11)
and (2.14) are related by a reparametrisation

$\displaystyle\tilde{g}_{t}:=\tfrac{1}{2}e^{2t}g_{(1-e^{-2t})},\qquad\tilde{I}_{t}=I_{(1-e^{2t})}.$

It follows directly from Lemma 2.9 that

$\frac{\partial}{\partial t}\tilde{F}=-2\rho^{B}_{J}(\tilde{g})+2\tilde{F},$

proving claim (1). To see claim (2), we first note that along a solution of
(2.11) one has that $\sigma$ is fixed in time ([31, Corollary 1.5] cf. also
[28, Proposition 9.29]). Claim (2) then follows easily from the
reparameterization relationships above. ∎

Thus the symplectic form $F$ remains in the canonical class, while the tensor
$\sigma$, which measures the deviation of a symplectic-type GK structure from
being Kähler, decays exponentially. Based on this, as well as the monotonicity
formulae described below in §5, one might expect the normalized GKRF (2.14) to
behave similarly to the much studied normalized Kähler-Ricci flow on Fano
manifolds. In particular, by a result of Cao [12] the solution of the
normalized Kähler-Ricci flow exists for all time, and, by results of Tian-Zhu
[67, 68], the global solution of the normalized Kähler-Ricci flow composed by
automorphisms of $(M,J)$ converges to a Kähler-Ricci soliton, provided that
the latter exists. Furthermore, by [74], in the toric case the reduced flow
will converge at the level of normalized potentials. It is natural to ask
whether or not similar results hold true for solutions of (2.14). More
precisely we have:

######  Conjecture 2.12.

Suppose $(g,I,J)$ is a symplectic type generalized Kähler structure on a Fano
manifold $(M,J)$, with $F\in 2\pi c_{1}(M,J)$. Then

  * •

The solution of (2.14) with this initial data exists for all time
$t\in[0,\infty)$.

  * •

If $(M,J)$ admits a Kähler-Einstein metric then the global solution of (2.14)
converges, in the $C^{\infty}(M)$ topology, to a Kähler-Einstein metric.

  * •

If $(M,J)$ admits a Kähler-Ricci soliton, with a soliton vector field $K$, and
$(g,I,J)$ is invariant by the torus action generated by $K$, then there exists
a smooth family of complex automorphisms $\tau_{t}$ of $(M,J)$ such that the
global solution $g_{t}$ of (2.14) pulled-back by $\tau_{t}$ converges, in the
$C^{\infty}(M)$ topology, to a Kähler-Ricci soliton metric in $c_{1}(M,J)$
with soliton vector field $K$.

  * •

If $(M,J)$ is a toric Fano variety and $(g,I,J)$ is invariant under the
maximal torus ${\mathbb{T}}$, then the reduced equation (1.5) has a global
solution $\phi_{t}(y)$ defined on $[0,\infty)\times{\mathbb{R}}^{m}$, and
there are families of points $y_{t}\in{\mathbb{R}}^{n}$ and real constants
$c_{t}$, such that $\tilde{\phi}_{t}(y):=\phi_{t}(y+y_{t})+c_{t}$ converges in
$C^{\infty}([0,\infty)\times{\mathbb{R}}^{m})$ to a smooth convex function
$\tilde{\phi}_{\infty}(y)$ on ${\mathbb{R}}^{m}$ which defines a
${\mathbb{T}}$-invariant Kähler Ricci soliton on $M$.

##  3\. Toric generalized Kähler structures

###  3.1. Toric Kähler structures

In this section we recall fundamental material on toric Kähler structures
going back to [2, 41]; we refer to [3, 25] for comprehensive surveys. Let
$(M,\omega_{0})$ be a smooth compact symplectic manifold of real dimension
$2m$, endowed with an effective Hamiltonian action of a compact
$m$-dimensional torus ${\mathbb{T}}$. We denote by ${\mathfrak{t}}$ the Lie
algebra of ${\mathbb{T}}$, and by ${\mathfrak{t}}^{*}$ the dual vector space.
Let $\mu:M\to{\mathfrak{t}}^{*}$ be a fixed momentum map for the action.
Delzant’s theorem [18] tells us that the image of $\mu(M)$ is a Delzant
polytope ${\mathrm{P}}\subset{\mathfrak{t}}^{*}$, which comes equipped with a
minimal set of defining hyperplanes

${{\mathrm{P}}}=\\{x\in{\mathfrak{t}}^{*}:L_{j}(x)=\langle
v_{j},x\rangle+\lambda_{j}\geq 0,\,j=1,\ldots,d\\},$

where $v_{j}$ are primitive elements of the lattice
$\Lambda\subset{\mathfrak{t}}$ of circle subgroups of ${\mathbb{T}}$, i.e.
${\mathbb{T}}={\mathfrak{t}}/2\pi\Lambda$. We denote by
${{\mathrm{L}}}=\\{L_{1}(x),\ldots,L_{d}(x)\\}$ the set of the defining
affine-linear functions of ${\mathrm{P}}$ as above, which sometimes is
referred to as a labelling of ${\mathrm{P}}$, see [49]. According to [18], the
data $({\mathrm{P}},{{\mathrm{L}}})$ in turn identify $(M,\omega_{0})$, up to
a ${\mathbb{T}}$-equivariant symplectomorphism, with the Kähler reduction of
${\mathbb{C}}^{d}$ (endowed with its flat Kähler structure) by a real torus of
dimension $d-m$. This gives rise to a canonical $\omega_{0}$-compatible
${\mathbb{T}}$-invariant Kähler structure $(g_{c},J_{c},\omega_{0})$ on $M$.

We next describe the set $\mathcal{K}^{{\mathbb{T}}}(M,\omega_{0})$ of
${\mathbb{T}}$-invariant, $\omega_{0}$-compatible complex structures on
$(M,\omega_{0})$, or equivalently, the set of ${\mathbb{T}}$-invariant Kähler
metrics $(g,J,\omega_{0})$ with fixed Kähler form $\omega_{0}$. Notice that
$J_{c}\in\mathcal{K}^{{\mathbb{T}}}(M,\omega_{0})$ by construction. On the
union $M^{0}:=\mu^{-1}(\mathring{{\mathrm{P}}})$ of the generic orbits of the
${\mathbb{T}}$-action (where $\mathring{{\mathrm{P}}}$ denotes the interior of
${\mathrm{P}}$), any Kähler metric $g$ determined by an element $J$ in
$\mathcal{K}^{{\mathbb{T}}}(M,\omega_{0})$ has a general expression due to V.
Guillemin [41]. In this description, the momentum map
$\mu:M^{0}\to\mathfrak{t}^{*}$ is supplemented by angular coordinates
(depending on $J$) $\theta:M^{0}\to{\mathfrak{t}}/2\pi\Lambda$, such that the
kernel of ${\mathrmsl d}\theta$ is orthogonal to the tangent space of the
torus orbits. These momentum-angular coordinates $(\mu,\theta)$ identify each
tangent space to $M^{0}$ with ${\mathfrak{t}}\oplus{\mathfrak{t}}^{*}$, and
the symplectic form $\omega_{0}$ and a compatible Kähler metric $g$ on $M^{0}$
have the form

(3.1) $\omega_{0}=\langle{\mathrmsl d}\mu,{\mathrmsl d}\theta\rangle,\qquad
g=\langle{\mathrmsl d}\mu,{\mathbf{G}}(\mu),{\mathrmsl
d}\mu\rangle+\langle{\mathrmsl d}\theta,{\mathbf{H}}(\mu),{\mathrmsl
d}\theta\rangle,$

where $\langle\cdot,\cdot\rangle$ denotes contraction of ${\mathfrak{t}}$ and
${\mathfrak{t}}^{*}$, ${\mathbf{G}}(\mu)$ is (the pull back by $\mu$) of a
positive definite $S^{2}{\mathfrak{t}}$-valued function on
$\mathring{{\mathrm{P}}}$ (with $S^{2}{\mathfrak{t}}$ denoting the symmetric
tensor product of ${\mathfrak{t}}$), ${\mathbf{H}}$ is its point-wise inverse
in $S^{2}{\mathfrak{t}}^{*}$, and $\langle\cdot,\cdot,\cdot\rangle$ denotes
the point-wise contraction ${\mathfrak{t}}^{*}\times
S^{2}{\mathfrak{t}}\times{\mathfrak{t}}^{*}\to{\mathbb{R}}$ or the dual
contraction. The corresponding complex structure $J$ is defined by

(3.2) $\displaystyle J{\mathrmsl d}\theta=-\langle{\mathbf{G}}(\mu),{\mathrmsl
d}\mu\rangle,\qquad J{\mathrmsl d}\mu=\langle{\mathbf{H}}(\mu),{\mathrmsl
d}\theta\rangle,$

and $J$ is integrable if and only if ${\mathbf{G}}={\rm Hess}(u)$ is the
Hessian of a smooth strictly convex function $u(x)$ on
$\mathring{{\mathrm{P}}}$, called _symplectic potential_ of $(g,J,\omega_{0})$
(cf. [2, 41]). This applies in particular to the canonical complex structure
$J_{c}\in\mathcal{K}^{{\mathbb{T}}}(M,\omega_{0})$ coming from the Delzant
construction, giving rise to canonical angular coordinates $\theta^{c}$ and
symplectic potential [41]

(3.3) $u_{c}(x):=\frac{1}{2}\sum_{j=1}^{d}L_{j}(x)\log L_{j}(x).$

Necessary and sufficient conditions for the symplectic potential $u$ to come
from a globally defined Kähler metric $(g,J)$ on $M$ are obtained in [2, 4,
23]. Here we mention the conditions established in [23]:

######  Proposition 3.1.

A smooth strictly convex function $u(x)$ on $\mathring{{\mathrm{P}}}$ is a
symplectic potential of a globally defined $\omega_{0}$-compatible Kähler
metric in $\mathcal{K}^{{\mathbb{T}}}(M,\omega_{0})$ if and only if $u$
satisfies the following boundary conditions:

  * •

$u(x)$ is smooth and strictly convex in the interior of ${\mathrm{P}}$, and
extends as a continuous function on ${\mathrm{P}}$ which is smooth and
strictly convex on the interior of each face of ${\mathrm{P}}$;

  * •

$u(x)-u_{c}(x)$ extends to a smooth function over ${\mathrm{P}}$, where
$u_{c}(x)$ is given by (3.3).

######  Definition 3.2.

We denote by $\mathcal{S}({\mathrm{P}},{{\mathrm{L}}})$ the space of functions
$u$ satisfying the conditions of Proposition 3.1.

A subtle point in the above description is the fact that the angular
coordinates $\theta$ depend upon $J$, but one can show [4, Lemma 3] that by
pulling back $J$ by a ${\mathbb{T}}$-equivariant symplectomorphism of
$(M,\omega_{0})$, we can assume that $\theta=\theta^{c}$, a normalization we
shall implicitly use below without further notice. It follows that the space
$\mathcal{K}^{{\mathbb{T}}}(M,\omega_{0})$ modulo pull-backs by
${\mathbb{T}}$-equivariant symplectomorphisms can be identified with the space
$\mathcal{S}({\mathrm{P}},{\mathrm{L}})$, modulo additions with affine-linear
functions. Because of this correspondence, for any
$u\in\mathcal{S}({\mathrm{P}},{\mathrm{L}})$, with a slight abuse of notation,
we denote by $(g_{u},J_{u})$ the corresponding ${\mathbb{T}}$-invariant Kähler
metric on $M$ (where we implicitly use the canonical angular coordinates
$\theta=\theta^{c}$ to define $(g_{u},J_{u})$).

It is a basic fact of the theory (see e.g. [49]) that any two elements of
$\mathcal{K}^{{\mathbb{T}}}(M,\omega_{0})$ are biholomorphic under a
${\mathbb{T}}$-equivariant diffeomorphism, which acts trivially on the
cohomology class of $[\omega_{0}]$, i.e. for any
$J\in\mathcal{K}^{{\mathbb{T}}}(M,\omega_{0})$, there exists a
${\mathbb{T}}$-equivariant diffeomorphism $\Phi$, such that $\Phi\cdot
J=J_{c}$ and $\Phi^{*}\omega_{0}$ is the Kähler form of a
${\mathbb{T}}$-invariant Kähler metric in the Kähler class $[\omega_{0}]$ on
the fixed complex manifold $(M,J_{c})$. The converse is also true by the
equivariant Moser lemma. These two equivalent descriptions are often referred
to as symplectic versus complex point of view, respectively, and will be used
in our study below. We thus give next an explicit description of this
correspondence, based on an observation from [41].

For any $u\in\mathcal{S}({\mathrm{P}},{{\mathrm{L}}})$, we let

(3.4) $\displaystyle y=\nabla u,\qquad\phi(y)+u(x)=\langle y,x\rangle$

be the Legendre transform of the strictly convex smooth function $u$ on
$\mathring{{\mathrm{P}}}$. Using the compactness of $\mathring{{\mathrm{P}}}$
and the strict convexity of $u$, one can see that

$y=\nabla u:\mathring{{\mathrm{P}}}\to{\mathfrak{t}}\cong{\mathbb{R}}^{m}$

is a diffeomorphism whereas $\phi(y)$ is a strictly convex smooth function on
${\mathbb{R}}^{m}$. We denote by $\mathcal{C}({{\mathbb{R}}}^{m})$ the space
of such functions on ${\mathbb{R}}^{m}$ and let

${\mathcal{T}}:\mathcal{S}({{\mathrm{P}}},{{\mathrm{L}}})\mapsto\mathcal{C}({{\mathbb{R}}}^{m}),\qquad\mathcal{T}(u(x))=\phi(y)$

be the induced map on the corresponding Frechét spaces. Notice that we can
recover $u(x)$ from its image $\phi(y)$ by letting $x=(\nabla\phi)(y)$ and
using (3.4). A simple computation using the definition (3.4) shows that the
differential of $\mathcal{T}$ is given by

(3.5) $\displaystyle(\delta_{u}\mathcal{T})(\dot{u})=-\dot{\phi}.$

It turns out that the Legendre transform $\mathcal{T}$ underlines the
geometric correspondence between the symplectic and complex points of view
mentioned above. To see this, for any
$J=J_{u}\in\mathcal{K}^{{\mathbb{T}}}(M,\omega_{0})$, we shall implicitly
identify the abstract variable $y=\nabla u\in{\mathfrak{t}}$ with the function
$y:M\to{\mathfrak{t}}$ given by the composition $y=\nabla u(\mu)$, and use
$\\{y(\mu),\theta^{c}\\}$ as a system of coordinates on $M^{0}$. Let us
introduce a basis $\\{K_{j}=\frac{\partial}{\partial\theta_{j}}\\}$ for
${\mathfrak{t}}$ (usually given by the generators of circle groups of
${\mathbb{T}}$) and write $\theta=(\theta_{1},\ldots,\theta_{m})$ and
$\mu=(\mu_{1},\ldots,\mu_{m})$ with respect to this and the dual basis. It is
easily seen that the $1$-forms $\\{{\mathrmsl d}y_{1},\ldots,{\mathrmsl
d}y_{m},{\mathrmsl d}\theta_{1},\ldots,{\mathrmsl d}\theta_{m}\\}$ form a dual
basis (at each point of $M^{0}$) of the basis of commuting real holomorphic
vector fields $\\{-JK_{1},\ldots,-JK_{m},K_{1},\ldots,K_{m}\\}$. We can thus
consider two systems of holomorphic coordinates on $(M^{0},J)$:
$\\{y_{1}+\sqrt{-1}\theta_{1},\ldots,y_{m}+\sqrt{-1}\theta_{m}\\}$ and
$\\{z_{1},\ldots,z_{m}\\}$ with $z_{j}:=e^{y_{j}+\sqrt{-1}\theta_{j}}$.
Geometrically, the action of ${\mathbb{T}}$ gives rise to a holomorphic action
of the complex torus ${\mathbb{T}}_{{\mathbb{C}}}\cong({\mathbb{C}}^{*})^{m}$
on $(M,J)$, which is the linear in the coordinates $\\{z_{1},\ldots,z_{m}\\}$
on $M^{0}$. In other words, letting $x_{u}\in{\mathrm{P}}^{0}$ be the pre-
image of $0\in{\mathfrak{t}}$ under $\nabla u$ (or, equivalently, the point of
minima of $u$) and $p_{u}$ the point in $M^{0}$ such that $\mu(p_{u})=x_{u}$
and $\theta(p_{u})=0$, we have the identification
$(M^{0},J)=({\mathbb{C}}^{*})^{m}\cdot p_{u}$ with the orbit of
${\mathbb{T}}_{{\mathbb{C}}}$ at $p_{u}$, with
$z_{j}=e^{y_{j}+\sqrt{-1}\theta_{j}}$ being the natural complex coordinate on
the $j$-th factor ${\mathbb{C}}^{*}$ (reflecting the fact that
$z_{j}(p_{u})=1$). The key observation in [41] is that the Kähler form
$\omega_{0}$ is written on $(M^{0},J)$ as

(3.6) $\omega_{0}={\mathrmsl d}{\mathrmsl
d}^{c}_{J}\phi(y(\mu))=\sum_{i,j=1}^{m}\phi_{,pq}(y){\mathrmsl
d}y_{p}\wedge{\mathrmsl d}\theta_{q},$

where $\phi=\mathcal{T}(u)$ and, by Legendre duality,
$\big{(}\phi_{,pq}(y)\big{)}=\big{(}u_{ij}(x)\big{)}^{-1}={\bf G}^{-1}(x)={\bf
H}(x)$.

The above description holds on $M^{0}$ in complex coordinates with respect to
$J=J_{u}$. To relate it to the canonical complex structure $J_{c}$ of $M$, let
$z^{c}=(z_{1}^{c},\ldots,z_{m}^{c})$ be the respective holomorphic coordinates
on $(M^{0},J_{c})$, obtained from the canonical potential $u_{c}(x)$ defined
by (3.3). We let $p_{0}:=p_{u_{0}}$ be the corresponding point in $M^{0}$ so
that $(M^{0},J_{c})=({\mathbb{C}}^{*})^{m}\cdot p_{0}$ with
$(z_{1}^{c},\ldots,z_{m}^{c})$ being the standard coordinates on
$({\mathbb{C}}^{*})^{m}$. Then, using the boundary conditions for $u$ in
Proposition 3.1, the following correspondence is established in [24, pp.
395-396]:

######  Proposition 3.3.

The Legendre transform $\mathcal{T}$ defines a bijective map from the space of
symplectic potentials ${\mathcal{S}}({\mathrm{P}},{{\mathrm{L}}})$ to the
space $\mathcal{K}({\mathbb{R}}^{m},\omega_{0})$ of strictly convex smooth
functions $\phi(y)$ on ${\mathbb{R}}^{m}$, satisfying the following
conditions:

  * •

the function defined on $({\mathbb{C}}^{*})^{m}\cdot p_{0}=(M^{0},J_{c})$ by

$\phi(z):=\phi\big{(}\tfrac{1}{2}\log|z^{c}_{1}|^{2},\ldots,\tfrac{1}{2}\log|z^{c}_{m}|^{2}\big{)}$

gives rise to a Kähler metric $\omega_{\phi}={\mathrmsl d}{\mathrmsl
d}^{c}\phi$ on $(M,J_{c})$.

  * •

$\phi(z^{c})-\phi_{c}(z^{c})$ extends to a smooth function on $(M,J_{c})$.

In particular, $\mathcal{T}$ gives rise to a bijection between the space of
${\mathbb{T}}$-invariant $\omega_{0}$-compatible Kähler metrics on $M$, modulo
the action of ${\mathbb{T}}$-equivariant symplectomorphisms of
$(M,\omega_{0})$, and the space of ${\mathbb{T}}$-invariant Kähler metrics in
the Kähler class $[\omega_{0}]$ on $(M,J_{c})$, modulo the action of
${\mathbb{T}}_{{\mathbb{C}}}=({\mathbb{C}}^{*})^{m}$.

###  3.2. Toric generalized Kähler structures

Inspired by the theory of toric Kähler structures, L. Boulanger [11] and Y.
Wang [70, 71] gave a similar description in momentum-angular coordinates of
the ${\mathbb{T}}$-invariant generalized Kähler structures compatible with a
symplectic $2$-form $F$ on a smooth compact toric symplectic manifold $(M,F)$.
In the notation of Section 3.1, we let $(M,\omega_{0},{\mathbb{T}})$ be a
given toric symplectic manifold with momentum map $\mu:M\to{\mathfrak{t}}^{*}$
and $\theta:M^{0}\to{\mathfrak{t}}/2\pi\Lambda$ standard angular coordinates
on $M^{0}$ (i.e. $\theta=\theta^{c}$); choosing a lattice basis of
${\mathfrak{t}}$, we write (see (3.1))

(3.7) $\omega_{0}=\sum_{j=1}^{m}{\mathrmsl d}\mu_{j}\wedge{\mathrmsl
d}\theta_{j}.$

####  3.2.1. Toric generalized Kähler deformations of type $A$

Following [11], we can construct a ${\mathbb{T}}$-invariant almost complex
structure $J$ on $M^{0}$ from a smooth non-degenerate field ${\bf\Psi}(x)$ of
bilinear forms on $\mathring{{\mathrm{P}}}$, with inverse ${\bf\Psi}^{-1}(x)$,
by letting

(3.8) $J:=\sum_{i,j=1}^{m}\Big{(}{\bf\Psi}_{ij}(\mu){\mathrmsl
d}\mu_{i}\otimes\frac{\partial}{\partial\theta_{j}}-({\bf\Psi}^{-1})_{ij}(\mu){\mathrmsl
d}\theta_{i}\otimes\frac{\partial}{\partial\mu_{j}}\Big{)},$

where $({\bf\Psi}_{ij}(x))$ is the Gram matrix (in the fixed basis of
${\mathfrak{t}}$ and ${\mathfrak{t}}^{*}$) of ${\bf\Psi}(x)$. This is
consistent with the description (3.2) of toric Kȧhler structures (which
correspond to the case ${\bf\Psi}(x)={\bf G}(x)={\rm Hess}(u(x))$).

The integrability of the almost complex structure $J$ given by (3.8) reads as
(see the proof of [11, Theorem 6])

${\bf\Psi}_{ij,k}={\bf\Psi}_{ik,j}.$

The almost-complex structure $I=-\omega_{0}^{-1}J^{*}\omega_{0}$ is expressed
in momentum-angular coordinates by

$I=\sum_{i,j=1}^{m}\Big{(}({\bf\Psi}^{\rm T})_{ij}{\mathrmsl
d}\mu_{i}\otimes\frac{\partial}{\partial\theta_{j}}-\big{(}{\bf\Psi}^{\rm
T}\big{)}_{ij}^{-1}{\mathrmsl
d}\theta_{i}\otimes\frac{\partial}{\partial\mu_{j}}\Big{)}.$

Furthermore, $\omega_{0}$ tames $J$ if and only if the the symmetric tensor

(3.9) $g:=-(\omega_{0}J)^{\rm s}=\sum_{i,j=1}^{m}\Big{(}({\bf\Psi}^{\rm
s})_{ij}{\mathrmsl d}\mu_{i}{\mathrmsl d}\mu_{j}+({\bf\Psi}^{-1})^{\rm
s}_{ij}{\mathrmsl d}\theta_{i}{\mathrmsl d}\theta_{j}\Big{)}$

is positive definite on $M^{0}$. In the above formula (and in what follows) we
use upper-indices ${\rm s}$ and ${\rm a}$ to denote respectively the symmetric
and skew-symmetric parts of bilinear forms and the corresponding Gram
matrices. We shall make an abundant use of the following identities which hold
for a non-degenerate matrix ${\bf\Psi}=({\bf\Psi}_{ij})$ (they can be checked
easily using the polar decomposition of ${\bf\Psi}$).

(3.10) $\begin{split}&{\bf\Psi}\big{(}{\bf\Psi}^{-1}\big{)}^{\rm
s}{\bf\Psi}^{\rm T}={\bf\Psi}^{\rm
s},\,\,{\bf\Psi}\big{(}{\bf\Psi}^{-1}\big{)}^{\rm a}{\bf\Psi}^{\rm
T}=-{\bf\Psi}^{\rm a};\\\ &{\bf\Psi}^{\rm s}\big{(}{\bf\Psi}^{-1}\big{)}^{\rm
a}+{\bf\Psi}^{\rm a}\big{(}{\bf\Psi}^{-1}\big{)}^{\rm s}=0,\,\,{\bf\Psi}^{\rm
s}\big{(}{\bf\Psi}^{-1}\big{)}^{\rm s}+{\bf\Psi}^{\rm
a}\big{(}{\bf\Psi}^{-1}\big{)}^{\rm a}={\rm Id},\end{split}$

where ${\bf\Psi}^{\rm T}$ stands for the transposed matrix. It follows from
(3.9) that $g$ is positive definite over $M^{0}$ if and only if
${\bf\Psi}^{\rm s}(x)$ is positive definite over $\mathring{{\mathrm{P}}}$.

A special case when the almost complex structures $J$ and $I$ are both
integrable and tamed by $\omega_{0}$ on $M^{0}$ is obtained by letting

(3.11) ${\bf\Psi}(x)={\rm Hess}(u(x))+A$

where $u(x)$ is a smooth strictly convex function on $\mathring{{\mathrm{P}}}$
and $A\in{}^{2}{\mathfrak{t}}$ is a (constant) skew-symmetric bilinear form on
${\mathfrak{t}}^{*}$, thus extending the Kähler setting which is obtained by
letting $A=0$.

Thus, (3.11) gives rise to a ${\mathbb{T}}$-invariant generalized structure on
$M^{0}$ which is compatible with the symplectic form $\omega_{0}$ in the sense
of (1.3), as mentioned in the introduction. The following result is
established in [11, Theorem 11] for $m=2$, and in [70, Theorems 4.9 & 4.11] in
general.

######  Proposition 3.4.

The complex structure $J_{A}$ defined on $M^{0}$ by (3.8) and ${\bf\Psi}(x)$
of the form (3.11) for some smooth strictly convex function $u(x)$ on
$\mathring{{\mathrm{P}}}$ and $A\in{}^{2}{\mathfrak{t}}$ extends to a globally
defined $\omega_{0}$-compatible generalized Kähler structure
$(g_{A},I_{A},J_{A})$ on $M$ if and only if
$u\in\mathcal{S}({\mathrm{P}},{{\mathrm{L}}})$.

######  Definition 3.5.

The GK structures $(g_{A},J_{A},I_{A})$ obtained from a toric
$\omega_{0}$-compatible Kähler structure $(g_{u},J_{u})$ via Proposition 3.4
are called _generalized Kähler deformations of type $A$_ of $(g_{u},J_{u})$.

For a toric generalized Kähler structure $(g_{A},J_{A},I_{A})$ as above, with
corresponding $u\in\mathcal{S}({\mathrm{P}},{{\mathrm{L}}})$ and
$A\in{}^{2}{\mathfrak{t}}$, we introduce pluriharmonic functions on
$(M^{0},J)$ by

(3.12) $\tilde{y}_{j}(\mu):=u_{,j}(\mu)+\sum_{k=1}^{m}A_{kj}\mu_{k},\qquad
j=1,\dots,m.$

Indeed, it is easily checked from (3.8) that, at each point, $\\{{\mathrmsl
d}\tilde{y}_{1},\ldots,{\mathrmsl d}\tilde{y}_{m},{\mathrmsl
d}\theta_{1},\ldots,{\mathrmsl d}\theta_{m}\\}$ is the dual basis of the basis
of commuting real holomorphic vector fields
$\\{-JK_{1},\ldots,-JK_{m},K_{1},\ldots,K_{m}\\}$. As in the case of
compatible toric Kähler structures, we can think of $\tilde{y}$ as abstract
variables on ${\mathfrak{t}}$, defined by the transformation

(3.13) ${\tilde{y}}:=(\nabla u)(x)+A(x),$

where $A$ is thought of as a linear map from
${\mathfrak{t}}^{*}\to{\mathfrak{t}}$. Using that $u$ is strictly convex and
$A$ is skew-symmetric, one can show as in the Legendre transform (3.4) that
${\tilde{y}}(x)$ is a diffeomorphism from $\mathring{{\mathrm{P}}}$ to
${\mathbb{R}}^{m}$. Similarly to the construction in the Kähler case, we let
$\tilde{x}_{u}$ be the pre-image of $0\in{\mathfrak{t}}$ under (3.13) and
$\tilde{p}_{u}$ the point in $M^{0}$ with $\mu(\tilde{p}_{u})=\tilde{x}_{u}$
and $\theta(\tilde{p}_{u})=0$ so that
$\tilde{z}_{j}=e^{\tilde{y}_{j}+\sqrt{-1}\theta_{j}}$ are identified with
standard holomorphic coordinates on the orbit
$({\mathbb{C}}^{m})\cdot\tilde{p}_{u}=(M^{0},J)$. With this understood, we
have the canonical biholomorphism between two complex structures given by
deformations of type $A$.

######  Lemma 3.6 (Canonical biholomorphism between $(M,J_{A})$ and
$(M,J_{A_{0}})$).

Let $J_{A}$ and $J_{A_{0}}$ be ${\mathbb{T}}$-invariant complex structures on
$M$, obtained by deformations of type $A$ of the Kähler structures
$(g_{u},J_{u})$ and $(g_{u_{0}},J_{u_{0}})$ in
$\mathcal{K}^{{\mathbb{T}}}(M,\omega_{0})$ with corresponding symplectic
potentials $u,u_{0}\in\mathcal{S}({\mathrm{P}},{{\mathrm{L}}})$, respectively.
Then, the ${\mathbb{T}}$-equivariant diffeomorphism $\tilde{\Phi}$ on $M^{0}$,
which sends the complex coordinates $(\tilde{z}_{1},\ldots,\tilde{z}_{m})$ of
$J_{A}$ to the respective complex coordinates
$(\tilde{z}_{1}^{0},\ldots,\tilde{z}_{m}^{0})$ of $J_{A_{0}}$, extends to a
$({\mathbb{C}}^{*})^{m}$-equivariant biholomorphism from $(M,J_{A})$ to
$(M,J_{A_{0}})$. In particular, independent of $u$ and $A$,
$(M,J_{A},{\mathbb{T}})$ is isomorphic, as a complex toric variety, to
$(M,J_{c},{\mathbb{T}})$, where $J_{c}$ is the canonical complex structure in
$\mathcal{K}^{{\mathbb{T}}}(M,\omega_{0})$.

###### Proof.

Given some $J=J_{A}$ as in the statement, we shall build a holomorphic
$({\mathbb{C}}^{*})^{m}$-equivariant atlas on $(M,J)$ consisting of complex
charts $\psi_{v}:M_{v}\to{\mathbb{C}}^{m}$ associated to each vertex $v$ of
${\mathrm{P}}$. To this end, we suppose that the chosen basis $e$ of
${\mathfrak{t}}$, defining a dual basis of ${\mathfrak{t}}^{*}$ and the
holomorphic coordinates $(\tilde{y}_{1},\ldots,\tilde{y}_{m})$ via (3.12) on
$(M^{0},J)$, is a ${\mathbb{Z}}$-basis of $\Lambda$. By Delzant theory [18],
each vertex $v\in{\mathrm{P}}$ corresponds to a point $p_{v}\in M$ fixed by
the action of ${\mathbb{T}}$. We change the momentum map $\mu^{v}:=\mu-v$ by a
translation, so that $\mu(p_{v})=0\in{\mathfrak{t}}^{*}$. Furthermore, we
change the initial ${\mathbb{Z}}$-basis of ${\mathfrak{t}}$ with a
${\mathbb{Z}}$-basis $e_{v}$ consisting of the primitive normals of the labels
$L_{j}\in{{\mathrm{L}}}$ vanishing at $v$, with corresponding transition
matrix of bases $N_{v}=(n_{ij})\in{\bf GL}(m,{\mathbb{Z}})$. Using the new
basis $e_{v}$ and the modified momentum map $\mu^{v}$, we define via (3.12),
new $J$-holomorphic coordinates $(\tilde{z}_{1}^{v},\ldots,\tilde{z}_{m}^{v})$
on $(M^{0},J)$ by
$\tilde{z}^{v}_{k}=e^{(\tilde{y}^{v}_{k}+\sqrt{-1}\theta^{v}_{k})}$. It
follows that on $(M^{0},J)\cong({\mathbb{C}}^{*})^{m}\cdot\tilde{p}_{u}$, the
coordinates $\tilde{z}$ and $\tilde{z}^{v}$ are related by

(3.14) $\displaystyle\tilde{z}^{v}=e^{-\lambda_{v}(A)}\tilde{z}^{N_{v}},$

which is an abbreviation to the map

$(\tilde{z}^{v})_{k}=e^{-(N_{v}A(v))_{j}}(\tilde{z}_{1})^{n_{k1}}(\tilde{z}_{2})^{n_{k2}}\cdots(\tilde{z}_{m})^{n_{km}},\,k=1,\ldots
m.$

We now show that $(\tilde{z}_{1}^{v},\ldots,\tilde{z}_{m}^{v})$ can be
extended from $(M^{0},J)$ to define a $({\mathbb{C}}^{*})^{m}$-equivariant
holomorphic chart on $(M,J)$ centered at $p_{v}$. To this end, notice that
with respect to the basis $e_{v}$ and with the normalization for the momentum
map $\mu^{v}$, the labels $L_{k}\in{{\mathrm{L}}}$ vanishing at
$v=\mu(p_{v})=0$ are $L_{k}(\mu^{v})=\mu^{v}_{k},k=1,\ldots m$. We will show
that $(\tilde{z}_{1}^{v},\ldots,\tilde{z}_{m}^{v})$ extend at any point in the
pre-image $M^{0}_{v}$ by $\mu^{v}$ of the union of $v$ and the interiors of
all faces of ${\mathrm{P}}$ containing the vertex $v$. Let us take a point
$p_{*}$ in the pre-image of the interior $\mathring{\Sigma}_{k}$ of the facet
defined by $\mu^{v}_{k}=0$. (The argument for a face of higher co-dimension is
similar.) Letting $x_{*}=\mu^{v}(p_{*})$, by the second boundary condition for
$u$ in Proposition 3.1, we have that
$u_{,k}(\mu^{v})=\frac{1}{2}\log(\mu^{v}_{k})+\varphi(\mu^{v})$ for a smooth
function $\varphi(x)$ defined in a closed ball $B_{*}$ around $x_{*}$; it thus
follows that on $(\mu^{v})^{-1}(B_{*})\cap M^{0}$, we have
$|\tilde{z}^{v}_{k}|^{2}=(\mu^{v}_{k})e^{\varphi(\mu^{v})}$, showing that
$\tilde{z}^{v}_{k}$ tends to zero for any sequence of points in $M^{0}$ which
converges to $p_{*}$. The same boundary condition also ensures that the
functions $\tilde{y}^{v}_{r}(x)$ with $r\neq k$ are extendable as smooth
functions on $\mathring{\Sigma}_{k}$. Using the first boundary condition of
Proposition 3.1 and the definition [18] of the canonical angular coordinates
$\theta_{r}$ (with $r\neq k$) on $\mu_{v}^{-1}(\mathring{\Sigma}_{k})$, it
follows that
$(\tilde{z}_{1},\ldots,\tilde{z}_{k-1},0,\tilde{z}_{k+1},\ldots,\tilde{z}_{m})$
gives rise to a $({\mathbb{C}}^{*})^{m}$-equivariant coordinate system on
$(\mu^{v})^{-1}(\mathring{\Sigma}_{k})$. Similar arguments on any face
containing $v$ yield that $(\tilde{z}_{1}^{v},\ldots,\tilde{z}_{m}^{v})$
defines a global $({\mathbb{C}}^{*})^{m}$-equivariant chart
$\psi_{v}:M_{v}\to{\mathbb{C}}^{m}$. Considering such charts for all vertices
of ${\mathrm{P}}$ gives rise to a holomorphic atlas on $(M,J)$.

We now let $\psi^{0}_{v}:(M_{v},J_{0})\to{\mathbb{C}}_{m}$ be the
corresponding chart at $p_{v}$ for the complex structure $J_{0}=J_{A_{0}}$
(obtained by a deformation of type $A$ from
$u_{0}\in\mathcal{S}({\mathrm{P}},{{\mathrm{L}}})$ and
$A_{0}\in{}^{2}{\mathfrak{t}}$). Using (3.14), it follows that in the charts
$\psi_{v},\psi_{v}^{0}$, the map $\Phi:(M^{0},J)\to(M^{0},J_{0})$ sending
$(\tilde{z}_{1},\ldots,\tilde{z}_{m})$ to
$(\tilde{z}_{1}^{0},\ldots,\tilde{z}_{m}^{0})$ becomes

$\tilde{z}^{v}\mapsto e^{\lambda_{v}(A_{0}-A)}(\tilde{z}^{0})^{v},$

which is well-defined on the whole ${\mathbb{C}}^{m}$. ∎

####  3.2.2. Toric generalized Kähler deformations of type $B$

Following [70], let $(g_{A},I_{A},J_{A})$ be a generalized Kähler deformation
of type $A$ on $(M,\omega_{0},{\mathbb{T}})$, defined via (3.8) and (3.11) by
$u\in\mathcal{S}({\mathrm{P}},{{\mathrm{L}}})$ and $A\in{}^{2}{\mathfrak{t}}$,
and take another element $B\in{}^{2}{\mathfrak{t}}$. Then the $2$-form

(3.15) $F:=\omega_{0}+\langle{\mathrmsl d}\mu,B,{\mathrmsl
d}\mu\rangle=\sum_{j=1}^{m}{\mathrmsl d}\mu_{i}\wedge{\mathrmsl
d}\theta_{i}+\sum_{i,j=1}^{m}B_{ij}{\mathrmsl d}\mu_{i}\wedge{\mathrmsl
d}\mu_{j}$

is ${\mathbb{T}}$-invariant and symplectic on $M$, with moment map $\mu$ and
associated Delzant polytope $(\Delta,{{\mathrm{L}}})$. In particular,
$(M,\omega_{0},{\mathbb{T}})$ and $(M,F,{\mathbb{T}})$ are isomorphic as
symplectic toric manifolds, but the momentum-angular coordinates of $F$ are
$(\mu,\theta_{F}:=\theta-B(\mu))$.

It is observed in [70] that for any $B\in{}^{2}{\mathfrak{t}}$, the almost
complex structure $I_{A,B}:=-(F)^{-1}(J_{A})^{*}F$ is integrable. Indeed, with
respect to the frame
$(\frac{\partial}{\partial\theta_{i}},\frac{\partial}{\partial\mu_{j}})$ on
$TM^{0}$, $J_{A}$ and $I_{A,B}$ are represented by matrices

(3.16) $J_{A}\sim\left(\begin{array}[]{cc}0&{\bf\Psi}^{\rm T}\\\
-\big{(}{\bf\Psi}^{\rm
T}\big{)}^{-1}&0\end{array}\right),\qquad\,\,I_{A,B}\sim\left(\begin{array}[]{cc}2B{\bf\Psi}^{-1}&{\bf\Psi}+4B{\bf\Psi}^{-1}B\\\
-{\bf\Psi}^{-1}&-2{\bf\Psi}^{-1}B\end{array}\right),$

where ${\bf\Psi}$ is given by (3.11). One can then check from the above
representations that

(3.17) $\overline{y}:=\nabla
u-A(\mu),\qquad\overline{\theta}:=\theta-2B(\mu)=\theta_{F}-B(\mu)$

are pluriharmonic coordinates with respect to $I_{A,B}$ and hence $I_{A,B}$ is
integrable.

Suppose now that $J_{A}$ is tamed by $F$. This gives rise to a
${\mathbb{T}}$-invariant generalized Kähler structure
$(g_{A,B},I_{A,B},J_{A})$, where the induced Riemannian metric
$g_{A,B}:=-\big{(}FJ_{A}\big{)}^{\rm s}$ is represented in the frame
$(\frac{\partial}{\partial\theta_{i}},\frac{\partial}{\partial\mu_{j}})$ on
$TM^{0}$ by the Gram matrix

(3.18)
$\begin{split}g_{A,B}\sim&\left(\begin{array}[]{cc}\big{(}{\bf\Psi}^{-1}\big{)}^{\rm
s}&{\bf\Psi}^{-1}B\\\ -B\big{(}{\bf\Psi}^{\rm T}\big{)}^{-1}&{\bf\Psi}^{\rm
s}\end{array}\right)=\left(\begin{array}[]{cc}{\bf\Psi}^{-1}&0\\\ 0&{\rm
Id}\end{array}\right)\left(\begin{array}[]{cc}{\bf\Psi}^{\rm s}&B\\\
-B&{\bf\Psi}^{\rm
s}\end{array}\right)\left(\begin{array}[]{cc}\big{(}{\bf\Psi}^{\rm
T}\big{)}^{-1}&0\\\ 0&{\rm Id}\end{array}\right).\end{split}$

The equality of matrices in the above formula is deduced by using (3.10). It
follows that $F$ tames $J_{A}$ on $M^{0}$ if and only if the Hermitian matrix

(3.19) $\displaystyle{\rm Hess}(u)+\sqrt{-1}B>0$

is positive definite at any point of $\mathring{{\mathrm{P}}}$. A similar
condition imposed on the interior of each face of ${\mathrm{P}}$ by virtue of
Proposition 3.1 guarantees that $F$ tames $J$ on $M$. Notice that this is
independent of the transformation of type $A$ used to obtain $J$.

######  Definition 3.7.

The generalized Kähler structure $(g_{A,B},I_{A,B},J_{A})$ on $M$ described
above will be called a generalized Kähler deformation of type $B$ of
$(g_{A},I_{A},J_{A})$.

By the results in [71, Sect. 4], we have the following generalization of the
description of toric Kähler metrics.

######  Proposition 3.8.

[71] Any ${\mathbb{T}}$-invariant generalized Kähler structure of symplectic
type on $M$, compatible with a ${\mathbb{T}}$-invariant symplectic form $F$ on
$M$ with associated labelled Delzant polytope $({\mathrm{P}},{{\mathrm{L}}})$,
is ${\mathbb{T}}$-equivariantly isomorphic to a generalized Kähler structure
$(g_{A,B},I_{A,B},J_{A})$, obtained from a toric Kähler metric
$(g_{u},J_{u},\omega_{0})$ corresponding to a symplectic potential
$u\in{\mathcal{S}}({\mathrm{P}},{{\mathrm{L}}})$, by generalized Kähler
transformations of type $A$ and $B$.

######  Convention 1.

In view of the above result, and to ease the notation, we shall omit the
indices $A,B$ and implicitly assume (without loss of generality) that a toric
generalized Kähler structure $(g,I,J)$ compatible with a
${\mathbb{T}}$-invariant symplectic form $F$ on $M$ and corresponding Delzant
polytope $({\mathrm{P}},{{\mathrm{L}}})$ is always obtained from a
${\mathbb{T}}$-invariant Kähler structure
$({\overline{g}}_{u},{\overline{J}}_{u})$, via generalized Kähler deformations
of type $A$ and $B$ for respective elements $A,B\in{}^{2}{\mathfrak{t}}$,
where $u\in\mathcal{S}({\mathrm{P}},{{\mathrm{L}}})$, $\omega_{0}$ is a
symplectic form associated to $({\mathrm{P}},{{\mathrm{L}}})$ via the Delzant
construction, and $({\overline{g}}_{u},{\overline{J}}_{u})$ is a
$\omega_{0}$-compatible Kähler metric with symplectic potential $u$ and
canonical momentum-angular coordinates $(\mu,\theta^{c})$ on
$(M^{0},\omega_{0})$.

The computations (3.13) and (3.17) of the corresponding pluriharmonic
coordinates of $J$ and $I$ underline the general symmetry of the construction:

(3.20) $(u,A,B)\longleftrightarrow(u,-A,-B)$

which, geometrically, corresponds to switching the roles of $J$ and $I$ of the
corresponding toric GK structure $(g,I,J)$. Indeed, this becomes apparent if
we express the construction in the momentum/angular coordinates
$(\mu,\theta_{F})$ of $F$. Then, the corresponding Kähler structures
$\overline{\omega}_{{\overline{J}}_{u}}$ and
$\overline{\omega}_{{\overline{I}}_{u}}$ are linked to the common symplectic
form $F$ via the relations

(3.21)
$\begin{split}\overline{\omega}_{{\overline{J}}_{u}}&=\sum_{j=1}^{m}{\mathrmsl
d}\mu_{j}\wedge{\mathrmsl d}\theta_{j}=F-\sum_{i,j=1}^{m}B_{ij}{\mathrmsl
d}\mu_{i}\wedge{\mathrmsl d}\mu_{j}\\\
\overline{\omega}_{{\overline{I}}_{u}}&=\sum_{j=1}^{m}{\mathrmsl
d}\mu_{j}\wedge{\mathrmsl
d}\overline{\theta}_{j}=F+\sum_{i,j=1}^{m}B_{ij}{\mathrmsl
d}\mu_{i}\wedge{\mathrmsl d}\mu_{j}.\\\
F&=\tfrac{1}{2}(\overline{\omega}_{{\overline{I}}_{u}}+\overline{\omega}_{{\overline{J}}_{u}}),\end{split}$

where we recall $\overline{\theta}$ are defined in (3.17).

####  3.2.3. The case $A=0$

When $A\neq 0$, the integrable almost complex structures $J=J_{A}$ and
${\overline{J}}_{u}$ associated respectively to the generalized Kähler
structure $(g,I,J)$ and the Kähler structure
$({\overline{g}}_{u},{\overline{J}}_{u})$ are different (even though they are
${\mathbb{T}}$-equivariantly biholomorphic by virtue of Lemma 3.6). In the
special case $A=0$, we have ${\overline{J}}_{u}=J$ and ${\overline{I}}_{u}=I$.
We shall later make use of this simplification, so we recast below some facts
specific to this case.

First, letting

(3.22)
$\Pi:=-\sum_{i,j=1}^{m}B_{ij}\frac{\partial}{\partial\theta_{i}}\wedge\frac{\partial}{\partial\theta_{j}}$

denote the real Poisson structure on $M$ determined by
$B\in{}^{2}{\mathfrak{t}}$, the formula (3.15) yields the following basic
relation between $\overline{\omega}$ and $\omega$ on $(M,J)$:

(3.23) $F=(\overline{\omega}^{-1}+\Pi)^{-1},\qquad\omega=F^{1,1}.$

It will be also convenient in this case to express the geometry in terms of
the $J$-pluriharmonic coordinates $(y_{j},\theta_{j})$ obtained via the
Legendre transform of $u$ (see (3.4) and Proposition 3.3). Setting
$y_{j}=u_{,j}(\mu)$ and $\phi(y)+u(\mu)=\sum_{j=1}^{m}\mu_{j}y_{j}$, we have
on $(M^{0},J)$ (see (3.6)):

$\overline{\omega}={\mathrmsl d}{\mathrmsl
d}^{c}\phi=\sum_{i,j=1}^{m}\Big{(}{\rm Hess}(\phi(y))\Big{)}_{ij}{\mathrmsl
d}y_{i}\wedge d\theta_{j},$

whereas the symplectic $2$-form $F$ and the fundamental $2$-form
$\omega=F^{1,1}$ are given by:

(3.24) $\begin{split}F&=\sum_{i,j=1}^{m}\Big{(}\phi_{,ij}{\mathrmsl
d}y_{i}\wedge{\mathrmsl d}\theta_{j}+\Big{[}\big{(}{\rm
Hess}(\phi)\big{)}B\big{(}{\rm Hess}(\phi)\big{)}\Big{]}_{ij}{\mathrmsl
d}y_{i}\wedge{\mathrmsl d}y_{j}\Big{)}\\\
&=\overline{\omega}+\overline{\omega}\Pi\overline{\omega};\\\
\omega&=\sum_{i,j=1}^{m}\Big{(}\phi_{,ij}{\mathrmsl d}y_{i}\wedge{\mathrmsl
d}\theta_{j}+\frac{1}{2}\Big{[}\big{(}{\rm Hess}(\phi)\big{)}B\big{(}{\rm
Hess}(\phi)\big{)}\Big{]}_{ij}({\mathrmsl d}y_{i}\wedge{\mathrmsl
d}y_{j}+{\mathrmsl d}\theta_{i}\wedge{\mathrmsl d}\theta_{j})\Big{)}\\\
&=\overline{\omega}+\overline{\omega}\left(\Pi^{1,1}\right)\overline{\omega}.\end{split}$

In this case, $(y_{j},\overline{\theta}_{j})$ are pluriharmonic coordinates of
$I$ (see (3.17)), so similar formulae hold with respect to $I$ (in which one
should replace $B$ by $-B$ and $\theta_{j}$ by ${\overline{\theta}}_{j}$). We
also notice that

(3.25) ${\mathrmsl d}{\mathrmsl
d}^{c}_{J}\phi=\overline{\omega}_{J},\qquad{\mathrmsl d}{\mathrmsl
d}^{c}_{I}\phi=\overline{\omega}_{I},$

so, by (3.21), we get

(3.26) $F=\tfrac{1}{2}({\mathrmsl d}{\mathrmsl d}^{c}_{J}\phi+{\mathrmsl
d}{\mathrmsl d}^{c}_{I}\phi).$

####  3.2.4. The holomorphic Poisson structure of a toric generalized Kähler
structure

The next result gives a geometric meaning of the parameters
$A,B\in{}^{2}{\mathfrak{t}}$. In particular, they completely determine the
associated holomorphic Poisson structures.

######  Proposition 3.9.

Let $(g,I,J)$ be a toric GK structure associated to the data $(u,A,B$). Then
the corresponding holomorphic Poisson structure on $(M,J)$ given by (1.4) is

$\sigma_{J}=2\sum_{i,j=1}^{m}\big{(}A_{ij}+\sqrt{-1}B_{ij}\big{)}\big{(}K_{i}-\sqrt{-1}JK_{i}\big{)}\wedge\big{(}K_{j}-\sqrt{-1}JK_{j}\big{)},$

where $K_{1},\ldots,K_{m}$ are the fundamental vector fields associated to a
basis of ${\mathfrak{t}}$, and $A_{ij}$ and $B_{ij}$ are the corresponding
components of $A$ and $B$ in this basis.

###### Proof.

The above formula follows by straightforward but somewhat tedious matrix
computations, using that $K_{j}=\frac{\partial}{\partial\theta_{j}}$, (3.16),
the identification (which in turn follows from (3.18))

$g^{-1}\sim\left(\begin{array}[]{cc}{\bf\Psi}^{\rm T}&0\\\ 0&{\rm
Id}\end{array}\right)\left(\begin{array}[]{cc}{\mathbf{X}}&{\mathbf{Y}}\\\
-{\mathbf{Y}}&{\mathbf{X}}\end{array}\right)\left(\begin{array}[]{cc}{\bf\Psi}&0\\\
0&{\rm Id}\end{array}\right),$

where

${\mathbf{X}}:=\Big{(}{\bf\Psi}^{\rm s}+B\big{(}{\bf\Psi}^{\rm
s}\big{)}^{-1}B\Big{)}^{-1},\,\ {\mathbf{Y}}:=-\big{(}{\bf\Psi}^{\rm
s}\big{)}^{-1}B{\mathbf{X}},$

and the identities (3.10). ∎

###  3.3. The Bismut-Ricci form of a toric generalized Kähler structure

M. Abreu [1] obtained the expression of the Ricci form $\rho_{J}$ of a Kähler
metric $(g,J)\in\mathcal{K}^{{\mathbb{T}}}(M,\omega_{0})$ in terms of the
momentum/angular coordinates and the corresponding
$u\in\mathcal{S}({\mathrm{P}},{{\mathrm{L}}})$ as follows

(3.27) $\rho_{J}=\tfrac{1}{2}{\mathrmsl d}{\mathrmsl
d}^{c}_{J}\Big{(}\log\det\big{(}{\rm
Hess}(u)\big{)}(\mu)\Big{)}=-\tfrac{1}{2}\sum_{i,j,k=1}^{m}({\bf
G}^{-1})_{ij,ik}(\mu){\mathrmsl d}\mu_{k}\wedge{\mathrmsl d}\theta_{j}.$

In this subsection we derive the Bismut-Ricci forms
$\rho^{B}_{I},\rho^{B}_{J}$ associated to a toric generalized Kähler structure
$(g,I,J)$, using momentum-angular coordinates of $\omega_{0}$, thus extending
(3.27).

######  Lemma 3.10.

Let $(g,I,J)$ be a toric generalized Kähler structure, associated to the data
$(u,A,B)$. Then, the Bismut-Ricci forms $\rho_{I}^{B},\rho^{B}_{J}$ of $(g,J)$
are given on $M^{0}$ by

$\begin{split}\rho^{B}_{I}=&\ \tfrac{1}{2}{\mathrmsl d}{\mathrmsl
d}^{c}_{J}\Big{(}\log\det\big{(}{\rm
Hess}(u)\big{)}(\mu)+\sqrt{-1}B\Big{)},\\\ \rho^{B}_{J}=&\
\tfrac{1}{2}{\mathrmsl d}{\mathrmsl d}^{c}_{I}\Big{(}\log\det\big{(}{\rm
Hess}(u)\big{)}(\mu)+\sqrt{-1}B\Big{)}.\end{split}$

###### Proof.

We are going to use Proposition 2.8 with respect to the holomorphic section
(defined on $M^{0}$)

$\Theta_{J}:=\left((K_{1}-\sqrt{-1}JK_{1})\wedge\cdots\wedge(K_{m}-\sqrt{-1}JK_{m})\right)^{-1}.$

To this end, recall that $K_{j}=\frac{\partial}{\partial\theta_{j}}$ and, by
(3.16) and (3.18), we have (in momentum/angular coordinates)

(3.28)
$\omega_{J}\sim\left(\begin{array}[]{cc}{\bf\Psi}^{-1}B\big{(}{\bf\Psi}^{\rm
T}\big{)}^{-1}&-{\bf\Psi}^{-1}{\bf\Psi}^{\rm s}\\\ {\bf\Psi}^{\rm
s}\big{(}{\bf\Psi}^{\rm T}\big{)}^{-1}&B\end{array}\right),$

so we compute, by using the above together with (3.18), (3.28), (3.10)

$\displaystyle\begin{split}2^{-m}\left(\frac{\omega_{J}^{[m]}}{v_{\Theta}}\right)&=\det\Big{(}g(K_{i},K_{j})-\sqrt{-1}\omega_{J}(K_{i},K_{j})\Big{)}\\\
&=\det\Big{(}\big{(}{\bf\Psi}^{\rm T}\big{)}^{-1}\big{(}{\bf\Psi}^{\rm
s}-\sqrt{-1}B\big{)}{\bf\Psi}^{-1}\Big{)}=\frac{\det\big{(}{\rm
Hess}(u)+\sqrt{-1}B\big{)}}{\big{(}\det{\bf\Psi}\big{)}^{2}},\end{split}$

As $\omega_{0}^{[m]}=F^{[m]}$ (see (3.7) and (3.15)), using (3.7) and (3.28),
we also get

$\left(\frac{F^{[m]}}{\omega_{J}^{[m]}}\right)=\left(\frac{\omega_{0}^{[m]}}{\omega_{J}^{[m]}}\right)=\frac{\det{\bf\Psi}}{\det\left(\mathrm{Hess}(u)+\sqrt{-1}B\right)},$

and therefore

(3.29)
$\begin{split}\left[\log\left(\frac{F^{[m]}}{v_{\Theta}}\right)+\log\left(\frac{F^{[m]}}{\omega_{J}^{[m]}}\right)\right]&=\left[2\log\left(\frac{F^{[m]}}{\omega_{J}^{[m]}}\right)+\log\left(\frac{\omega_{J}^{[m]}}{v_{\Theta}}\right)\right]\\\
&=-\log\det\left(\mathrm{Hess}(u)+\sqrt{-1}B\right)+\mbox{const}.\end{split}$

This yields the formula for $\rho^{B}_{I}$ via Proposition 2.8. Using the
symmetry (3.20) and that $\det({\rm Hess}(u)+\sqrt{-1}B)=\det({\rm
Hess}(u)-\sqrt{-1}B)$, we get the formula for $\rho^{B}_{J}$. ∎

We now restrict to the special case when the compact toric symplectic manifold
$(M,\omega_{0},{\mathbb{T}})$ satisfies the (topological) Fano condition $2\pi
c_{1}(M,J_{u})=[\omega_{0}]$ for any
$J_{u}\in\mathcal{K}^{{\mathbb{T}}}(M,\omega_{0})$. It is well-known that this
is equivalent to the assumption that $({\mathrm{P}},{{\mathrm{L}}})$ satisfies

(3.30) $\displaystyle L_{j}(0)=1,\quad\,j=1,\ldots d.$

We refer to a Delzant polytope $({\mathrm{P}},{{\mathrm{L}}})$ satisfying
(3.30) as barycentred. In this setting, by (3.27) and (3.6), we have for any
$\omega_{0}$-compatible toric Kähler structure $(g_{u},J_{u})$

(3.31) $\rho_{J_{u}}-\omega_{0}={\mathrmsl d}{\mathrmsl
d}^{c}\Big{(}\tfrac{1}{2}\log\det\big{(}{\rm
Hess}(u(\mu))\big{)}-(-u(\mu)+\sum_{i=1}^{m}\mu_{i}u_{,i}(\mu))\Big{)}.$

Using the boundary conditions for $u$ in Proposition 3.1 (which imply that
$\det{\rm Hess}(u)=\frac{\delta(x)}{\prod_{j=1}^{d}L_{j}(x)}$ for a positive
smooth function $\delta$ on ${\mathrm{P}}$, see [2]) and the Fano condition
(3.30), the function

(3.32) $h_{u}(\mu):=\tfrac{1}{2}\log\det\big{(}{\rm
Hess}(u(\mu))\big{)}-(-u(\mu)+\sum_{i=1}^{m}\mu_{i}u_{,i}(\mu))$

appearing at the RHS of (3.31) is smooth on $M$. It thus follows that
$\rho_{J_{u}}-\omega_{0}={\mathrmsl d}{\mathrmsl d}^{c}_{J_{u}}h_{u}$
_globally_ on $M$.

Similar to the Kähler toric Fano case, a generalized Kähler toric structure
corresponding to data $(u,A,B)$ also admits a relative Ricci potential:

######  Lemma 3.11.

Suppose $({\mathrm{P}},{{\mathrm{L}}})$ is a barycentred Delzant polytope
corresponding to a toric Fano variety. Then, for any toric GK structure
$(g,I,J)$ corresponding to the data $(u,A,B)$, the function

$\begin{split}{{h}}_{u,B}(x)&:=\tfrac{1}{2}\log\det\left({\rm
Hess}(u(x))+\sqrt{-1}B\right)-\big{(}-u(x)+\sum_{j=1}^{m}x_{j}u_{,j}(x)\big{)}\\\
&=\tfrac{1}{2}\log\det\left(\big{(}{\rm
Hess}(\phi(y))\big{)}^{-1}+\sqrt{-1}B\right)-\phi(y)\end{split}$

pulls-back by the momentum map $\mu$ of $F$ to a smooth function on $M$.
Furthermore, if $A=0$ we have

$\begin{split}&\rho^{B}_{J}-\overline{\omega}_{I}={\mathrmsl d}{\mathrmsl
d}^{c}_{I}(h_{u,B}(\mu)),\qquad\rho^{B}_{I}-\overline{\omega}_{J}={\mathrmsl
d}{\mathrmsl d}^{c}_{J}(h_{u,B}(\mu)),\\\
&\tfrac{1}{2}(\rho^{B}_{J}+\rho^{B}_{I})-F=\tfrac{1}{2}\big{(}{\mathrmsl
d}{\mathrmsl d}^{c}_{I}(h_{u,B}(\mu))+{\mathrmsl d}{\mathrmsl
d}^{c}_{J}(h_{u,B}(\mu))\big{)}.\end{split}$

###### Proof.

The smoothness of ${{h}}_{u,B}(\mu)$ follows from the smoothness of
$h_{u}(\mu)$ in (3.32), and the fact that $\det({\rm Hess}(u))/\det({\rm
Hess}(u)+\sqrt{-1}B)$ is smooth and positive on $M$ as it computes the ratio
$F^{m}/\omega_{J}^{m}$, where $(F,\omega_{J})$ is the GK structure obtained by
a $B$-transformation of the Kähler structure
$({\overline{g}}_{u},\omega_{0})$. The Bismut-Ricci curvature formulas follow
from Lemma 3.10, (3.25), and (3.26). ∎

##  4\. The normalized generalized Kähler-Ricci flow on a toric Fano variety

###  4.1. Scalar Reduction

We now describe the NGKRF (1.2) with $\lambda=1$, starting from a toric
generalized Kähler structure $(g,I,J)$ on a smooth toric Fano manifold
$(M,\omega_{0})$. We suppose that the initial symplectic form $F$ belongs to
$2\pi c_{1}(M,J_{0})$, and that $(g,I,J)$ satisfies Convention 1 (so we have
$\overline{\omega}=\omega_{0}\in 2\pi c_{1}(M,J)$ as well). The solution of
(1.2) is defined in terms of Hermitian metrics $(g_{t},J)$ on a fixed complex
manifold $(M,J)$, whereas Convention 1 normalizes (via the action of a
${\mathbb{T}}$-equivariant diffeomorphism) the toric GK structures on $M$ to
be obtained from a Kähler structure $({\overline{g}}_{u},{\overline{J}}_{u})$
compatible with the fixed symplectic form $\omega_{0}$ and canonical angular
coordinates via deformations of type $A$ and $B$. This is similar to the
symplectic versus complex viewpoints in the Kähler case (see Proposition 3.3),
and the ${\mathbb{T}}$-equivariant diffeomorphism in Lemma 3.6 connects the
two points of view in the generalized Kähler case. The next proposition makes
this explicit.

######  Proposition 4.1.

Let $(g_{0},I_{0},J_{0})$ be a toric GK structure on a Fano toric variety,
corresponding to the data $(u_{0},A_{0},B_{0})$ with respect to a symplectic
form $\omega_{0}\in 2\pi c_{1}(M,J_{0})$. Denote by $(g_{t},I_{t},J_{t})$ the
family of GK structures corresponding to the solution $(u_{t},A_{t},B_{t})$ of

(4.1) $\begin{split}\frac{\partial}{\partial t}u=&\
\log\Big{(}{\det}\big{(}{\rm
Hess}(u)+\sqrt{-1}e^{-2t}B_{0}\big{)}\Big{)}-2\Big{(}-u+\sum_{j=1}^{m}x_{j}u_{,j}\Big{)},\\\
A_{t}=&\ e^{-2t}A_{0},\qquad B_{t}=e^{-2t}B_{0}.\end{split}$

Then $\Phi_{t}^{*}(g_{t},I_{t},J_{t})$ solves (1.2), where $\Phi_{t}^{*}$ is
the canonical biholomorphism between $(M,J_{0})$ and $(M,J_{t})$ from Lemma
3.6.

###### Proof.

We shall do computations on $M^{0}$, using by Lemma 3.11 that for
$u_{t}\in\mathcal{S}({\mathrm{P}},{{\mathrm{L}}})$ the RHS of (4.1) is a
smooth function on ${\mathrm{P}}$, so (4.1) is a flow of elements
$u_{t}\in\mathcal{S}({\mathrm{P}},{{\mathrm{L}}})$ defined for $t\in[0,T)$,
thus giving rise to a smooth path of GK structures
$(g_{t},I_{t},J_{t}),t\in[0,T)$ on $M$, obtained as deformations of type $A$
and $B$ of the Kähler structures $(g_{u_{t}},J_{u_{t}})$ in
$\mathcal{K}^{{\mathbb{T}}}(M,\omega_{0})$ with the $2$-vectors
$A_{t},B_{t}\in{}^{2}{\mathfrak{t}}$.

We denote by $J_{t}$ the complex structure defined by the deformation of type
$A$ of the complex structures
$J_{u_{t}}\in\mathcal{K}^{{\mathbb{T}}}(M,\omega_{0})$ with the $2$-vector
$A_{t}=e^{-2t}A_{0}\in{}^{2}{\mathfrak{t}}$, and by
$F_{t}=\omega_{0}+e^{-2t}\langle{\mathrmsl d}\mu,B_{0},d\mu\rangle$ the
symplectic form of $(g_{t},J_{t},I_{t})$, with $I_{t}$ being the
$F_{t}$-conjugate of the almost complex structure $J_{t}$. Denote by
$\tilde{y}_{t}$ the $J_{t}$-pluriharmonic variables (3.12), and consider the
${\mathbb{T}}$-equivariant diffeomorphism $\Phi_{t}$ defined in Lemma 3.6,
which satisfies $\Phi_{t}(\tilde{y}_{t})=\tilde{y}_{0}$ and
$\Phi_{t}^{*}(J_{t})=J_{0}=:J$. We let $\tilde{F}_{t}:=\Phi_{t}^{*}(F_{t})$,
so that $\tilde{F}_{t}$ is a symplectic form on $(M,J)$ which gives rise to a
GK structure $(\tilde{g}_{t},\tilde{I}_{t},J)$ compatible with
$\tilde{F}_{t}$. We will check that with $(u_{t},A_{t},B_{t})$ satisfying
(4.1), the symplectic forms $\tilde{F}_{t}$ satisfy

(4.2) $\frac{\partial}{\partial
t}\tilde{F}_{t}=-2\rho_{J}^{B}(\tilde{g}_{t})+2\tilde{F}_{t}.$

Taking the $(1,1)$ part with respect to $J$ of (4.2) yields (1.2) with
$\lambda=1$.

For a toric GK structure $(g,I,J)$, corresponding to the data $(u,A,B)$, we
consider the pluriharmonic variables $\tilde{y}$ with respect to $J$, defined
in (3.12), and a function $\tilde{\phi}(\tilde{y})=\phi(y)$, introduced on
$M^{0}$ by

(4.3)
$\tilde{\phi}(\tilde{y}):=-u(\mu)+\langle\mu,\tilde{y}\rangle=-u(\mu)+\langle\mu,y\rangle=\phi(y),$

where $y_{j}=u_{,j}(\mu)$, and $\phi(y)$ is the Legendre transform (3.4) of
$u$. Using (3.16) we obtain the following formula (the detailed verification
of which we leave to the reader)

(4.4) ${\mathrmsl d}{\mathrmsl d}^{c}_{J}\tilde{\phi}=F+{\mathrmsl
d}J\Big{(}\sum_{i,j=1}^{m}A_{ij}\mu_{i}{\mathrmsl d}\mu_{j}\Big{)}-{\mathrmsl
d}\Big{(}\sum_{i,j=1}^{m}B_{ij}\mu_{i}\wedge{\mathrmsl d}\mu_{j}\Big{)}.\\\ $

Using (4.4), we have that

$\begin{split}\tilde{F}_{t}=&\ \Phi_{t}^{*}\Big{(}{\mathrmsl d}{\mathrmsl
d}^{c}_{J_{t}}{\tilde{\phi}}_{t}({\tilde{y}}_{t})-{\mathrmsl
d}J_{t}\big{(}\sum_{i,j=1}^{m}\big{(}A_{t})_{ij}\mu_{i}{\mathrmsl
d}\mu_{j}\big{)}+\sum_{i,j=1}^{m}(B_{t})_{ij}{\mathrmsl
d}\mu_{i}\wedge{\mathrmsl d}\mu_{j}\Big{)}\\\ =&\ {\mathrmsl d}{\mathrmsl
d}^{c}_{J}{\tilde{\phi}}_{t}({\tilde{y}}_{0})-{\mathrmsl
d}J\big{(}\sum_{i,j=1}^{m}\big{(}A_{t})_{ij}\mu^{t}_{i}{\mathrmsl
d}\mu^{t}_{j}\big{)}+\sum_{i,j=1}^{m}(B_{t})_{ij}{\mathrmsl
d}\mu^{t}_{i}\wedge{\mathrmsl d}\mu^{t}_{j}\Big{)},\end{split}$

where $\mu^{t}:=\mu\circ\Phi_{t}$. Using (4.1), we get

(4.5) $\begin{split}\frac{\partial}{\partial t}\tilde{F}_{t}=&\
\frac{\partial}{\partial t}\Big{(}{\mathrmsl d}{\mathrmsl
d}^{c}_{J}\tilde{\phi}_{t}({\tilde{y}}_{0})-{\mathrmsl
d}J\big{(}\sum_{i,j=1}^{m}(A_{t})_{ij}\mu_{i}^{t}{\mathrmsl
d}\mu_{j}^{t}\big{)}+\sum_{i,j=1}^{m}(B_{t})_{ij}{\mathrmsl
d}\mu^{t}_{i}\wedge{\mathrmsl d}\mu^{t}_{j}\Big{)}\\\ =&\ {\mathrmsl
d}J{\mathrmsl d}\Big{(}\frac{\partial\tilde{\phi}_{t}}{\partial
t}(\tilde{y}_{0})\Big{)}+2{\mathrmsl
d}J\Big{(}\sum_{i,j=1}^{m}(A_{t})_{ij}\mu_{i}^{t}{\mathrmsl
d}\mu_{j}^{t}\Big{)}-2\sum_{i,j=1}^{m}(B_{t})_{ij}{\mathrmsl
d}\mu^{t}_{i}\wedge{\mathrmsl d}\mu^{t}_{j}\\\ &\ \qquad-{\mathrmsl
d}J\Big{[}\sum_{i,j=1}^{m}\Big{(}(A_{t})_{ij}\big{(}\frac{\partial{\mu}_{i}^{t}}{\partial
t}\big{)}{\mathrmsl d}\mu_{j}^{t}+(A_{t})_{ij}{\mu}_{i}^{t}{\mathrmsl
d}\big{(}\frac{\partial{\mu}_{j}^{t}}{\partial t}\big{)}\Big{)}\Big{]}\\\ &\
\qquad+{\mathrmsl
d}\Big{[}\sum_{i,j=1}^{m}\Big{(}(B_{t})_{ij}\big{(}\frac{\partial{\mu}_{i}^{t}}{\partial
t}\big{)}{\mathrmsl d}\mu_{j}^{t}+(B_{t})_{ij}{\mu}_{i}^{t}{\mathrmsl
d}\big{(}\frac{\partial{\mu}_{j}^{t}}{\partial
t}\big{)}\Big{)}\Big{]}.\end{split}$

We thus need to compute the $t$ derivatives of $\tilde{\phi}_{t}$ and
$\tilde{\mu}^{t}$. We start with computing the $t$ derivative of
$\tilde{\phi}_{t}$. By (3.12), (4.3) and (4.1), we first compute

(4.6) $\frac{\partial\tilde{y}_{r}^{t}}{\partial t}=\big{(}\log\det\big{(}{\rm
Hess}(u_{t})+\sqrt{-1}B_{t}\big{)}\big{)}_{,r}-2\sum_{\ell=1}^{m}\big{(}{\bf\Psi}_{t}^{\rm
T}\big{)}_{r\ell}\mu_{\ell}.$

Therefore, we have

(4.7) $\begin{split}\frac{\partial\tilde{\phi}_{t}}{\partial
t}(\tilde{y}_{t})=&\
-\sum_{r=1}^{m}\Big{(}\frac{\partial{\tilde{\phi}}_{t}}{\partial{\tilde{y}}_{r}}\Big{)}\Big{(}\frac{\partial\tilde{y}_{r}}{\partial
t}\Big{)}-\frac{\partial u_{t}}{\partial
t}(\mu)+\sum_{r=1}^{m}\mu_{r}\Big{(}\frac{\partial\tilde{y}_{r}}{\partial
t}\Big{)}\\\ =&\ -\frac{\partial u_{t}}{\partial
t}(\mu)-\sum_{k,r=1}^{m}\big{(}A_{t}\big{(}{\bf\Psi}_{t}^{\rm
T}\big{)}^{-1}\big{)}_{kr}\big{(}\log\det\big{(}{\rm
Hess}(u_{t})+\sqrt{-1}B_{t}\big{)}\big{)}_{,r}\mu_{k}\\\ =&\
-\log\det\big{(}{\rm
Hess}(u_{t})+\sqrt{-1}B_{t}\big{)}+2\tilde{\phi}_{t}(\tilde{y}_{t})\\\ &\
\qquad-\sum_{k,r=1}^{m}\big{(}A_{t}{\big{(}\bf\Psi}_{t}^{\rm
T}\big{)}^{-1}\big{)}_{kr}\big{(}\log\det\big{(}{\rm
Hess}(u_{t})+\sqrt{-1}B_{t}\big{)}\big{)}_{,r}\mu_{k},\end{split}$

where for passing from the first line to the second we have used (4.6) and the
fact that $A_{ij}$ is skew-symmetric, and for passing from the second line to
the third we have used (4.1) and the fact that
$\tilde{\phi}_{t}(\tilde{y}_{t})=\phi_{t}(y_{t})$ (see (4.3)).

We now compute the derivative of $\mu^{t}=\mu\circ\Phi_{t}$. Using that
$\mu_{j}^{t}(\tilde{y}_{t})=\mu_{j}$ and (4.6), we get

(4.8) $\begin{split}\frac{\partial{\mu}_{j}^{t}}{\partial
t}(\tilde{y}_{t})&=2\mu_{j}-\sum_{r=1}^{m}\big{(}{\bf\Psi}_{t}^{\rm
T}\big{)}^{-1}_{jr}\big{(}\log{\det}\big{(}{\rm
Hess}(u_{t})+\sqrt{-1}B_{t}\big{)}\big{)}_{,r}.\end{split}$

Thus, substituting back in (4.5) and using (4.4), we obtain

$\begin{split}&\big{(}\Phi_{t}^{*})^{-1}\Big{(}\frac{\partial}{\partial
t}\tilde{F}_{t}\Big{)}\\\ &\qquad={\mathrmsl d}{\mathrmsl
d}^{c}_{J_{t}}\Big{[}2\tilde{\phi}_{t}-\log\det\big{(}{\rm
Hess}(u_{t})+\sqrt{-1}B_{t}\big{)}-\sum_{k,r=1}^{m}\big{(}A_{t}\big{(}{\bf\Psi}_{t}^{\rm
T}\big{)}^{-1}\big{)}_{kr}\mu_{k}\big{(}\log\big{(}\det{\rm
Hess}(u_{t})+\sqrt{-1}B_{t}\big{)}\big{)}_{,r}\Big{]}\\\
&\qquad\quad-2{\mathrmsl
d}J_{t}\Big{[}\sum_{k,r=1}^{m}(A_{t})_{kr}\mu_{k}{\mathrmsl
d}\mu_{r}\Big{]}+2{\mathrmsl
d}\Big{[}\sum_{k,r=1}^{m}(B_{t})_{kr}\mu_{k}{\mathrmsl d}\mu_{r}\Big{]}\\\
&\qquad\quad-{\mathrmsl
d}J_{t}\Big{[}\sum_{k,r=1}^{m}\big{(}A_{t}\big{(}{\bf\Psi}_{t}^{\rm
T}\big{)}^{-1}\big{)}_{kr}\big{(}\log\det\big{(}{\rm
Hess}(u_{t})+\sqrt{-1}B_{t}\big{)}\big{)}_{,r}{\mathrmsl d}\mu_{k}\Big{]}\\\
&\qquad\quad+{\mathrmsl d}J_{t}\Big{[}\sum_{k,r=1}^{m}\mu_{k}{\mathrmsl
d}\Big{(}\big{(}A_{t}\big{(}{\bf\Psi}_{t}^{\rm
T}\big{)}^{-1}\big{)}_{kr}\big{(}\log\det\big{(}{\rm
Hess}(u_{t})+\sqrt{-1}B_{t}\big{)}\big{)}_{,r}\Big{)}\Big{]}\\\
&\qquad\quad+2{\mathrmsl
d}\Big{[}\sum_{k,r=1}^{m}\big{(}B_{t}\big{(}{\bf\Psi}_{t}^{\rm
T}\big{)}^{-1}\big{)}_{kr}\big{(}\log\det\big{(}{\rm
Hess}(u_{t})+\sqrt{-1}B_{t}\big{)}\big{)}_{,r}{\mathrmsl
d}\mu_{k}\Big{]}.\end{split}$

Furthermore, using Lemma 3.10 and (3.16), we finally have

$\begin{split}&\big{(}\Phi_{t}^{*})^{-1}\Big{(}\frac{\partial}{\partial
t}\tilde{F}_{t}\Big{)}=2F_{t}-{\mathrmsl d}J_{t}{\mathrmsl
d}\Big{[}\log\det\big{(}{\rm Hess}(u_{t})+\sqrt{-1}B_{t}\big{)}\Big{]}\\\
&\qquad\quad-2{\mathrmsl
d}\Big{[}\sum_{\ell,r=1}^{m}\Big{(}{\bf\Psi}_{t}^{-1}A_{t}\big{(}{\bf\Psi}_{t}^{\rm
T}\big{)}^{-1}\big{)}_{\ell r}\big{(}\log\det\big{(}{\rm
Hess}(u_{t})+\sqrt{-1}B_{t}\big{)}\big{)}_{,r}{\mathrmsl
d}\theta_{\ell}\Big{)}\Big{]}\\\ &\qquad\quad+2{\mathrmsl
d}\Big{[}\sum_{k,r=1}^{m}\big{(}B_{t}\big{(}{\bf\Psi}_{t}^{\rm
T}\big{)}^{-1}\big{)}_{kr}\big{(}\log\det\big{(}{\rm
Hess}(u_{t})+\sqrt{-1}B_{t}\big{)}\big{)}_{,r}{\mathrmsl d}\mu_{k}\Big{]}\\\
&\qquad=2F_{t}-{\mathrmsl d}J_{t}{\mathrmsl d}\Big{[}\log\det\big{(}{\rm
Hess}(u_{t})+\sqrt{-1}B_{t}\big{)}\Big{]}-2{\mathrmsl
d}\Big{[}\sum_{\ell,r=1}^{m}\big{(}{\bf\Psi}_{t}^{-1}\big{)}^{\rm
a}_{r\ell}\big{(}\log\det{\rm
Hess}(u_{t})+\sqrt{-1}B_{t}\big{)}_{,r}{\mathrmsl
d}\theta_{\ell}\Big{)}\Big{]}\\\ &\qquad\quad-2{\mathrmsl
d}\Big{[}\sum_{k,r=1}^{m}\big{(}{\bf\Psi}_{t}^{-1}B_{t}\big{)}_{rk}\big{(}\log\det\big{(}{\rm
Hess}(u_{t})+\sqrt{-1}B_{t}\big{)}\big{)}_{,r}{\mathrmsl d}\mu_{k}\Big{]}\\\
&\qquad=2F_{t}-{\mathrmsl d}{I_{t}}{\mathrmsl d}\Big{[}\log\det\big{(}{\rm
Hess}(u_{t})+\sqrt{-1}B_{t}\big{)}\Big{]}=2F_{t}-2\rho^{B}_{J_{t}}(g_{t}),\end{split}$

as required. ∎

We apply the Legendre transform (3.4) to $u_{t}(x)$ and use the formula (3.5)
to rewrite the evolution equation (4.1) of symplectic potentials $u_{t}(x)$ as
an evolution equation of convex functions $\phi_{t}(y)$ on ${\mathbb{R}}^{m}$
(similar to the reduction (4.10) of the normalized Kähler-Ricci flow):

######  Proposition 4.2.

Let $(u_{t},A_{t},B_{t})$ denote a solution to (4.1) on a toric Fano manifold.
The associated one-parameter family of potentials $\phi_{t}(y)$ on
$\mathbb{R}^{m}$ satisfies

(4.9) $\frac{\partial}{\partial t}\phi_{t}=\log{\det}\Big{(}\big{(}{\rm
Hess}(\phi_{t})\big{)}^{-1}+\sqrt{-1}e^{-2t}B_{0}\Big{)}^{-1}+2\phi_{t}.$

###### Proof.

This follows from (4.1) and the basic properties of the Legendre transform:
${\rm Hess}(\phi_{t}(y))=\Big{(}{\rm Hess}(u_{t}(x))\Big{)}^{-1}$ and (3.5). ∎

The results of Proposition 4.1 and 4.2 yield that the flow of $A$ completely
decouples from the evolution equations for $u$ and $B$. Because of this it
suffices to study the flow in the case $A=0$, an observation which greatly
simplifies the technicalities to follow.

######  Corollary 4.3.

Let $(g_{0},I_{0},J_{0})$ be a toric GK structure on a Fano toric variety,
corresponding to the data $(u_{0},A_{0},B_{0})$ with respect to a symplectic
form $\omega_{0}\in 2\pi c_{1}(M,J_{0})$. Let $(u_{t},0,e^{-2t}B_{0})$ be the
unique maximal solution of (4.1) with the initial data $(u_{0},0,B_{0})$.
Denote by $(g_{t},I_{t},J_{t})$ the generalized Kähler structure corresponding
to the one-parameter family $(u_{t},e^{-2t}A_{0},e^{-2t}B_{0})$, and by
$\Phi_{t}^{*}$ the canonical biholomorphism between $(M,J_{0})$ and
$(M,J_{t})$ from Lemma 3.6. Then $\Phi_{t}^{*}(g_{t},I_{t},J_{t})$ is the
solution to (1.2) with initial data $(g_{0},I_{0},J_{0})$.

We first notice that when $B_{0}=0$, (4.9) reduces to the scalar reduction of
the normalized Kähler Ricci flow on a toric Fano manifold (cf. [74])

(4.10) $\frac{\partial}{\partial t}\phi_{t}(y)=\log\det\big{(}{\rm
Hess}(\phi_{t}(y))\big{)}+2\phi_{t}(y).$

In this case it follows from the fundamental works (cf. [67, 68, 69, 74]) that
the solution to (4.10) exists for all time and converges in $C^{\infty}$ after
translations to a smooth function $\psi_{s}$ defining a Kähler-Ricci soliton.

To study the general case when $B\neq 0$, by now restricting to the case $A=0$
using Corollary 4.3, we are able to give an explicit description of the
pluriclosed metric ${\omega}_{t}$ along the generalized Kähler Ricci flow in
the toric setting, as a certain transformation of a flow of associated Kähler
metrics $\overline{\omega}_{t}=\overline{\omega}_{\phi_{t}}={\mathrmsl
d}{\mathrmsl d}_{J}^{c}\phi_{t}$ on a fixed complex manifold $(M,J)$. The
precise relation is given by (3.24), i.e.

(4.11)
$\omega_{t}=\overline{\omega}_{t}+e^{-2t}\left[\overline{\omega}_{t}\left(\Pi_{B_{0}}^{1,1}\right)\overline{\omega}_{t}\right],$

where $\overline{\omega}_{t}:=\overline{\omega}_{\phi_{t}}$ is the flow of
Kähler metrics corresponding to a solution $\phi_{t}$ of (4.9) and
$\Pi_{B_{0}}^{1,1}$ denotes the $(1,1)$-part of the real Poisson tensor
(3.22). We next recapture the Kähler flow $\overline{\omega}_{t}$ in terms of
$\overline{\omega}_{0}$-relative Kähler potentials (where
$\overline{\omega}_{0}$ can be the initial Kähler metric, or any other
reference ${\mathbb{T}}$-invariant Kähler metric in $2\pi c_{1}(M,J)$).

######  Proposition 4.4.

Let $(g_{0},I_{0},J)$ denote a toric generalized Kähler structure on $(M,J)$,
with corresponding data $(u_{0},0,B_{0})$, and let $\overline{\omega}_{0}\in
2\pi c_{1}(M,J)$ denote an initial toric Kähler metric with Ricci potential
$h_{0}$. Suppose $\varphi_{t}$ satisfies

(4.12) $\begin{split}&\frac{\partial}{\partial
t}\varphi_{t}=\log\left(\frac{\overline{\omega}_{\varphi_{t}}^{[m]}}{\overline{\omega}_{0}^{[m]}}\right)+\log\left(\frac{\overline{\omega}_{\varphi_{t}}^{[m]}}{\omega_{\varphi_{t}}^{[m]}}\right)+2\varphi_{t}-2h_{0}+c_{t},\qquad\varphi(0)=0,\\\
&\overline{\omega}_{\varphi_{t}}=\overline{\omega}_{0}+dd^{c}_{J}\varphi_{t},\qquad\omega_{\varphi_{t}}=\overline{\omega}_{\varphi_{t}}+e^{-2t}\left(\overline{\omega}_{\varphi_{t}}\Pi_{B_{0}}^{1,1}\overline{\omega}_{\varphi_{t}}\right).\end{split}$

Then, $\overline{\omega}_{t}:=\overline{\omega}_{\varphi_{t}}$ is a flow of
Kähler metrics in $2\pi c_{1}(M,J)$, and $\omega_{t}=\omega_{\varphi_{t}}$ an
associated flow of pluriclosed Hermitian metrics $g_{t}=-\omega_{t}J$ on
$(M,J)$, defining GK structures $(g_{t},I_{t},J)$ with corresponding
symplectic form

$F_{t}:=\overline{\omega}_{t}+e^{-2t}\overline{\omega}_{t}\Pi_{B_{0}}\overline{\omega}_{t},$

and satisfying the evolution equations

(4.13) $\frac{\partial}{\partial
t}\overline{\omega}_{t}=-2(\rho^{B}_{I_{t}}(g_{t})-\overline{\omega}_{t}),\qquad\frac{\partial}{\partial
t}\omega_{t}=-2\left(\rho^{B}_{J}(g_{t})^{1,1}-\omega_{t}\right).$

###### Proof.

We let $\varphi_{t}(y):=\phi_{t}(y)-\phi_{0}(y)$ which, by Proposition 3.3,
pulls back to a smooth function on $M$ (still denoted by $\varphi_{t}$) such
that $\overline{\omega}_{t}=\overline{\omega}_{0}+{\mathrmsl d}{\mathrmsl
d}^{c}_{J}\varphi_{t}$. Using that in the $(y_{i},\theta_{j})$ coordinates we
have (see (3.24))

$\frac{\omega^{[m]}}{\overline{\omega}^{[m]}}=\frac{\big{(}\det\big{(}{\bf
G}+\sqrt{-1}B\big{)}\big{)}\big{(}\det{\bf H}\big{)}^{2}}{\det{\bf
H}_{0}},\qquad\frac{\overline{\omega}^{[m]}}{\overline{\omega}_{0}^{[m]}}=\frac{\det{\bf
H}}{\det{\bf H}_{0}},$

where $\overline{\omega}_{0}$ is any background toric Kähler metric (and ${\bf
H}_{0}=\overline{\omega}_{0}(K_{i},JK_{j})=\mathrm{Hess}(\phi_{0})$ is the
corresponding matrix), we can rewrite the flow (4.9) in terms of $\varphi_{t}$
as (4.12). The remaining claims follow from Corollary 4.3 and (4.11). ∎

###  4.2. Evolution equations

Here we determine the evolution of various quantities along NGKRF. To begin we
recall the Chern Laplacian and recall the explicit form for its action on
$(p,0)$-forms.

######  Definition 4.5.

Let $(M,g,J)$ denote a Hermitian manifold, with associated Chern connection.
Let $\Delta^{C}_{g}:\Gamma({}^{p,q}(M))\to\Gamma({}^{p,q}(M))$, denote the
_Chern Laplacian_ , defined by

$\Delta^{C}_{g}\psi=\sum_{i=1}^{m}(\nabla^{C})^{2}_{E_{i},\overline{E_{i}}}\psi,$

where $E_{i}:=e_{i}-\sqrt{-1}Je_{i},\,i=1,\ldots m$ for a $J$-adapted
orthonormal frame $\\{e_{i},Je_{i}\\}$.

######  Lemma 4.6.

Fix a Hermitian manifold $(M^{2n},g,J)$ with Chern connection $\nabla^{c}$ and
let $H=-d^{c}\omega$. For $\psi\in{}^{p,0}$, $\partial\psi=0$, and
$Z_{1},\ldots Z_{p}\in T^{1,0}M$, one has

$\displaystyle(\Delta^{C}_{g}\psi)(Z_{1},\ldots,Z_{p})=\Big{(}\Delta_{g}\psi+H\star({\mathrmsl
d}\psi)+{\mathrmsl
d}\left(H\star\psi\right)-{\mathcal{L}}_{\theta^{\sharp}}\psi\Big{)}_{Z_{1},\ldots,Z_{p}},$

where for a $(q+1)$-form $\psi$, we let

$(H\star\psi)_{X_{1},\ldots
X_{q}}:=\frac{1}{2}\sum_{i,k=1}^{2m}\sum_{j=1}^{q}(-1)^{j}H(e_{i},e_{k},X_{j})\psi(e_{i},e_{k},X_{1},\ldots,\hat{X}_{j},\ldots,X_{q}).$

###### Proof.

The proof is a straightforward computation left as an exercise. ∎

######  Lemma 4.7.

Let $\overline{\omega}_{\phi}={\mathrmsl d}{\mathrmsl d}^{c}_{J}\phi$ be a
toric Kähler metric on $(M,J)$, described by its toric Kähler potential $\phi$
on ${\mathbb{R}}^{m}$, associated to a toric GK structure
$(g_{\phi},I_{\phi},J)$ corresponding to $A=0,B\in{}^{2}{\mathfrak{t}}$. Then,
for a variation $\phi_{s}=\phi_{0}+s\dot{\phi}$ of $\phi$ one has

$\displaystyle\left.\frac{\partial}{\partial
s}\right|_{s=0}\log\det\left(\left({\rm
Hess}(\phi_{s})\right)^{-1}+\sqrt{-1}B\right)^{-1}=\Delta^{C}_{g_{\phi}}\dot{\phi}.$

###### Proof.

We denote by ${\bf H}:={\rm Hess}(\phi(y))=\big{(}{\rm
Hess}(u(x))\big{)}^{-1}$ and notice that

$\big{(}{\bf
H}^{-1}+\sqrt{-1}B\big{)}^{-1}={\mathbf{X}}+\sqrt{-1}{\mathbf{Y}}$

with ${\mathbf{X}}=\big{(}{\bf H}^{-1}+B{\bf
H}B\big{)}^{-1},\,\,{\mathbf{Y}}=-{\bf H}B{\mathbf{X}}$. Thus, we have

$\begin{split}&\delta_{\phi}\Big{(}\log\det\big{(}{\bf
H}^{-1}+\sqrt{-1}B\big{)}\Big{)}(\dot{\phi})=-{\rm
tr}\Big{(}\big{(}{\mathbf{X}}+\sqrt{-1}{\mathbf{Y}}\big{)}{\bf H}^{-1}\dot{\bf
H}{\bf H}^{-1}\Big{)}\\\ &\,=-{\rm tr}\Big{(}{\bf H}^{-1}{\mathbf{X}}{\bf
H}^{-1}{\rm Hess}(\dot{\phi})\Big{)}=-{\rm tr}\Big{(}\big{[}{\bf H}[{\bf
H}^{-1}+B{\bf H}B]{\bf H}\big{]}^{-1}{\rm Hess}(\dot{\phi})\Big{)}\\\
&\,=-{\rm tr}\Big{(}\big{[}{\bf H}+{\bf H}B{\bf H}B{\bf H}\big{]}^{-1}{\rm
Hess}(\dot{\phi})\Big{)}.\ \end{split}$

Notice that for any ${\mathbb{T}}$-invariant smooth function $f$ on $M$,
viewed as a function $f(y)$ in the holomorphic coordinates
$(y_{j}+\sqrt{-1}\theta_{j})$, we have

${\mathrmsl d}{\mathrmsl d}^{c}_{J}f(y)=\sum_{i,j=1}^{m}\big{(}{\rm
Hess}(f(y))\big{)}_{ij}{\mathrmsl d}y_{i}\wedge{\mathrmsl d}\theta_{j}.$

whereas the inverse of $\omega_{\phi}$ is computed from (3.24) to be

$(\omega_{\phi})^{-1}=-\sum_{i,j=1}^{m}\tilde{{\mathbf{X}}}_{ij}\Big{(}\frac{\partial}{\partial
y_{i}}\wedge\frac{\partial}{\partial\theta_{j}}\Big{)}+\frac{1}{2}\tilde{{\mathbf{Y}}}_{ij}\Big{(}\frac{\partial}{\partial
y_{i}}\wedge\frac{\partial}{\partial
y_{j}}+\frac{\partial}{\partial\theta_{i}}\wedge\frac{\partial}{\partial\theta_{j}}\Big{)}$

with $\tilde{{\mathbf{X}}}=\big{[}{\bf H}+{\bf H}B{\bf H}B{\bf
H}\big{]}^{-1},\,\tilde{{\mathbf{Y}}}=B{\bf H}\tilde{{\mathbf{X}}}$. We thus
get

$\Delta^{C}_{g_{\phi}}f=-{\rm tr}\Big{(}\big{[}{\bf H}+{\bf H}B{\bf H}B{\bf
H}\big{]}^{-1}{\rm Hess}(f(y))\Big{)}$

and the claim follows. ∎

######  Lemma 4.8.

Let $\varphi_{t}$ be a solution of (4.12). Then one has

(4.14) $\displaystyle\begin{split}\frac{\partial}{\partial
t}\dot{\varphi}_{t}&=\Delta^{C}_{g_{t}}\dot{\varphi}_{t}+2\dot{\varphi}_{t}-2|b_{t}|^{2}_{g_{t}}.\end{split}$

###### Proof.

Notice that by Proposition 3.3, $\varphi_{t}(y)=\phi_{t}(y)-\phi_{0}(y)$
extends to a smooth function on $M$, thus so does
$\dot{\varphi}_{t}=\dot{\phi}_{t}$. We take derivative with respect to $t$ in
(4.9) and use Lemma 4.7 to compute

$\begin{split}\frac{\partial}{\partial t}\dot{\phi}_{t}=&\
\Delta^{C}_{g_{t}}\dot{\phi}_{t}+2{\rm tr}\big{[}B_{t}{\bf
H}_{t}B_{t}{\mathbf{X}}_{t}\big{]}+2\dot{\phi}_{t}=\
\Delta^{C}_{g_{t}}\dot{\phi}_{t}+2m-2{\rm tr}\big{[}{\bf
H}_{t}^{-1}{\mathbf{X}}_{t}\big{]}+2\dot{\phi}_{t}\\\ =&\
\Delta^{C}_{g_{t}}\dot{\phi}_{t}+2m-2{\rm tr}\big{[}{\rm Id}+{\bf
H}_{t}B_{t}{\bf H}_{t}B_{t}\big{]}^{-1}+2\dot{\phi}_{t},\end{split}$

where we have set $B_{t}=e^{-2t}B_{0}$, and ${\bf H}_{t}$ and
${\mathbf{X}}_{t}$ are the objects defined in the proof of Lemma 4.7 with
respect to $\phi_{t}$.

As $F_{t}J=-g_{t}-b_{t}$, we have

$|F_{t}|^{2}_{g_{t}}=\tfrac{1}{2}|F_{t}J|^{2}_{g_{t}}=m+|b_{t}|^{2}_{g_{t}},$

noting that, by our convention, the norm of a $2$-form is half the norm of the
corresponding tensor. Using that $F_{t}=-2{g}_{t}(J+I_{t})^{-1}$ we obtain

$2m+2|b_{t}|^{2}_{g_{t}}=4|(J+I_{t})^{-1}|^{2}_{g_{t}}=-4{\rm
tr}(I_{t}+J)^{-2}$

whereas by (3.16) (recalling that $A=0$ and ${\bf H}={\bf\Psi}^{-1}$ in our
case) we compute

$(J+I_{t})^{2}\sim-4\left(\begin{array}[]{cc}{\rm Id}+B_{t}{\bf
H}_{t}B_{t}{\bf H}_{t}&0\\\ 0&{\rm Id}+B_{t}{\bf H}_{t}B_{t}{\bf
H}_{t}\end{array}\right),$

so that

$m+|b_{t}|^{2}_{g_{t}}={\rm tr}\big{[}{\rm Id}+B_{t}{\bf H}_{t}B_{t}{\bf
H}_{t}\big{]}^{-1}.$

The claim follows. ∎

Using Lemma 4.6, we can refine the result of Lemma 2.9, giving a useful
evolution equation for the $(2,0)$ piece of $b$, which yields a key
differential inequality.

######  Proposition 4.9.

Let $(M^{2n},g_{t},I_{t},J)$ be a solution to GKRF with symplectic-type
initial data, and set $\beta_{J}=\pi_{{}^{2,0}_{J}}b$. Then

$\displaystyle\frac{\partial}{\partial t}\beta_{J}=$ $\displaystyle\
\Delta^{C}_{g_{t}}\beta_{J},\qquad\frac{\partial}{\partial
t}\left|\beta_{J}\right|^{2}=\
\Delta^{C}_{g_{t}}\left|\beta_{J}\right|^{2}-\left|\nabla\beta_{J}\right|-\left|\overline{\partial}\beta_{J}\right|^{2}-\left|\beta_{J}\righthalfcup
H^{2,1}\right|^{2}.$

###### Proof.

We apply Lemma 4.6 to $\beta_{J}$. In this case, $H={\mathrmsl d}b$ and
${\mathrmsl d}\beta=H^{2,1}$, so that the term $(H\star d\beta)_{Z_{1},Z_{2}}$
is zero. By Lemma 2.6, the term $b\righthalfcup H$ is exact, and thus
$({\mathrmsl d}(H\star\beta))_{Z_{1},Z_{2}}=(d(b\righthalfcup
H))_{Z_{1},Z_{2}}=0$. We conclude that

$\Delta^{C}_{g}\beta_{J}=\Big{(}\Delta_{g}\beta_{J}-\mathcal{L}_{\theta_{J}^{\sharp}}\beta_{J}\Big{)}^{2,0}.$

As $\Delta_{g}-{\mathcal{L}}_{\theta_{J}^{\sharp}}$ is a real operator, it
follows from Lemma 2.9 that the evolution equation of $\beta_{t}$ under the
GKRF is

$\frac{\partial}{\partial t}\beta_{J}=\Delta^{C}_{g_{t}}\beta_{J},$

which is the formula obtained in [55], noticing that in this context
$\beta_{J}=\partial\alpha$ and $b=\Re(\partial\alpha)$ in the notation of
[55]. The evolution of $\left|\beta_{J}\right|^{2}$ then follows from the
established evolution for $\beta_{J}$ and a Bochner formula (cf. [56] Lemma
4.7). ∎

###  4.3. Long-time existence

In this subsection we give the proof of Theorem 1.1. We first derive an
estimate for the tensor $b$ which holds without any symmetry hypotheses. Next
we derive two key a priori estimates for the potential $\varphi$ in the toric
setting. Using these and further maximum principle arguments we prove the long
time existence.

######  Proposition 4.10.

Let $(M^{2n},g_{t},I_{t},J)$ be a solution to GKRF with symplectic-type
initial data defined on $[0,T)$. Then

$\displaystyle\sup_{M\times[0,T)}\left|b\right|^{2}\leq\sup_{M\times\\{0\\}}\left|b\right|^{2}.$

###### Proof.

This follows by applying the maximum principle to the second relation in
Proposition 4.9, noting that $b=\Re(\beta_{J})$. ∎

######  Proposition 4.11.

Let $\varphi_{t}$ denote a solution to (4.12) on a toric Fano manifold. There
exists a constant $C>0$ so that

$\displaystyle\sup_{M\times\\{t\\}}\left(\left|\varphi\right|+\left|\dot{\varphi}\right|\right)\leq
Ce^{2t}.$

###### Proof.

Applying the maximum principle directly to (4.14) yields the upper bound
$\dot{\varphi}\leq Ce^{2t}$. By Proposition 4.10), we have
$|b_{t}|^{2}_{g_{t}}\leq C$. Using this and applying the maximum principle to
(4.14) yield the lower bound $\dot{\varphi}\geqslant-Ce^{2t}$. Using the
estimate of $\dot{\varphi}$ the bound for $\varphi$ follows by integration in
time. ∎

###### Proof of Theorem 1.1.

By Proposition 3.8 the given invariant data is equivalent to a triple
$(u_{0},A_{0},B_{0})$. By Corollary 4.3 the solution to NGKRF is described by
a triple $(u_{t},0,e^{-2t}B_{0})$, where $u_{t}$ is a solution to (4.1).
Taking Legendre transform, by Proposition 4.4 this solution is equivalently
described by $\varphi_{t}$ which solves (4.12). We will prove the long time
existence of the flow $\varphi_{t}$ which in turn proves the long time
existence of (1.2) by the discussion above.

Equation (4.12) is strictly parabolic for $\varphi$, thus the short-time
existence is guaranteed by general theory. To establish global existence, we
first aim to establish uniform parabolicity of the equation, i.e. uniform
equivalence of the time-varying metrics. To obtain the upper bound we first
recall that by [6, Lemma 6.9], we have

(4.15) $\displaystyle\bigg{(}\frac{\partial}{\partial
t}-\Delta^{C}_{{g}_{t}}\bigg{)}\log\mathop{\mathrm{tr}}\nolimits_{\overline{\omega}_{0}}\omega_{t}\leq|T^{c}_{t}|^{2}_{g_{t}}+C\mathop{\mathrm{tr}}\nolimits_{\omega_{t}}(\overline{\omega}_{0}),$

where $T^{c}$ is the torsion of the Chern connection. To control the two terms
on the right hand side of this inequality, we will augment our test function
with two auxiliary terms. We first observe that by Proposition 4.9, using that
$db=H$ and dropping some negative terms we have

(4.16) $\displaystyle\left(\frac{\partial}{\partial
t}-\Delta^{C}_{g_{t}}\right)\left|b_{t}\right|^{2}_{g_{t}}\leq-\left|T^{c}\right|^{2}_{g_{t}}.$

Furthermore, observe that

(4.17) $\displaystyle\begin{split}\bigg{(}\frac{\partial}{\partial
t}-\Delta^{C}_{g_{t}}\bigg{)}\varphi=&\
\dot{\varphi}-\mathop{\mathrm{tr}}\nolimits_{\omega_{t}}{\mathrmsl
d}{\mathrmsl d}^{c}_{J}\varphi=\
\dot{\varphi}-\mathop{\mathrm{tr}}\nolimits_{\omega_{t}}\overline{\omega}_{t}+\mathop{\mathrm{tr}}\nolimits_{\omega_{t}}\overline{\omega}_{0}.\end{split}$

Notice that

$\mathop{\mathrm{tr}}\nolimits_{\omega_{t}}(\overline{\omega}_{t})=\mathop{\mathrm{tr}}\nolimits\big{[}{\rm
Id}+{\bf H}_{t}B_{t}{\bf H}_{t}B_{t}\big{]}^{-1}=m+|b_{t}|_{g_{t}}^{2}\leq C,$

where the last inequality follows from Proposition 4.10. Using this together
with Proposition 4.11 in (4.17) yields

(4.18)
$\displaystyle-C+\operatorname{tr}_{\omega_{t}}\overline{\omega}_{0}\leq\left(\frac{\partial}{\partial
t}-\Delta_{g_{t}}^{C}\right)\varphi\leq
C+\operatorname{tr}_{\omega_{t}}\overline{\omega}_{0}.$

Now set

$\Psi_{1}:=\log\mathop{\mathrm{tr}}\nolimits_{\overline{\omega}_{0}}\omega_{t}+A\Big{(}|b_{t}|^{2}_{g_{t}}+\varphi\Big{)}.$

For $A$ chosen sufficiently large, it follows from (4.15), (4.16), and (4.18)
that

$\bigg{(}\frac{\partial}{\partial t}-\Delta^{C}_{g_{t}}\bigg{)}\Psi_{1}\leq
CA.$

Applying the maximum principle, and using that $\left|\varphi\right|$ is
already bounded by Proposition 4.11 we conclude that for any $T>0$ there is a
constant $C$ so that

$\sup_{M\times[0,T]}\mathop{\mathrm{tr}}\nolimits_{\overline{\omega}_{0}}\omega_{t}\leq
C.$

To obtain the lower bound we get a lower bound for the volume form. First, by
[6, Lemma 6.7], we have

(4.19) $\displaystyle\left(\frac{\partial}{\partial
t}-\Delta^{C}_{g_{t}}\right)\log\frac{\omega_{t}^{m}}{\overline{\omega}_{0}^{m}}\geqslant-C\operatorname{tr}_{\omega_{t}}\overline{\omega}_{0}.$

Now set

$\displaystyle\Psi_{2}=\log\frac{\omega_{t}^{m}}{\overline{\omega}_{0}^{m}}+A\varphi.$

By (4.18) and (4.19) we obtain

$\displaystyle\left(\frac{\partial}{\partial
t}-\Delta^{C}_{g_{t}}\right)\Psi_{2}\geqslant-C.$

Applying the maximum principle establishes the lower bound for the
determinant. It follows that $C^{-1}g_{0}\leq g_{t}\leq Cg_{0}$ on any finite
time interval. As $|b_{t}|^{2}_{g_{t}}$ is bounded, we can now apply [48,
Theorem 1.2] to obtain the higher order regularity and hence the long time
existence. ∎

##  5\. Monotone functionals and behavior at infinity

In this section we analyze the behavior of NGKRF at infinity. First we derive
an extension of Perelman’s shrinker entropy to the GKRF (without toric
symmetry), and use it to derive the uniform $\kappa$-noncollapsing estimate
and convergence of nonsingular solutions of Theorem 1.2. Next we derive an
extension of the Mabuchi $K$-energy to the GK setting, and show that it is
monotone along the GKRF with toric symmetry. This monotonicity is the key
point behind the convergence statements of Theorem 1.3.

###  5.1. Entropy monotonicity

In this subsection we observe that a modification of Perelman’s entropy
functional [50] is monotone along generalized Kähler-Ricci flow for structures
of symplectic type. Our discussion is a direct adaptation of [28, Chapter 6],
(cf. also [60]). There the monotonicity formulae and applications are
contingent upon a further monotone quantity dubbed a torsion-bounding
subsolution. In the setting of generalized Kähler structures of symplectic
type the torsion potential $b$ plays this role, and we make this explicit
below

######  Definition 5.1.

Let $(M^{2m},g,I,J)$ be a generalized Kähler structure of symplectic type. Fix
$f\in C^{\infty}(M,{\mathbb{R}})$ and $\tau>0$. The _shrinker entropy_
associated to this data is

$\displaystyle\mathcal{W}(g,H,f,\tau):=\int_{M}\left[\tau\left(\left|\nabla
f\right|^{2}+R-\frac{1}{12}\left|H\right|^{2}\right)-\left|b\right|^{2}+f-2m\right](4\pi\tau)^{-m}e^{-f}dV.$

Furthermore, let

$\displaystyle\mu\left(g,H,\tau\right):=\inf_{\left\\{f\
|\int_{M}(4\pi\tau)^{-m}e^{-f}dV=1\right\\}}\mathcal{W}\left(g,H,f,\tau\right),$

######  Definition 5.2.

Let $(M^{2m},g_{t},I_{t},J)$ be a solution to GKRF with symplectic-type
initial data. The associated _conjugate heat operator_ is

$\displaystyle\square^{*}=-\frac{\partial}{\partial
t}-\Delta_{g_{t}}-\tfrac{1}{2}\operatorname{tr}_{g}\dot{g}$

######  Proposition 5.3.

Let $(M^{2m},g_{t},I_{t},J)$ be a solution to generalized Kähler-Ricci flow on
a compact manifold with symplectic-type initial data. Fix some $\tau^{*}>0$
and set $\tau(t)=\tau^{*}-t$. Let $u_{t}$ denote a solution of the conjugate
heat equation $\square^{*}u=0$, and define $f$ via
$u=\frac{e^{-f}}{(4\pi\tau)^{m}}$.

$\displaystyle\frac{d}{dt}$ $\displaystyle\mathcal{W}(g,H,f,\tau)=$
$\displaystyle\int_{M}\left[2\tau\left|\operatorname{Rc}-\frac{1}{4}H^{2}+\nabla^{2}f-\frac{1}{2\tau}g\right|^{2}+\frac{\tau}{6}\left|d^{*}H+i_{\nabla
f}H\right|^{2}+\frac{5}{6}\left|H\right|^{2}\right](4\pi\tau)^{-m}e^{-f}dV.$

###### Proof.

This follows immediately from ([28] Proposition 6.26), noting that
$\left|b\right|^{2}$ is a ‘torsion-bounding subsolution’, which follows from
Proposition 4.9 and the fact that
$\left|\overline{\partial}\beta_{J}\right|^{2}=\left|T^{c}\right|^{2}=\left|H\right|^{2}$
by Definition 2.1. ∎

######  Remark 5.4.

Proposition 5.3 shows that the only self-similar solutions of the normalized
GKRF with symplectic-type initial data are actually Kähler-Ricci solitons. In
contrast to the shrinking case, there exist non-Kähler steady solitons for
GKRF [53, 62, 63].

###### Proof of Theorem 1.2.

Using the entropy monotonicity of Proposition 5.3 the uniform
$\kappa$-noncollapsing estimate follows from Perelman’s original argument
([50], cf. [28] Theorem 6.29). Now assume we have a solution to the normalized
flow on $[0,\infty)$ which satisfies a uniform curvature estimate. By the
uniform $\kappa$-noncollapsing it follows that the volumes of unit balls are
uniformly bounded below along the flow, and there is a uniform lower bound on
the injectivity radius. Since the volume form is bounded above pointwise by
$F_{t}^{m}$ by (2.9), the total volume is bounded above by
$\int_{M}F_{t}^{m}=(2\pi c_{1}(M,J))^{m}$. It follows that there is a uniform
upper bound for the diameter along the flow. With these estimates in place, it
follows from Cheeger-Gromov compactness theory (cf. [42] Theorem 2.3 for our
simplified case) that for any sequence $\\{t_{j}\\}\to\infty$ there exists a
subsequence, still denoted $\\{t_{j}\\}$, such that $(M,g_{t_{j}})$ converges
in Cheeger-Gromov sense to a smooth limiting Riemannian manifold
$(M,g_{\infty})$, where the limiting manifold is still $M$ due to the diameter
upper bound. Also, it follows from ([57], [28] Ch. 5) that $H$ and all its
covariant derivatives are uniformly bounded along the flow, and thus by
choosing a further subsequence $H_{t_{j}}$ converges to $H_{\infty}$.

We claim that by choosing a further subsequence the GK structure also
converges, which requires establishing $C^{\infty}$ estimates for $I$ and $J$.
First, the $g$-norms of $I$ and $J$ are uniformly bounded since $g$ is
compatible with both. By a standard argument using that $\nabla^{B,I}$
preserves $I$, and that the coefficients of this connection are already
converging by the discussion above, it follows that there are uniform
$C^{\infty}$ estimates for $I$, and the same argument using $\nabla^{B,J}$
yields estimates for $J$. It follows that we can choose a further subsequence
such that $(M,g_{t_{j}},I_{t_{j}},J)$ converges in Cheeger-Gromov sense to a
limiting GK manifold $(M,g_{\infty},I_{\infty},J_{\infty})$. By the uniform
upper bound of $|b_{t}|^{2}_{g_{t}}$ established in Proposition 4.10,
$|F_{t}|^{2}_{g_{t}}=m+|b_{t}|^{2}_{g_{t}}$ is uniformly bounded. As
$F_{t}=-2g_{t}(I_{t}+J_{t})^{-1}$, it follows that
$\left|\det(I_{t}+J_{t})\right|$ is uniformly bounded from below by a positive
constant, showing that $(g_{\infty},I_{\infty},J_{\infty})$ is symplectic type
with corresponding symplectic form
$F_{\infty}=-2g_{\infty}(I_{\infty}+J_{\infty})^{-1}$.

Now note that, by rescaling, the result of Proposition 5.3 holds for the
normalized flow, setting $\tau\equiv\tfrac{1}{2}$. This also implies that the
corresponding functional $\mu$ is monotone increasing. Since the solution is
nonsingular, there are uniform bounds on $\mu$ and all of its derivatives. It
thus follows that for an arbitrary sequences of times $\\{t_{j}\\}\to\infty$
one has

$\displaystyle\lim_{j\to\infty}\left|\frac{d}{dt}\mu(g_{t_{j}},H_{t_{j}})\right|=0.$

For such a sequence $\\{t_{j}\\}$, let $(M,g_{\infty},I_{\infty},J_{\infty})$
denote the GK limit constructed as above. It follows from the entropy
monotonicity formula that the limiting torsion $H_{\infty}$ must vanish, and
furthermore $g_{\infty}$ is a shrinking soliton. Since $H_{\infty}$ vanishes,
the pairs $(g_{\infty},I_{\infty})$ and $(g_{\infty},J_{\infty})$ are both
Kähler, hence the symplectic $2$-form $F_{\infty}$ taming $I_{\infty}$ and
$J_{\infty}$ is parallel with respect to $g_{\infty}$, and hence must be a
$(1,1)$-form by the Fano condition of $(M,J_{\infty})$. It then follows that
we must have $I_{\infty}=J_{\infty}$. ∎

######  Remark 5.5.

In the toric setting, it is natural to expect that by adapting the ideas of
[74] the convergence can be improved to show that $J_{\infty}=J$ and that the
scalar potentials $\varphi$ converge.

###  5.2. The extended Mabuchi functional

In this section we give an extension of the Mabuchi functional to the setting
of GK structures of symplectic type on a Kähler background. To begin we first
observe that it is possible to canonically associate a Kähler metric to every
symplectic-type GK structure on a Kähler manifold. First, using Hodge
decomposition, the deRham class $\alpha=[F]$ defines an element
$\overline{\alpha}\in H^{1,1}(M,{\mathbb{R}})$. Using that $F$ tames $J$ and a
result of Demailly-Paun [19], it follows that $\overline{\alpha}$ is a Kähler
class of $(M,J)$. By Yau’s theorem [73], we can find a unique Kähler metric
$\overline{\omega}\in\overline{\alpha}$ with the property

(5.1) $\overline{\omega}^{[m]}=cF^{[m]},\qquad
c=\frac{\overline{\alpha}^{[m]}}{[\alpha]^{[m]}}.$

######  Definition 5.6.

Given $(M,J)$ Kähler and $\alpha\in H^{2}(M,\mathbb{R})$, let ${\bf
GK}(M,J,\alpha)$ be the space of symplectic-type GK structures $(g,I,J)$ such
that $[F]=\alpha$. Given $(g,I,J)\in{\bf GK}(M,J,\alpha)$, the _Kähler
reduction_ is the unique Kähler metric $\overline{\omega}$ satisfying (5.1).

######  Remark 5.7.

Kähler reductions exist in smooth families along a solution to GKRF of
symplectic type. In general it might not be possible to derive an explicit
formula for the evolution of $\overline{\omega}$, but in the toric Fano case
this is what is achieved in Proposition 4.4.

We now recall the definitions of the Aubin-Mabuchi functionals ${\bf I}$ and
${\bf I}^{\rho_{0}}$ which are defined on the space of
$\overline{\omega}_{0}$-relative Kähler potentials $\varphi$ by

(5.2) $\begin{split}(\delta_{\varphi}{\bf
I})(\dot{\varphi})&=\int_{M}\dot{\varphi}\overline{\omega}_{\varphi}^{[m]},\hskip
34.14322pt\qquad{\bf I}(0)=0,\\\ (\delta_{\varphi}{\bf
I}^{\rho_{0}})(\dot{\varphi})&=\int_{M}\dot{\varphi}\rho_{0}\wedge\overline{\omega}_{\varphi}^{[m-1]},\qquad{\bf
I}^{\rho_{0}}(0)=0\end{split}$

where $\rho_{0}$ is the Ricci form of $\overline{\omega}_{0}$. As $\rho_{0}\in
2\pi c_{1}(M,J)$, letting
$a:=4\pi\frac{c_{1}(M,J)\cdot[\overline{\alpha}]^{[m-1]}}{[\overline{\alpha}]^{[m]}}$
the expression $a{\bf I}(\varphi)-2{\bf I}^{\rho_{0}}(\varphi)$ does not
change if we add to $\varphi$ a real constant, i.e. it introduces a
functional, denoted by $\left(a{\bf I}-2{\bf
I}^{\rho_{0}}\right)(\overline{\omega}_{\varphi})$, acting on the Kähler
metrics $\overline{\omega}_{\varphi}\in\overline{\alpha}$.

######  Definition 5.8.

Fix $(M,J)$ Kähler and $\alpha\in H^{2}(M,\mathbb{R})$ such that
$\overline{\alpha}\in H^{1,1}(M,\mathbb{R})$ is a Kähler class with
representative $\overline{\omega}_{0}\in\overline{\alpha}$. Fix
$(g,I,J)\in{\bf GK}(M,J,\alpha)$ and let $\overline{\omega}_{\varphi}$ denote
its Kähler reduction. Define the _extended Mabuchi energy_ by

(5.3) $\begin{split}{\bf
M}(F)&:=\left(\frac{\overline{\alpha}^{m}}{[\alpha]^{m}}\right)\int_{M}\left[\log\left(\frac{F^{[m]}}{\overline{\omega}_{0}^{[m]}}\right)+\log\left(\frac{F^{[m]}}{\omega_{J}^{[m]}}\right)\right]F^{[m]}+\left(a{\bf
I}-2{\bf I}^{\rho_{0}}\right)(\overline{\omega}_{\varphi}),\\\
a&:=4\pi\frac{c_{1}(M,J)\cdot\overline{\alpha}^{[m-1]}}{\overline{\alpha}^{[m]}}.\end{split}$

######  Remark 5.9.

In the case when $\alpha=\overline{\alpha}$ is a Kähler class, one has that
$\overline{\omega}_{\varphi}^{[m]}=F^{[m]}\geq\omega_{J}^{[m]}$. This implies
the basic inequality

(5.4) ${\bf M}(F)\geq{\bf M}(\overline{\omega}_{\varphi}),$

where ${\bf M}(\overline{\omega}_{\varphi})$ is the usual Mabuchi energy of
the Kähler metric $\overline{\omega}_{\varphi}\in\alpha$, see e.g. [13].

###  5.3. The extended Mabuchi energy in the toric case

On a compact toric Kähler manifold $(M,J,\omega_{0},{\mathbb{T}})$ with
Delzant polytope $({\mathrm{P}},{{\mathrm{L}}})$, Donaldson [22] found concise
expressions for the Futaki invariant and the Mabuchi energy in terms of
$({\mathrm{P}},{\mathrm{L}})$ and the space
${\mathcal{S}}({\mathrm{P}},{\mathrm{L}})$. We use this point of view in order
to express the extended Mabuchi energy as a functional acting on the data
$(u,A,B)$ associated to $({\mathrm{P}},{\mathrm{L}})$. Following [22], we
introduce a linear functional defined on the space of continuous functions on
${\mathrm{P}}$ by

(5.5) ${\bf F}(f):=-a\int_{{\mathrm{P}}}f(x){\mathrmsl
d}x+2\int_{\partial{\mathrm{P}}}f(x){\mathrmsl d}\sigma_{{\mathrm{L}}},\qquad
a:=2\frac{\operatorname{Vol}\left({\partial{\mathrm{P}}},{\mathrmsl
d}\sigma_{{\mathrm{L}}}\right)}{\operatorname{Vol}({\mathrm{P}},{\mathrmsl
d}x)},$

where, in the above expression, ${\mathrmsl d}x$ is a fixed Lebesgue measure
on ${\mathfrak{t}}^{*}$ (associated with a chosen basis of ${\mathfrak{t}}$)
and ${\mathrmsl d}\sigma_{{\mathrm{L}}}$ is the measure induced by ${\mathrmsl
d}x$ and the inward normal ${\mathrmsl d}L_{j}$ on each facet
${\mathrm{P}}_{j}\subset\partial{\mathrm{P}}$. If we take
$f(x)=\langle\xi,x\rangle+a$ to be affine-linear, (5.5) computes, up to a
dimensional multiplicative constant, the Futaki invariant of the holomorphic
vector $JK_{\xi}$ on $(M,J,[\omega_{0}])$.

######  Lemma 5.10.

Let $(g,I,J)$ be a toric GK structure, corresponding to the data $(u,A,B)$
with respect to $\omega_{0}$. Then

$\frac{1}{(2\pi)^{m}}{\bf M}(F)={\bf
F}(u)-\int_{{\mathrm{P}}}\log\det\Big{(}{\rm
Hess}(u)+\sqrt{-1}B\big{)}{\mathrmsl
d}x+\int_{{\mathrm{P}}}\log\det\Big{(}{\rm Hess}(u_{0})\big{)}{\mathrmsl d}x,$

where $u_{0}$ is the symplectic potential of a background toric Kähler metric.

###### Proof.

On a toric manifold $H^{2}(M,{\mathbb{C}})=H^{1,1}(M,{\mathbb{C}})$, so we are
in the case $\alpha=\overline{\alpha}$ with
$F^{[m]}=\overline{\omega}_{\varphi}^{[m]}$. By Definition 5.8 and (3.29), we
compute

$\begin{split}{\bf M}(F)&={\bf
M}(\overline{\omega}_{\varphi})+(2\pi)^{m}\int_{{\mathrm{P}}}\log\left(\frac{\det\mathrm{Hess}(u)}{\det(\mathrm{Hess}(u)+\sqrt{-1}B)}\right){\mathrmsl
d}x.\end{split}$

The claim follows by the above, taking in mind that
$\overline{\omega}_{\varphi}={\mathrmsl d}{\mathrmsl
d}^{c}_{J_{A}}\phi(\tilde{y})$, where $\phi$ is the Legendre transform of
$u\in{\mathcal{S}}({\mathrm{P}},{\mathrm{L}})$ (see (3.4)) and $\tilde{y}$ are
the pluriharmonic coordinates of $J_{A}$ (see (3.13)), and the fact that (see
[22])

${\bf M}(\overline{\omega}_{\varphi})=(2\pi)^{m}\left({\bf
F}(u)-\int_{{\mathrm{P}}}\log\det(\mathrm{Hess}(u)){\mathrmsl
d}x+\int_{{\mathrm{P}}}\log\det(\mathrm{Hess}(u_{0})){\mathrmsl d}x\right).$

∎

We next derive fundamental variational properties for the extended Mabuchi
energy. Recall our expression for the metric

$\displaystyle{\bf X}=\Re\big{[}{\rm Hess}(u)+\sqrt{-1}B\big{]}^{-1}.$

It turns out that the quantity

(5.6) $\kappa(u,A,B):=-\sum_{i.j=1}^{m}{\bf X}_{ij,ij}$

is identified in [11, 72] with the momentum map for the action of the group of
${\mathbb{T}}$-equivariant Hamiltonian symplectomorphisms of $(M,\omega_{0})$
on the space of compatible toric GK structures, and thus $\kappa$ extends the
notion of scalar curvature to the GK context, via the momentum map picture of
[21, 27]. We call $\kappa$ the _generalized scalar curvature_. This point of
view is extended to GK manifolds of symplectic type (without toric symmetry)
in [33]. We thus conclude that the critical points of ${\bf M}$ correspond to
toric GK structures for which ${\rm Gscal}(g)=\kappa(u,A,B)=a$.

######  Proposition 5.11.

Given a one-parameter family of symplectic potentials $u_{s}=u+s\dot{u}$ and
fixed matrices $A,B$, we obtain

$(\delta_{u}{\bf
M})(\dot{u})=\int_{{\mathrm{P}}}\dot{u}\left(\kappa(u,A,B)-a\right){\mathrmsl
d}x.$

In particular, the critical points of ${\bf M}$ on the space of toric
generalized Kähler structures of symplectic type on $(M,J)$ with fixed
cohomology class and holomorphic Poisson tensor have constant generalized
Kähler scalar curvature. Furthermore,

$(\delta^{2}_{u}{\bf M})(\dot{u},\dot{u})=\int_{{\mathrm{P}}}{\rm
tr}\Big{[}\Big{(}\big{(}{\rm Hess}(u)+\sqrt{-1}B\big{)}^{-1}{\rm
Hess}(\dot{u})\Big{)}^{2}\Big{]}{\mathrmsl d}x.$

###### Proof.

These follow by observing that the linearization of the entropy term is

$\int_{{\mathrm{P}}}{\rm tr}\Big{[}\big{(}{\rm
Hess}(u)+\sqrt{-1}B\big{)}^{-1}\circ{\rm Hess}(\dot{u})\Big{]}{\mathrmsl
d}x=\int_{{\mathrm{P}}}{\rm tr}\big{(}{\bf X}\circ{\rm
Hess}(\dot{u})\big{)}{\mathrmsl d}x.$

Using that ${\bf F}$ is linear, and differentiating one more time the above
expression yields the formula for the Hessian of ${\bf M}$. To obtain the
first formula, we need to integrate by parts the RHS of the above expression
as in [22]. To this end, we observe that ${\bf X}=\big{[}{\rm
Hess}(u)+B\big{(}{\rm Hess}(u)\big{)}^{-1}B\big{]}^{-1}$ is a symmetric matrix
on ${\mathrm{P}}$, which satisfies the first order boundary conditions of [4,
Prop. 1] (see also Remark 4 in that reference). The second variation follows
easily by passing another $s$ derivative through the variation formula above
and using that $\ddot{u}=0$. ∎

Lastly in this subsection we establish the monotonicity of the extended
Mabuchi energy along NGKRF in the toric Fano setting. We build this up through
a series of lemmas.

######  Lemma 5.12.

On a toric Fano manifold, along the flow (4.12) one has

$\frac{d}{dt}{\bf
M}(F_{t})=\left[-\int_{M}|d\dot{\varphi}_{t}|^{2}_{\overline{\omega}_{t}}F_{t}^{[m]}+\int_{M}\left(s_{J}^{B}(g_{\varphi_{t}})-2m\right)F_{t}^{[m]}\right],$

where $s_{J}^{B}(g_{t})$ denotes the Bismut scalar curvature of the Hermitian
structure $(g_{t},J)$.

###### Proof.

Referring to (5.3) (with $\alpha=\overline{\alpha}$ and $a=2m$ according to
our normalization), $F_{t}^{m}=\overline{\omega}_{t}^{m}$ and using
(4.12)-(4.13) (with $c(t)\equiv 0$), we compute

$\begin{split}\frac{d}{dt}{\bf M}(F_{t})=&\
2\int_{M}\operatorname{tr}_{\omega_{t}}\left(\rho^{B}_{J}(g_{t})-\overline{\omega}_{t}\right)\overline{\omega}_{t}^{[m]}+\int_{M}\big{(}\dot{\varphi}_{t}-2\varphi_{t}+2h_{0}\big{)}({\mathrmsl
d}{\mathrmsl d}^{c}\dot{\varphi}_{t})\wedge\overline{\omega}_{t}^{[m-1]}\\\ &\
\qquad+2m\int_{M}\dot{\varphi}_{t}\overline{\omega}_{t}^{[m]}-2\int_{M}\dot{\varphi}_{t}\rho_{0}\wedge\overline{\omega}_{t}^{[m-1]}\\\
=&\
\int_{M}\left(s_{J}^{B}(g_{t})-2m\right)\overline{\omega}_{t}^{[m]}-\int_{M}|d\dot{\varphi}_{t}|^{2}_{\overline{\omega}_{t}}\overline{\omega}_{t}^{[m]}\\\
&\
\qquad-2\int_{M}\dot{\varphi}_{t}(\overline{\omega}_{t}-\overline{\omega}_{0})\wedge\overline{\omega}_{t}^{[m-1]}+2\int_{M}\dot{\varphi}_{t}(\rho_{0}-\overline{\omega}_{0})\wedge\overline{\omega}_{t}^{[m-1]}\\\
&\
\qquad+2m\int_{M}\dot{\varphi}_{t}\overline{\omega}_{t}^{[m]}-2\int_{M}\dot{\varphi}_{t}\rho_{0}\wedge\overline{\omega}_{t}^{[m-1]}\\\
=&\
\int_{M}\left(s_{J}^{B}(g_{t})-2m\right)\overline{\omega}_{t}^{[m]}-\int_{M}|d\dot{\varphi}_{t}|^{2}_{\overline{\omega}_{t}}\overline{\omega}_{t}^{[m]}.\end{split}$

The claim follows by using $\overline{\omega}_{t}^{[m]}=F_{t}^{[m]}$ again. ∎

######  Lemma 5.13.

Suppose $(M,g,I,J)$ is a compact GK manifold of symplectic type. Then

$\int_{M}\left[\operatorname{tr}_{\omega_{J}}(\rho^{B}_{J})-\operatorname{tr}_{F}(\rho^{B}_{J})\right]F^{[m]}=-\frac{1}{8}\int_{M}\left[\Big{|}{\mathrmsl
d}\log\det(I+J)-\theta_{I}-\theta_{J}\Big{|}^{2}_{g}+|\theta_{I}-\theta_{J}|^{2}_{g}\right]F^{[m]}.$

###### Proof.

We denote by $s_{J}^{B}$ and $s_{I}^{B}$ the Bismut-Ricci scalar curvatures of
$(g,J)$ and $(g,I)$, respectively. By Proposition 2.7 and [47], we know that

$\rho^{B}_{J}-\rho^{B}_{I}=\tfrac{1}{2}{\mathcal{L}}_{(\theta_{J}^{\sharp}-\theta_{I}^{\sharp})}F,\qquad
s_{J}^{B}-s_{I}^{B}=-\delta^{g}(\theta_{J}-\theta_{I}).$

Using that $F=-2g(I+J)^{-1}$ and the above formula, we compute

$\begin{split}&2\left[\operatorname{tr}_{\omega_{J}}(\rho^{B}_{J})-\operatorname{tr}_{F}(\rho^{B}_{J})\right]=s_{J}^{B}-\big{\langle}\rho^{B}_{J},\omega_{J}+\omega_{I}\big{\rangle}_{g}=\tfrac{1}{2}s_{J}^{B}-\big{\langle}\rho^{B}_{J},\omega_{I}\big{\rangle}_{g}\\\
&\qquad=\tfrac{1}{2}(s_{J}^{B}-s_{I}^{B})-\tfrac{1}{2}\Big{\langle}{\mathcal{L}}_{(\theta_{J}^{\sharp}-\theta_{I}^{\sharp})}F,\omega_{I}\Big{\rangle}_{g}=-\tfrac{1}{2}\delta^{g}(\theta_{J}-\theta_{I})-\tfrac{1}{2}\Big{\langle}{\mathcal{L}}_{(\theta_{J}^{\sharp}-\theta_{I}^{\sharp})}F,\omega_{I}\Big{\rangle}_{g}.\end{split}$

We next note that, by (2.8),

${\mathcal{L}}_{(\theta_{J}^{\sharp}-\theta_{I}^{\sharp})}F={\mathrmsl
d}\left(F(\theta_{J}^{\sharp}-\theta_{I}^{\sharp})\right)={\mathrmsl
d}\left(I(\theta_{J}-\theta_{I})+Ib(\theta_{J}^{\sharp}-\theta_{I}^{\sharp})\right).$

Thus

$\begin{split}2\left[\operatorname{tr}_{\omega_{J}}(\rho^{B}_{J})-\operatorname{tr}_{F}(\rho^{B}_{J})\right]&=-\tfrac{1}{2}\delta^{g}(\theta_{J}-\theta_{I})-\tfrac{1}{2}\Big{\langle}{\mathrmsl
d}\left(I\left((\theta_{J}-\theta_{I})+b(\theta_{J}^{\sharp}-\theta_{I}^{\sharp})\right)\right),\omega_{I}\Big{\rangle}_{g}\\\
&=\tfrac{1}{2}\left(\delta^{g}\left(b(\theta_{J}^{\sharp}-\theta_{I}^{\sharp})\right)+\big{\langle}\theta_{J}-\theta_{I}+b(\theta_{J}^{\sharp}-\theta_{I}^{\sharp}),\theta_{I}\big{\rangle}_{g}\right),\end{split}$

where we have used the identity $\langle{\mathrmsl
d}I\alpha,\omega_{I}\rangle_{g}=-\delta^{g}\alpha-\langle\alpha,\theta_{I}\rangle_{g}$.
We next use the identity (which follows from Lemma 2.6)

${\mathrmsl
d}\log\det(I+J)=\theta_{I}+\theta_{J}+b(\theta_{J}^{\sharp}-\theta_{I}^{\sharp})$

in order to express the terms containing
$b(\theta_{J}^{\sharp}-\theta_{I}^{\sharp})$. We thus get

$\begin{split}2\left[\operatorname{tr}_{\omega_{J}}(\rho^{B}_{J})-\operatorname{tr}_{F}(\rho^{B}_{J})\right]&=\tfrac{1}{2}\left(-\Delta^{g}\log\det(I+J)-\delta^{g}(\theta_{I}+\theta_{J})+\big{\langle}-2\theta_{I}+{\mathrmsl
d}\log\det(I+J),\theta_{I}\big{\rangle}_{g}\right)\\\
&=\tfrac{1}{2}\left(-\Delta^{g}\log\det(I+J)-\delta^{g}(\theta_{I}+\theta_{J})-2|\theta_{I}|^{2}_{g}+\big{\langle}{\mathrmsl
d}\log\det(I+J),\theta_{I}\big{\rangle}_{g}\right).\end{split}$

We finally use that (as $F=-2g(I+J)^{-1}$)

$F^{[m]}=2^{m}\det(I+J)^{-\frac{1}{2}}\,\omega_{J}^{[m]}$

and integration by parts to get

$\begin{split}&\int_{M}\left[\operatorname{tr}_{\omega_{J}}(\rho^{B}_{J})-\operatorname{tr}_{F}(\rho^{B}_{J})\right]F^{[m]}\\\
&=2^{m-2}\int_{M}\left(-\Delta^{g}\log\det(I+J)+\big{\langle}{\mathrmsl
d}\log\det(I+J),\theta_{I}\big{\rangle}_{g}-\delta^{g}(\theta_{I}+\theta_{J})-2|\theta_{I}|^{2}_{g}\right)\det(I+J)^{-\frac{1}{2}}\,\omega_{J}^{[m]}\\\
&=2^{m-2}\int_{M}\left(-\Delta^{g}\log\det(I+J)+\big{\langle}{\mathrmsl
d}\log\det(I+J),\theta_{I}\big{\rangle}_{g}-|\theta_{I}|^{2}_{g}-|\theta_{J}|^{2}_{g}-2\delta^{g}\theta_{J}\right)\det(I+J)^{-\frac{1}{2}}\,\omega_{J}^{[m]}\\\
&=2^{m-2}\int_{M}\left[-\frac{1}{2}\Big{|}{\mathrmsl
d}\log\det(I+J)\Big{|}^{2}_{g}+\big{\langle}{\mathrmsl
d}\log\det(I+J),\theta_{I}+\theta_{J}\big{\rangle}_{g}-|\theta_{I}|^{2}_{g}-|\theta_{J}|^{2}_{g}\right]\det(I+J)^{-\frac{1}{2}}\,\omega_{J}^{[m]}\\\
&=-\frac{1}{8}\int_{M}\left[\Big{|}{\mathrmsl
d}\log\det(I+J)-\theta_{I}-\theta_{J}\Big{|}^{2}_{g}+|\theta_{I}-\theta_{J}|^{2}_{g}\right]F^{[m]}\end{split}$

where we have used the relation
$\delta^{g}\theta_{I}+|\theta_{I}|^{2}_{g}=\frac{1}{6}|H|_{g}^{2}=\delta^{g}\theta_{J}+|\theta_{J}|^{2}_{g}$
from [47]. ∎

######  Corollary 5.14.

On a toric Fano manifold, the Mabuchi functional ${\bf M}(F_{\varphi_{t}})$ is
monotone decreasing along NGKRF.

###### Proof.

This follows directly from Lemma 5.12 and Lemma 5.13, noting that
$\operatorname{tr}_{\omega}(\rho^{B}_{J}(g))=\frac{s_{J}^{B}}{2}$ and

$\int_{M}\operatorname{tr}_{F}(\rho_{J}^{B}(g))F^{[m]}=\int_{M}\rho^{B}_{J}(g)\wedge
F^{[m-1]}=m$

by the Fano condition and the normalization $[F]\in 2\pi c_{1}(M,J)$. ∎

###  5.4. Weak convergence

We suppose through this section that $(M,J,{\mathbb{T}})$ is a smooth toric
Fano manifold, $\overline{\omega}_{0}\in 2\pi c_{1}(M,J)$ is an initial Kähler
form with corresponding barycentered Fano polytope
$({\mathrm{P}},{{\mathrm{L}}})$, and we further assume that the Futaki
invariant of $(M,J,[\overline{\omega}_{0}])$ vanishes. In this case, by [69],
$(M,J)$ admits a Kähler-Einstein metric and, by [17, 64], the Mabuchi energy
is coercive relative to $\mathrmsl{Aut}_{0}(M,J)$. We first recall what this
precisely means. Denote by

${\mathcal{H}}(M,\overline{\omega}_{0})=\left\\{\varphi\in
C^{\infty}(M)\,|\,\overline{\omega}_{\varphi}:=\overline{\omega}_{0}+{\mathrmsl
d}{\mathrmsl d}^{c}\varphi>0\right\\}$

the space of $\overline{\omega}_{0}$-relative Kähler potentials on $M$, and
consider the Aubin-Mabuchi functional ${\bf I}$ on
${\mathcal{H}}(M,\overline{\omega}_{0})$, defined in (5.2). Notice that ${\bf
I}(\varphi+c)={\bf I}(\varphi)+c\operatorname{Vol}(M,\overline{\omega}_{0})$,
so we can introduce the slice

$\mathring{{\mathcal{H}}}(M,\overline{\omega}_{0}):={\mathcal{H}}(M,\overline{\omega}_{0})\cap{\bf
I}^{-1}(0),$

and consider the induced action of ${\mathbb{G}}:=\mathrmsl{Aut}_{0}(M,J)$ on
$\mathring{{\mathcal{H}}}(M,\overline{\omega}_{0})$ via the natural action of
${\mathbb{G}}$ on the space of ${\mathbb{T}}$-invariant Kähler metrics in
$[\overline{\omega}_{0}]$ by pullbacks, i.e. for any
$\varphi\in\mathring{{\mathcal{H}}}(M,\overline{\omega}_{0})$ and any
$\tau\in{\mathbb{G}}$, we let $\tau[\varphi]$ denote the uniquely determined
$\overline{\omega}_{0}$-relative Kähler potential in
$\mathring{{\mathcal{H}}}(M,\overline{\omega}_{0})$ of the Kähler metric
$\tau^{*}(\omega_{\varphi})$. We thus have

$\tau[\varphi]=\tau[0]+\varphi\circ\tau.$

We next consider the $d_{1}$-distance on
${\mathcal{H}}(M,\overline{\omega}_{0})$, introduced via the $L^{1}$-length of
a smooth curve $\varphi(t)\in{\mathcal{H}}(M,\overline{\omega}_{0})$:

$l_{1}(\varphi(t))=\int_{0}^{1}\int_{M}|{\dot{\varphi}}(t)|\,(\overline{\omega}_{\varphi(t)})^{[m]}.$

By the results in [16],

$d_{1}(\varphi_{0},\varphi_{1}):=\inf\Big{\\{}l_{1}(\varphi(t))\,\Big{|}\,\varphi(t,x)\in
C^{\infty}([0,1]\times
M),\,\varphi(t,\cdot)\in{\mathcal{H}}(M,\overline{\omega}_{0}),\,\varphi(0)=\varphi_{0},\,\varphi(1)=\varphi_{1}\Big{\\}}$

defines a distance. We then have

######  Theorem 5.15.

[17] If $(M,J,[\overline{\omega}_{0}])$ admits a Kähler-Einstein metric, then
the Mabuchi energy ${\bf M}$ is ${\mathbb{G}}$-invariant and
${\mathbb{G}}$-coercive on ${\mathcal{H}}(M,\overline{\omega}_{0})$, i.e.
there are uniform constants $\lambda>0,\delta$ such that

${\bf
M}(\varphi)\geq\lambda\inf_{\tau\in{\mathbb{G}}}d_{1}(0,\tau[\varphi])-\delta,\qquad\forall\varphi\in\mathring{{\mathcal{H}}}(M,\overline{\omega}_{0}).$

An important ingredient in proving the above result is the weak compactness,
established in [7, 8]. To state it, we denote by
$\mathrm{PSH}(M,\overline{\omega}_{0})$ the space of
$\overline{\omega}_{0}$-relative plurisubharmonic functions on $M$, i.e. the
usc functions $\varphi\in L^{1}(M,\overline{\omega}_{0})$ such that
$\overline{\omega}_{\varphi}=\overline{\omega}_{0}+{\mathrmsl d}{\mathrmsl
d}^{c}\varphi\geq 0$ in the sense of currents. Furthermore, we denote by
${\mathcal{E}}(M,\overline{\omega}_{0})\subset\mathrm{PSH}(M,\overline{\omega}_{0})$
the subspace of elements of _full Monge-Ampère mass_ , i.e.