# Deformation classes in generalized Kähler geometry

Matthew Gibson  Rowland Hall
University of California
Irvine, CA 92617<EMAIL_ADDRESS>and  Jeffrey Streets  Rowland Hall
University of California
Irvine, CA 92617<EMAIL_ADDRESS>
(Date: May 5th, 2020)

###### Abstract.

We introduce natural deformation classes of generalized Kähler structures
using the Courant symmetry group. We show that these yield natural extensions
of the notions of Kähler class and Kähler cone to generalized Kähler geometry.
Lastly we show that the generalized Kähler-Ricci flow preserves this
generalized Kähler cone, and the underlying real Poisson tensor.

##  1\. Introduction

A rudimentary notion of Kähler geometry is that of the Kähler class: given
$(M^{2n},\omega,J)$ a Kähler manifold, the Kähler form $\omega$ is closed, and
$[\omega]\in H^{1,1}_{\mathbb{R}}$ is the associated Kähler class. Fixing the
complex structure $J$, the space of all Kähler classes defines an open cone in
$H^{1,1}_{\mathbb{R}}$ (the Kähler cone), and the fundamental result of
Demailly-Paun [5] gives a characterization of this cone in terms of pairing
against complex subvarieties. The space of Kähler metrics within a given
Kähler class is an open infinite dimensional cone in $C^{\infty}(M)$ using the
$\sqrt{-1}\partial\overline{\partial}$-lemma. Thus the basic structure of the
space of Kähler metrics compatible with a fixed complex structure $J$ is
fairly well understood. If we instead ask for the space of all Kähler pairs
$(g,J)$ on a given smooth manifold $M$, the question becomes decidedly more
delicate. The global structure can be quite wild, with disconnected components
of arbitrarily large dimension (cf. [4]).

Understanding the space of _generalized Kähler_ structures on a given manifold
$M$ becomes even more delicate. Originally discovered by Gates-Hull-Rocek [6],
a generalized Kähler structure on a smooth manifold $M$ is a triple $(g,I,J)$
consisting of a Riemannian metric $g$ compatible with two integrable complex
structures $I,J$ further satisfying

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

Later, Gualtieri [8] gave a natural description of this geometry using the
language of Hitchin’s generalized complex structures [9], in particular in
terms of a pair of generalized complex structures
$(\mathbf{J}_{1},\mathbf{J}_{2})$ satisfying some natural conditions (cf.
§2.1). A fundamental question is to understand the degrees of freedom, moduli,
and topology of the space of generalized Kähler structures on a given smooth
manifold.

Whereas in the Kähler setting we can roughly speaking divide the problem of
understanding the space of Kähler metrics into the space of possible complex
structures and then to consider the space of compatible Kähler metrics, in the
generalized Kähler setting such a decomposition is not really possible.
Indeed, in many settings, given two complex structures $I,J$, there is at most
one compatible metric which defines a generalized Kähler structure.
Nonetheless, many different classes of deformations of generalized Kähler
structure have been constructed. Joyce gave the first examples of nontrivial
(i.e. non-Kähler) generalized Kähler structures by deforming away from
hyperKähler structures (cf. [1]), specifically using an action of
diffeomorphisms which are Hamiltonian with respect to an associated
holomorphic symplectic structure. Later Hitchin produced nontrivial
generalized Kähler structures on del Pezzo surfaces, with a choice of
holomorphic Poisson structure playing a key role [10]. Also, Goto [7] has
extended the stability result of Kodaira-Spencer to the generalized Kähler
setting, with the restriction that one of the generalized complex structures
be defined by a pure spinor.

Our main purpose in this work is to describe a class of deformations which
generalizes and unifies different notions of “Kähler class” arising in the
different flavors of generalized Kähler geometry. It is well-known that two-
forms ($B$-fields) can act on generalized complex structures by conjugation,
with the integrability condition being preserved if and only if $B$ is closed.
Our deformations exploit a different, and moreover infinitessimal, action of
$B$-fields. In particular, we will say (cf. Definition 3.1) that a one-
parameter family of generalized Kähler structures is a _canonical deformation_
if there exists a one parameter family $K_{t}\in\Lambda^{2}$ such that for all
times $t$ where defined, one has

$\displaystyle\frac{\partial}{\partial
t}\mathbf{J}_{1}=[\mathbf{J}_{1},e^{K}\mathbf{J}_{1}],\qquad\frac{\partial}{\partial
t}\mathbf{J}_{2}=[\mathbf{J}_{2},e^{K}\mathbf{J}_{2}].$

Equivalence classes of canonical deformations lead to natural definitions of
generalized Kähler class and generalized Kähler cone (cf. §3). These
definitions make nonobvious departures from the classical idea of Kähler class
and Kähler cone. The first is the use of infinitessimal deformations as
opposed to ‘large’ deformations. Whereas any two metrics in the same Kähler
class admit an explicit relationship using the
$\sqrt{-1}\partial\overline{\partial}$-lemma, we can no longer expect such an
explicit relationship in general. For instance, as described above Joyce’s
construction of nontrivial GK structure uses diffeomorphisms which are
Hamiltonian with respect to the associated holomorphic symplectic structure,
and in general these cannot be described by a single potential function.
Instead, as is typical of Hamiltonian diffeomorphisms, we expect to be able to
explicitly describe their infinitessimal deformations. Moreover, given that in
the Kähler setting, deformations in the Kähler class involve freezing the
complex structure and varying the Kähler form, it is natural to imagine that
one should deform while fixing either $\mathbf{J}_{1},\mathbf{J}_{2}$.
Nonetheless through careful consideration of natural variational classes of
different flavors of generalized Kähler metrics it emerges that varying
$\mathbf{J}_{1}$ and $\mathbf{J}_{2}$ simultaneously will correctly capture
various existing notions of Kähler class in GK geometry.

A fundamental first step in unpacking this definition is to derive the
algebraic and differential conditions which are imposed on $K$ to preserve the
compatibility and integrability conditions for the pair
$(\mathbf{J}_{1},\mathbf{J}_{2})$. Through careful computations, it turns out
that the answer is pleasingly simple:

######  Theorem 1.1.

Given $M$ a smooth manifold, suppose $(\mathbf{J}_{1},\mathbf{J}_{2})$ is a
generalized Kähler structure. Suppose
$(\mathbf{J}_{1}^{t},\mathbf{J}_{2}^{t})$ is a one-parameter family of
generalized almost complex structures such that

$\displaystyle\frac{\partial}{\partial
t}\mathbf{J}_{1}=[\mathbf{J}_{1},e^{K}\mathbf{J}_{1}],\qquad\frac{\partial}{\partial
t}\mathbf{J}_{2}=[\mathbf{J}_{2},e^{K}\mathbf{J}_{2}],$

for some one parameter family $K_{t}\in\Lambda^{2}$. Then
$(\mathbf{J}_{1}^{t},\mathbf{J}_{2}^{t})$ is a one-parameter family of
generalized Kähler structures if and only if for all $t$ one has

  1. (1)

$K_{t}\in\Lambda^{1,1}_{J^{t}}$,

  2. (2)

$dK_{t}=0$,

  3. (3)

$\left<-\mathbf{J}_{1}^{t}\mathbf{J}_{2}^{t}\cdot,\cdot\right>>0$.

where $J^{t}$ is determined via the Gualtieri map, and $\left<,\right>$
denotes the symmetric neutral inner product on $T\oplus T^{*}$ (cf. §3.1).

In particular, this theorem exhibits that the canonical deformations are, as
is true in the Kähler setting, determined infinitessimally by a closed form
which is $(1,1)$ with respect to $J$. We emphasize here that the condition
that $dK=0$ does _not_ follow from the known fact that the conjugation action
of $B$-fields on generalized complex structures preserves integrability if and
only if $dB=0$. For instance, if we consider our infinitessimal action on a
single generalized complex structure, the condition to preserve integrability
is _strictly weaker_ than $dK=0$ (cf. Proposition 2.3). It is only in the
context of preserving the integrability conditions of generalized _Kähler_
geometry that one derives $dK=0$.

Despite the simplicity of the conditions of Theorem 1.1 and the apparent
simplicity of canonical deformations from the point of view of generalized
geometry, the deformations induced on the classical bihermitian data $(g,I,J)$
are delicate. Remarkably, these canonical deformations unify all previously
known instances of “Kähler class” in generalized geometry, specifically the
classical notion of Kähler class, the modified Kähler classes implicit in
Apostolov-Gualtieri ([2] Proposition 5, cf. also [6]) in the commuting GK
case, as well as Joyce’s Hamiltonian deformation construction in the
nondegenerate case. We state this for emphasis (cf. §3.6 for notation):

######  Proposition 1.2.

The following hold:

  1. (1)

Given $(M^{2n},g,J)$ a Kähler manifold, and $u\in C^{\infty}(M)$ such that
$\omega+\sqrt{-1}\partial\overline{\partial}u>0$, the one-parameter family

$\displaystyle(\omega_{I})_{t}=\omega+tdIdu,\qquad I_{t}=J,\qquad J_{t}=J$

arises as a canonical deformation of generalized Kähler structures for $0\leq
t\leq 1$ defined by

$\displaystyle K_{t}=dJdu.$

  2. (2)

Given $(M^{2n},g,I,J)$ a generalized Kähler manifold such that $[I,J]=0$, and
$u\in C^{\infty}(M)$ such that
$\omega_{I}+\sqrt{-1}\left(\partial_{+}\overline{\partial}_{+}-\partial_{-}\overline{\partial}_{-}\right)u>0$,
the one-parameter family

$\displaystyle(\omega_{I})_{t}=\omega_{I}+t\sqrt{-1}\left(\partial_{+}\overline{\partial}_{+}-\partial_{-}\overline{\partial}_{-}\right)u,\qquad
I_{t}=I,\qquad J_{t}=J$

arises as a canonical deformation of generalized Kähler structures for $0\leq
t\leq 1$ defined by

$\displaystyle K_{t}=dJdu.$

  3. (3)

Let $(M^{2n},g,I,J)$ be a generalized Kähler manifold such that the Poisson
structure $\sigma=\tfrac{1}{2}[I,J]g^{-1}$ is nondegenerate, with
$\Omega=\sigma^{-1}$. Given $u_{t}\in C^{\infty}(M)$ a family of smooth
functions, let $\phi_{t}$ denote the one-parameter family of
$\Omega$-Hamiltonian diffeomorphisms generated by $u_{t}$. Then, for all $t$
such that $-\operatorname{Im}\pi_{\Lambda_{I}^{1,1}}\Omega_{J}>0$, the one-
parameter family of generalized Kähler structures determined by

$\displaystyle\Omega_{t}=\Omega,\qquad I_{t}=I,\qquad J_{t}=\phi_{t}^{*}J$

defines a canonical deformation of generalized Kähler structures determined by

$\displaystyle K_{t}=dJ_{t}du_{t}.$

As a final point to contextualize these deformations, we recall that various
interesting deformation classes of generalized Kähler structure have been
produced using holomorphic Poisson structures. Given a generalized Kähler
structure $(g,I,J)$, there is a Poisson tensor

$\displaystyle\sigma=\tfrac{1}{2}[I,J]g^{-1}$

which is the real part of a holomorphic Poisson tensor with respect to both
$I$ and $J$. By choosing an appropriate deformation of $\sigma$, Hitchin [10]
produced deformations of Kähler metrics on del Pezzo surfaces to strictly
generalized Kähler structures. Also the deformation theory of Goto [7] changes
this underlying Poisson tensor. As it turns out our deformations fix $\sigma$
and $I$, so occur against a fixed background of a holomorphic Poisson
structure.

######  Corollary 1.3.

Given $M$ a smooth manifold, suppose $(\mathbf{J}_{1}^{t},\mathbf{J}_{2}^{t})$
is a canonical deformation of generalized Kähler structures. Then for all $t$,

$\displaystyle I^{t}\equiv I^{0},\qquad\sigma^{t}\equiv\sigma^{0}$

As an application, we are able to express the generalized Kähler-Ricci flow in
a simple way using canonical deformations. The equation is an extension of
Kähler-Ricci flow to the setting of generalized Kähler geometry, introduced by
the second author and Tian [13]. Recently this flow has been used to study the
global topology of the (nonlinear) space of generalized Kähler structures in
certain settings [3]. To describe this flow, fix $(g,I,J)$ a generalized
Kähler structure. Associated to the Hermitian structure $(g,I)$ is the Bismut
connection

$\displaystyle\nabla^{I}=D+\tfrac{1}{2}Hg^{-1},$

where $H=d^{c}_{I}\omega_{I}$, and $D$ denotes the Levi-Civita connection.
This is a Hermitian connection, and if $\Omega_{I}$ denotes its curvature, we
obtain a representative of the first Chern class via contraction, called the
Bismut-Ricci tensor:

$\displaystyle\rho_{I}=$ $\displaystyle\
\tfrac{1}{2}\operatorname{tr}\Omega_{I}I.$

From the Bianchi identity we know that $d\rho_{I}=0$, but it is not in general
true that $\rho_{I}\in\Lambda^{1,1}_{I}$, and we will let $\rho_{I}^{1,1}$
denote its $(1,1)$ projection. Furthermore, associated to $(g,I)$ we obtain
the _$I$ -Lee form_, defined by

$\displaystyle\theta_{I}(X)=d^{*}\omega_{I}(IX).$

Similarly we obtain the Lee form $\theta_{J}$ associated to $(g,J)$. With this
background in place, we can describe the generalized Kähler-Ricci flow in the
$I$-fixed gauge simply by

(1.1) $\displaystyle\begin{split}\frac{\partial}{\partial t}\omega_{I}=&\
-\rho_{I}^{1,1},\qquad\frac{\partial}{\partial
t}J=L_{\tfrac{1}{2}\left(\theta_{J}^{\sharp}-\theta_{I}^{\sharp}\right)}J.\end{split}$

The evolution of the complex structure $J$ is derived in [13], arising from
delicate gauge manipulations and curvature identities. On the other hand it
has been shown in several special cases (cf. §4 below) that the generalized
Kähler-Ricci is driven entirely by $\rho_{I}$. Using our description of
canonical deformations, and a further subtle curvature identity for
generalized Kähler manifolds (Proposition 4.2), we confirm that this is true
in full generality, and give a very simple description of generalized Kähler-
Ricci flow in terms of the associated generalized complex structures.

######  Theorem 1.4.

Let $(M^{2n},g_{t},I,J_{t})$ be a solution of generalized Kähler-Ricci flow in
the $I$-fixed gauge. The one parameter family of pairs of associated
generalized complex structures $(\mathbf{J}_{1}^{t},\mathbf{J}_{2}^{t})$
evolve by

(1.2) $\displaystyle\begin{split}\frac{\partial}{\partial t}\mathbf{J}_{1}=&\
[\mathbf{J}_{1},e^{\rho_{I}}\mathbf{J}_{1}],\qquad\frac{\partial}{\partial
t}\mathbf{J}_{2}=[\mathbf{J}_{2},e^{\rho_{I}}\mathbf{J}_{2}].\end{split}$

In other words, the generalized Kähler-Ricci flow is the canonical deformation
driven by the $I$-Bismut-Ricci tensor.

Immediately following from Theorem 1.4 and Corollary 1.3 is that fact that
generalized Kähler-Ricci flow preserves the underlying real Poisson tensor
$\sigma$, and moreover preserves the generalized Kähler cone associated to the
initial data.

######  Corollary 1.5.

Let $(M^{2n},g_{t},I,(J)_{t})$ be a solution of generalized Kähler-Ricci flow
in the $I$-fixed gauge. The associated one-parameter families of generalized
complex structures $(\mathbf{J}_{1},\mathbf{J}_{2})$ lies in the generalized
Kähler cone associated to the initial data. In particular, for all $t$ such
that the flow is defined,

$\displaystyle\sigma^{t}\equiv\sigma^{0},$

in other words, the real Poisson tensor $\sigma$ is fixed along the flow.

##  2\. Formal deformations of generalized complex structure

###  2.1. Background

Given $M$ a smooth manifold, the generalized tangent bundle is given by
$T\oplus T^{*}$. This bundle comes equipped with a family of natural brackets
determined by a closed three-form $H$. In particular, given
$H\in\Lambda^{3}T^{*}$, $dH=0$, define the twisted Courant bracket $[,]$ for
sections of $T\oplus T^{*}$ via

(2.1) $\displaystyle[X+\xi,Y+\eta]=$ $\displaystyle\ [X,Y]+L_{X}\eta-
L_{Y}\xi+\tfrac{1}{2}d\left(\xi(Y)-\eta(X)\right)+i_{Y}i_{X}H.$

A generalized complex structure $\mathbf{J}$ is then an almost complex
structure on $T\oplus T^{*}$, whose $\sqrt{-1}$-eigenbundle, denoted by $L$,
is integrable with respect to the the twisted Courant bracket. This condition
is naturally captured by a corresponding version of the Nijenhuis tensor,
where for a given almost complex structure we associate the natural projection
maps $\pi_{0,1},\pi_{1,0}$ and then for $\vec{x},\vec{y}\in T\oplus T^{*}$ we
have

(2.2) $\displaystyle
N_{\mathbf{J}}(\vec{x},\vec{y})=\pi_{0,1}[\pi_{1,0}(\vec{x}),\pi_{1,0}(\vec{y})].$

Direct computations show that this is tensorial, and vanishes if and only if
the associated almost generalized complex structure is integrable. See [8] for
further discussion.

###  2.2. Variations of generalized complex structure

To begin we define an action of $B$-fields on generalized complex structures.

######  Definition 2.1.

Given a smooth manifold $M$ and $K\in\Lambda^{2}$, define

$\displaystyle\Phi_{K}$ $\displaystyle:\operatorname{End}(T\oplus
T^{*})\to\operatorname{End}(T\oplus T^{*}),$ $\displaystyle\Phi_{K}$
$\displaystyle(\mathbf{J})=[\mathbf{J},e^{K}\mathbf{J}],$

where

$\displaystyle e^{K}=\left(\begin{matrix}1&0\\\
K&1\end{matrix}\right)\in\operatorname{End}(T\oplus T^{*}).$

For a given $\mathbf{J}$, we intend to use $\Phi_{K}(\mathbf{J})$ as a tangent
vector to a one-parameter variation of $\mathbf{J}$ through generalized
complex structures. We first note that variations of this kind will indeed
preserve the space of generalized almost complex structures.

######  Lemma 2.2.

Let $\mathbf{J}_{t}$ be a one-parameter family of endomorphisms of $T\oplus
T^{*}$ such that $\mathbf{J}_{0}$ is an almost generalized complex structure
and

$\displaystyle\frac{\partial}{\partial t}\mathbf{J}_{t}=$ $\displaystyle\
[\mathbf{J}_{t},e^{K_{t}}\mathbf{J}_{t}].$

Then $\mathbf{J}_{t}^{2}=-1$ for each $t$, i.e. $\mathbf{J}_{t}$ is a family
of generalized almost complex structures.

###### Proof.

Differentiating the expression $\mathbf{J}_{t}^{2}$ yields

$\displaystyle\frac{\partial}{\partial
t}\mathbf{J}_{t}^{2}=\left(\frac{\partial}{\partial
t}\mathbf{J}\right)\mathbf{J}+\mathbf{J}\left(\frac{\partial}{\partial
t}\mathbf{J}\right)=\left(\mathbf{J}e^{K}\mathbf{J}+e^{K}\right)\mathbf{J}+\mathbf{J}\left(\mathbf{J}e^{K}\mathbf{J}+e^{K}\right)=0.$

Thus $\frac{\partial}{\partial t}\mathbf{J}_{t}^{2}=0$, and since
$\mathbf{J}_{0}^{2}=-1$ the lemma follows. ∎

Next we can characterize the condition for these deformations to preserve
integrability of $\mathbf{J}_{t}$.

######  Proposition 2.3.

Let $\mathbf{J}_{t}$ be a one-parameter family of generalized almost complex
structures such that

$\displaystyle\left.\frac{\partial}{\partial
t}\mathbf{J}_{t}\right|_{t=0}=\Phi_{K}(\mathbf{J}_{0}).$

Then for $\vec{x},\vec{y}\in T\oplus T^{*}$,

$\displaystyle\left.\frac{\partial}{\partial
t}\right|_{t=0}N_{\mathbf{J}_{t}}(\vec{x},\vec{y})=$ $\displaystyle\
\sqrt{-1}\pi_{0,1}(dK\left(\pi_{T}\pi_{1,0}(\vec{y}),\pi_{T}\pi_{1,0}(\vec{x}),\cdot\right))$
$\displaystyle\
-\sqrt{-1}N_{\mathbf{J}_{0}}(\vec{x},\vec{y})+\mathbf{J}_{0}e^{K}N_{\mathbf{J}_{0}}(\vec{x},\vec{y})+N_{\mathbf{J}_{0}}(e^{K}\mathbf{J}_{0}(\vec{x}),\vec{y})+N_{\mathbf{J}_{0}}(\vec{x},e^{K}\mathbf{J}_{0}\vec{y}).$

###### Proof.

For notational simplicity we set $\mathbf{J}=\mathbf{J}_{0}$. We differentiate
the formula (2.2) at $t=0$, using the explicit formulae for the projection
maps, to obtain

(2.3) $\displaystyle\begin{split}\tfrac{\partial}{\partial
t}&(\pi^{t}_{0,1}\left[\pi^{t}_{1,0}(\vec{x}),\pi^{t}_{1,0}(\vec{y})\right])\\\
=&\ \tfrac{\partial}{\partial
t}(\pi^{t}_{0,1})\left[\pi^{t}_{1,0}(\vec{x}),\pi^{t}_{1,0}(\vec{y})\right]+\pi^{t}_{0,1}\left[\tfrac{\partial}{\partial
t}\pi^{t}_{1,0}(\vec{x}),\pi^{t}_{1,0}(\vec{y})\right]+\pi^{t}_{0,1}\left[\pi^{t}_{1,0}(\vec{x}),\tfrac{\partial}{\partial
t}\pi^{t}_{1,0}(\vec{y})\right]\\\ =&\
\tfrac{\sqrt{-1}}{2}\left\\{\Phi_{K}(\mathbf{J})[\pi_{1,0}(\vec{x}),\pi_{1,0}(\vec{y})]-\pi_{0,1}[\Phi_{K}(\mathbf{J})(\vec{x}),\pi_{1,0}(\vec{y})]-\pi_{0,1}[\pi_{1,0}(\vec{x}),\Phi_{K}(\mathbf{J})(\vec{y})]\right\\}.\end{split}$

Note that for any $\vec{z}\in T\oplus T^{*}$ we may write
$\vec{z}=2\pi_{1,0}(\vec{z})+\sqrt{-1}\mathbf{J}(\vec{z})$ which leads to

$\displaystyle\Phi_{K}(\mathbf{J})(\vec{z})$
$\displaystyle=(\mathbf{J}e^{K}\mathbf{J}+e^{K})(\vec{z})$
$\displaystyle=\mathbf{J}e^{K}\mathbf{J}(\vec{z})+e^{K}(2\pi_{1,0}(\vec{z})+\sqrt{-1}\mathbf{J}(\vec{z}))$
$\displaystyle=2e^{K}\pi_{1,0}(\vec{z})+\sqrt{-1}\left(e^{K}\mathbf{J}\vec{z}-\sqrt{-1}\mathbf{J}(e^{K}\mathbf{J}\vec{z})\right)$
$\displaystyle=2e^{K}\pi_{1,0}(\vec{z})+2\sqrt{-1}\pi_{1,0}\left(e^{K}\mathbf{J}(\vec{z})\right).$

This observation allows us to rewrite the final line of 2.3 as

$\displaystyle...=$ $\displaystyle\
\sqrt{-1}e^{K}\pi_{1,0}[\pi_{1,0}(\vec{x}),\pi_{1,0}(\vec{y})]-\pi_{1,0}e^{K}\mathbf{J}[\pi_{1,0}(\vec{x}),\pi_{1,0}(\vec{y})]$
$\displaystyle\
-\sqrt{-1}\pi_{0,1}[e^{K}\pi_{1,0}(\vec{x})+\sqrt{-1}\pi_{1,0}e^{K}\mathbf{J}(\vec{x}),\pi_{1,0}(\vec{y})]$
$\displaystyle\
-\sqrt{-1}[\pi_{1,0}(\vec{x}),e^{K}\pi_{1,0}(\vec{y})+\sqrt{-1}\pi_{1,0}e^{K}\mathbf{J}(\vec{y})]$
$\displaystyle=$ $\displaystyle\
\sqrt{-1}e^{K}\pi_{1,0}[\pi_{1,0}(\vec{x}),\pi_{1,0}(\vec{y})]-\pi_{1,0}e^{K}\mathbf{J}[\pi_{1,0}(\vec{x}),\pi_{1,0}(\vec{y})]$
$\displaystyle\
-\sqrt{-1}\pi_{0,1}[e^{K}\pi_{1,0}(\vec{x}),\pi_{1,0}(\vec{y})]-\sqrt{-1}\pi_{0,1}[\pi_{1,0}(\vec{x}),e^{K}\pi_{1,0}(\vec{y})]$
$\displaystyle\
+\pi_{0,1}\left([\pi_{1,0}e^{K}\mathbf{J}(\vec{x}),\pi_{1,0}(\vec{y})]+[\pi_{1,0}(\vec{x}),\pi_{1,0}(e^{K}\mathbf{J}\vec{y})]\right).$

Focusing on the first two terms, letting
$\vec{z}=[\pi_{1,0}(\vec{x}),\pi_{1,0}(\vec{y})]$,

$\displaystyle\sqrt{-1}e^{K}\pi_{1,0}\vec{z}-\pi_{1,0}e^{K}\mathbf{J}\vec{z}$
$\displaystyle=\frac{\sqrt{-1}}{2}e^{K}(\vec{z}-\sqrt{-1}\mathbf{J}\vec{z})-\tfrac{1}{2}(e^{K}\mathbf{J}\vec{z}-\sqrt{-1}\mathbf{J}e^{K}\mathbf{J}\vec{z})$
$\displaystyle=\frac{\sqrt{-1}}{2}(e^{K}\vec{z}+\mathbf{J}e^{K}\mathbf{J}\vec{z})$
$\displaystyle=\sqrt{-1}\pi_{0,1}e^{K}\vec{z}+\tfrac{1}{2}\mathbf{J}e^{K}\vec{z}+\frac{\sqrt{-1}}{2}\mathbf{J}e^{K}\mathbf{J}\vec{z}$
$\displaystyle=\sqrt{-1}\pi_{0,1}(e^{K}\vec{z})+\tfrac{1}{2}\mathbf{J}e^{K}(\vec{z}+\sqrt{-1}\mathbf{J}\vec{z})$
$\displaystyle=\sqrt{-1}\pi_{0,1}(e^{K}\vec{z})+\mathbf{J}e^{K}\pi_{0,1}(\vec{z}).$

Equation 2.3 now simplifies further to

(2.4) $\displaystyle\begin{split}...=&\
\sqrt{-1}\pi_{0,1}\left(e^{K}[\pi_{1,0}(\vec{x}),\pi_{1,0}(\vec{y})]-[e^{K}\pi_{1,0}(\vec{x}),\pi_{1,0}(\vec{y})]-[\pi_{1,0}(\vec{x}),e^{K}\pi_{1,0}(\vec{y})]\right)\\\
&\
+\mathbf{J}e^{K}\pi_{0,1}[\pi_{1,0}(\vec{x}),\pi_{1,0}(\vec{y})]+\pi_{0,1}\left([\pi_{1,0}e^{K}\mathbf{J}(\vec{x}),\pi_{1,0}(\vec{y})]+[\pi_{1,0}(\vec{x}),\pi_{1,0}(e^{K}\mathbf{J}\vec{y})]\right)\\\
=&\
\sqrt{-1}\pi_{0,1}\left(e^{K}[\pi_{1,0}(\vec{x}),\pi_{1,0}(\vec{y})]-[e^{K}\pi_{1,0}(\vec{x}),\pi_{1,0}(\vec{y})]-[\pi_{1,0}(\vec{x}),e^{K}\pi_{1,0}(\vec{y})]\right)\\\
&\
+\mathbf{J}e^{K}N(\vec{x},\vec{y})+N(e^{K}\mathbf{J}(\vec{x}),\vec{y})+N(\vec{x},e^{K}\mathbf{J}\vec{y}).\end{split}$

The Courant bracket satisfies
$e^{K}[X+\xi,Y+\eta]=[e^{K}(X+\xi),e^{K}(Y+\eta)]+\iota_{X}\iota_{Y}dK$. Using
this together with the fact that the Courant bracket involving a section with
no tangent vector component vanishes we see that the first term of (2.4)
becomes

$\displaystyle\sqrt{-1}\pi_{0,1}$
$\displaystyle\left([e^{K}\pi_{1,0}(\vec{x}),e^{K}\pi_{1,0}(\vec{y})]-[e^{K}\pi_{1,0}(\vec{x}),\pi_{1,0}(\vec{y})]-[\pi_{1,0}(\vec{x}),e^{K}\pi_{1,0}(\vec{y})]+dK\left(\pi_{T}\pi_{1,0}(\vec{y}),\pi_{T}\pi_{1,0}(\vec{x}),\cdot\right)\right)$
$\displaystyle=$ $\displaystyle\
\sqrt{-1}\pi_{0,1}\left(-[\pi_{1,0}(\vec{x}),\pi_{1,0}(\vec{y})]+dK\left(\pi_{T}\pi_{1,0}(\vec{y}),\pi_{T}\pi_{1,0}(\vec{x}),\cdot\right)\right)$
$\displaystyle=$ $\displaystyle\
-\sqrt{-1}N(\vec{x},\vec{y})+\sqrt{-1}\pi_{0,1}(dK\left(\pi_{T}\pi_{1,0}(\vec{y}),\pi_{T}\pi_{1,0}(\vec{x}),\cdot\right)).$

Collecting these computations gives the result. ∎

######  Corollary 2.4.

Let $\mathbf{J}_{t}$ be a one-parameter family of generalized almost complex
structures such that $\mathbf{J}_{0}$ is integrable and for all $t$ one has

$\displaystyle\frac{\partial}{\partial t}\mathbf{J}=\Phi_{K_{t}}(\mathbf{J}),$

where furthermore for all $\vec{x},\vec{y}\in T\oplus T^{*}$ one has

(2.5)
$\displaystyle\pi_{0,1}^{t}dK_{t}\left(\pi_{T}\pi^{t}_{1,0}(\vec{y}),\pi_{T}\pi^{t}_{1,0}(\vec{x}),\cdot\right)=$
$\displaystyle\ 0.$

Then $\mathbf{J}_{t}$ are integrable for each $t$.

###### Proof.

Choosing any Hermitian metric on $(T\oplus T^{*})\otimes\mathbb{C}$, using
Proposition 2.3 and the hypothesis (2.5) one directly derives for all $t$

$\displaystyle\frac{\partial}{\partial
t}\left|N_{\mathbf{J}_{t}}\right|^{2}\leq$ $\displaystyle\
C(K,\mathbf{J})\left|N_{\mathbf{J}_{t}}\right|^{2}.$

Since $N_{\mathbf{J}_{0}}=0$ the result follows from Gronwall’s inequality. ∎

######  Remark 2.5.

We may also formulate integrability of generalized complex structures in terms
of its $-\sqrt{-1}$-eigenbundle, $\overline{L}_{t}$, being Courant integrable.
Replicating the above arguments with the roles of $\pi_{1,0}$ and $\pi_{0,1}$
reversed we obtain that the relevant condition on $K_{t}$ is

$\displaystyle
0=\pi_{0,1}^{t}dK_{t}\left(\pi_{T}\pi_{0,1}(\vec{y}),\pi_{T}\pi_{0,1}(\vec{x}),\cdot\right).$

##  3\. Variations of generalized Kähler structure

Having defined certain variations of generalized complex structure, we now
extend this to defining variations of generalized Kähler structure. A naive
guess would be that we should simply take a variation of one of the underlying
generalized complex structures and seek the further integrability conditions.
However, for reasons to be illuminated by the examples below, it is much more
natural to vary _both_ generalized complex structures by a single $B$-field as
described in §2.2.

###  3.1. Background

A generalized Kähler structure is a pair of commuting generalized complex
structures $\mathbf{J}_{1},\mathbf{J}_{2}$ such that
$\mathbf{G}=-\mathbf{J}_{1}\mathbf{J}_{2}$ defines a generalized metric, i.e.
$\left<\mathbf{G}\cdot,\cdot\right>$ is a positive definite inner product on
$T\oplus T^{*}$, where $\left<,\right>$ denotes the symmetric neutral inner
product on $T\oplus T^{*}$, i.e.

$\displaystyle\left<X+\xi,Y+\eta\right>=$ $\displaystyle\
\tfrac{1}{2}\left(\xi(Y)+\eta(X)\right).$

A fundamental theorem of Gualtieri ([8] Chapter 6) says that a generalized
Kähler structure $(\mathbf{J}_{1},\mathbf{J}_{2})$ as defined here corresponds
to a bihermitian structure $(g,I,J,b)$, with Kähler forms
$\omega_{I},\omega_{J}$, as described in the introduction. The explicit
relationship is given by

(3.1) $\mathbf{J}_{1/2}=\tfrac{1}{2}e^{b}\left(\begin{array}[]{cc}I\pm
J&-(\omega_{I}^{-1}\mp\omega_{J}^{-1})\\\\[2.0pt]
\omega_{I}\mp\omega_{J}&-(I^{*}\pm J^{*})\end{array}\right)e^{-b}.$

We recall that a generalized Kähler structure induces a fourfold decomposition
of the complexified generalized tangent bundle. Specifically, letting $L_{i}$
and $\overline{L}_{i}$ denote the $\pm\sqrt{-1}$-eigenbundles of
$\mathbf{J}_{i}$ respectively, we have the following decomposition:

$(T\oplus T^{*})\otimes\mathbb{C}=L_{1}^{+}\oplus
L_{1}^{-}\oplus\overline{L_{1}^{-}}\oplus\overline{L_{1}^{+}}:=(L_{1}\cap
L_{2})\oplus(L_{1}\cap\overline{L}_{2})\oplus(\overline{L_{1}}\cap
L_{2})\oplus(\overline{L}_{1}\cap\overline{L}_{2}).$

###  3.2. Definitions

######  Definition 3.1.

A one-parameter family of generalized Kähler structures
$(\mathbf{J}_{1}^{t},\mathbf{J}_{2}^{t})$ is a _canonical family_ if for all
$t$, there exists $K_{t}\in\Lambda^{2}$ such that

$\displaystyle\frac{\partial}{\partial t}\mathbf{J}_{1}^{t}=$ $\displaystyle\
\Phi_{K_{t}}(\mathbf{J}^{t}_{1}),\qquad\frac{\partial}{\partial
t}\mathbf{J}_{2}^{t}=\Phi_{K_{t}}(\mathbf{J}^{t}_{2}).$

Given $\widetilde{\mathbf{J}}_{1},\widetilde{\mathbf{J}}_{2}$ another
generalized Kähler structure, we define an equivalence relation where

$\displaystyle(\widetilde{\mathbf{J}}_{1},\widetilde{\mathbf{J}}_{2})\sim(\mathbf{J}_{1},\mathbf{J}_{2})$

if and only if there exists a canonical family
$(\mathbf{J}_{1}^{t},\mathbf{J}_{2}^{t})$, $t\in[0,1]$, such that
$(\mathbf{J}_{1}^{0},\mathbf{J}_{2}^{0})=(\mathbf{J}_{1},\mathbf{J}_{2})$,
$(\mathbf{J}_{1}^{1},{\mathbf{J}}_{2}^{1})=(\widetilde{\mathbf{J}}_{1},\widetilde{\mathbf{J}}_{2})$.
Furthermore, the _generalized Kähler cone associated to
$(\mathbf{J}_{1},\mathbf{J}_{2})$_ is

$\displaystyle\mathcal{GK}(\mathbf{J}_{1},\mathbf{J}_{2})=\\{(\widetilde{\mathbf{J}}_{1},\widetilde{\mathbf{J}}_{2})\mbox{
generalized K\"{a}hler }|\
(\widetilde{\mathbf{J}}_{1},\widetilde{\mathbf{J}}_{2})\sim(\mathbf{J}_{1},\mathbf{J}_{2})\\}.$

###  3.3. Compatibility Condition

We first address the condition required for a canonical deformation to
preserve the algebraic compatibility condition of generalized Kähler
structures. We first prove a formal lemma reducing this to an algebraic
condition on $K$, then analyze this explicitly using the Gualtieri map.

######  Lemma 3.2.

Let $\mathbf{J}_{1}^{t},\mathbf{J}_{2}^{t}$ be one-parameter families of
generalized almost complex structures, with
$[\mathbf{J}_{1}^{0},\mathbf{J}_{2}^{0}]=0$, which satisfy

$\displaystyle\frac{\partial}{\partial
t}\mathbf{J}_{1}^{t}=\Phi_{K_{t}}(\mathbf{J}_{1}),\qquad\frac{\partial}{\partial
t}\mathbf{J}_{2}^{t}=\Phi_{K_{t}}(\mathbf{J}_{2}).$

Then $[\mathbf{J}_{1}^{t},\mathbf{J}_{2}^{t}]=0$ for all $t$ if and only if

$\displaystyle\begin{split}[\Phi_{K}(\mathbf{J}_{1}),\mathbf{J}_{2}]=[\Phi_{K}(\mathbf{J}_{2}),\mathbf{J}_{1}]\end{split}$

for all $t$.

###### Proof.

Differentiating $[\mathbf{J}_{1}^{t},\mathbf{J}_{2}^{t}]$ at any time $t$
shows

$\displaystyle\frac{\partial}{\partial
t}[\mathbf{J}_{1}^{t},\mathbf{J}_{2}^{t}]$
$\displaystyle=\Phi_{K}(\mathbf{J}_{1})\mathbf{J}_{2}+\mathbf{J}_{1}\Phi_{K}(\mathbf{J}_{2})-\Phi_{K}(\mathbf{J}_{2})\mathbf{J}_{1}-\mathbf{J}_{2}\Phi_{K}(\mathbf{J}_{1})$
$\displaystyle=[\Phi_{K}(\mathbf{J}_{1}),\mathbf{J}_{2}]-[\Phi_{K}(\mathbf{J}_{2}),\mathbf{J}_{1}].$

Since $[\mathbf{J}_{1}^{0},\mathbf{J}_{2}^{0}]=0$, the result follows. ∎

Here we reformulate the compatibility condition of Lemma 3.2 by expanding the
necessary equation in terms of the Gualtieri map and analyzing the result,
which simplifies dramatically.

######  Proposition 3.3.

Given $M$ a smooth manifold and $(\mathbf{J}_{1},\mathbf{J}_{2})$ a
generalized Kähler structure, for $K\in\Lambda^{2}$ one has

$\displaystyle[\Phi_{K}(\mathbf{J}_{1}),\mathbf{J}_{2}]=[\Phi_{K}(\mathbf{J}_{2}),\mathbf{J}_{1}]$

if and only if

$\displaystyle K\in\Lambda^{1,1}_{J}.$

###### Proof.

Let $\Upsilon_{1/2}=\tfrac{1}{2}\begin{pmatrix}(I\pm
J)&-(\omega_{I}^{-1}\mp\omega_{J}^{-1})\\\
(\omega_{I}\mp\omega_{J})&-(I^{*}\pm J^{*})\end{pmatrix}$, so that 3.1 can be
expressed as $\mathbf{J}_{1/2}=e^{b}\Upsilon_{1/2}e^{-b}$. Using this notation
and the fact that $e^{K}$ and $e^{b}$ commute, it follows easily that

$\displaystyle[\Phi_{K}(\mathbf{J}_{1}),\mathbf{J}_{2}]=$ $\displaystyle\
e^{b}[\Phi_{K}(\Upsilon_{1}),\Upsilon_{2}]e^{-b},$
$\displaystyle[\Phi_{K}(\mathbf{J}_{2}),\mathbf{J}_{1})]=$ $\displaystyle\
e^{b}[\Phi_{K}(\Upsilon_{2}),\Upsilon_{1}]e^{-b}.$

Hence
$[\Phi_{K}(\mathbf{J}_{1}),\mathbf{J}_{2}]=[\Phi_{K}(\mathbf{J}_{2}),\mathbf{J}_{1})]$
reduces to the condition
$[\Phi_{K}(\Upsilon_{1}),\Upsilon_{2}]=[\Phi_{K}(\Upsilon_{2}),\Upsilon_{1}]$.
As a first step, we record the simplified forms of $\Phi_{K}(\Upsilon_{1/2})$
obtained through a direct computation:

$\Phi_{K}(\Upsilon_{1/2})=\frac{1}{4}\left(\begin{matrix}-(\omega_{I}^{-1}\mp\omega_{J}^{-1})K(I\pm
J)&(\omega_{I}^{-1}\mp\omega_{J}^{-1})K(\omega_{I}^{-1}\mp\omega_{J}^{-1})\\\\[6.0pt]
4K-(I^{*}\pm J^{*})K(I\pm J)&(I^{*}\pm
J^{*})K(\omega_{I}^{-1}\mp\omega_{J}^{-1})\\\ \end{matrix}\right).$

Further tedious computation yields

$\displaystyle[\Phi_{K}(\mathbf{J}_{1}),\mathbf{J}_{2}]$
$\displaystyle=\tfrac{1}{2}\begin{pmatrix}(\omega_{I}^{-1}+\omega_{J}^{-1})K+g^{-1}K(I+J)&(\omega_{I}^{-1}-\omega_{J}^{-1})Kg^{-1}-g^{-1}K(\omega_{I}^{-1}-\omega_{J}^{-1})\\\\[6.0pt]
K(I-J)+(I^{*}-J^{*})K&(I^{*}+J^{*})Kg^{-1}-K(\omega_{I}^{-1}+\omega_{J}^{-1})\end{pmatrix},$
$\displaystyle[\Phi_{K}(\mathbf{J}_{2}),\mathbf{J}_{1}]$
$\displaystyle=\tfrac{1}{2}\begin{pmatrix}(\omega_{I}^{-1}-\omega_{J}^{-1})K+g^{-1}K(I-J)&(\omega_{I}^{-1}+\omega_{J}^{-1})Kg^{-1}-g^{-1}K(\omega_{I}^{-1}+\omega_{J}^{-1})\\\\[6.0pt]
K(I+J)+(I^{*}+J^{*})K&(I^{*}-J^{*})Kg^{-1}-K(\omega_{I}^{-1}-\omega_{J}^{-1})\end{pmatrix}.$

By comparing each entry of the matrices above, we see equality holds if and
only if $KJ=-J^{*}K$, as required. ∎

###  3.4. Integrability Condition

We next address the integrability condition. Since our deformations should
preserve integrability of each generalized complex structure $\mathbf{J}_{i}$,
Proposition 2.3 yields two partial integrability conditions which $K$ must
satisfy. We again emphasize that neither of these conditions alone will force
$dK=0$, while somewhat surprisingly the combination of the two conditions
does.

######  Proposition 3.4.

Given $M$ a smooth manifold and $(\mathbf{J}_{1},\mathbf{J}_{2})$ a
generalized Kähler structure, for $K\in\Lambda^{2}$ one has

$\displaystyle\pi_{0,1}^{\mathbf{J}_{i}}\left(\iota_{X}\iota_{Y}dK\right)=$
$\displaystyle\ 0\quad\text{\ for all \ }X,Y\in\pi_{T}(L_{i}),\quad i=1,2.$

if and only if

$\displaystyle dK=0.$

###### Proof.

The sufficiency of $dK=0$ is obvious, we prove it is necessary. Note that for
a pure covector $\xi$, one has
$\mathbf{J}_{1/2}\xi=e^{b}\Upsilon_{1/2}\xi=-\tfrac{1}{2}\begin{pmatrix}(\omega_{I}^{-1}\mp\omega_{J}^{-1})\xi\\\
b(\omega_{I}^{-1}\mp\omega_{J}^{-1})\xi+(I^{*}\pm J^{*})\xi\end{pmatrix}$ and
so

(3.2) $\displaystyle\pi_{0,1}^{1/2}(\xi)$
$\displaystyle=\begin{pmatrix}\frac{-\sqrt{-1}}{2}(\omega_{I}^{-1}\mp\omega_{J}^{-1})\xi\\\
\xi-\frac{\sqrt{-1}}{2}\left(b(\omega_{I}^{-1}\mp\omega_{J}^{-1})\xi+(I^{*}\pm
J^{*})\xi\right)\end{pmatrix}.$

Fix vectors $X,Y\in T^{1,0}_{I}$, and then choose lifts $X_{+},Y_{+}$ to
$C_{+}$, the $+1$-eigenspace of $\mathbf{G}$. Using the representation of
$\mathbf{J}_{i}$ with respect to the $\pm 1$-eigenspace decomposition induced
by $\mathbf{G}$ ([8] Proposition 6.12), it follows that $X_{+},Y_{+}\in
L_{1}\cap L_{2}=L_{1}^{+}$. Now let $\xi=i_{Y}i_{X}dK$, and note that
Proposition 2.3 applied to both $\mathbf{J}_{1}$ and $\mathbf{J}_{2}$ implies
that

$\displaystyle\pi_{0,1}^{1/2}(\xi)=0.$

Comparing against equation (3.2) we obtain
$(\omega_{I}^{-1}\mp\omega_{J}^{-1})(\xi)=0$. Therefore
$\iota_{Y}\iota_{X}dK=0$, for all $X,Y\in T^{1,0}_{I}$. Since $K$ is real it
follows that $dK=0$. ∎

###### Proof of Theorem 1.1.

Fix $(\mathbf{J}_{1},\mathbf{J}_{2})$ generalized Kähler and fix
$(\mathbf{J}_{1}^{t},\mathbf{J}_{2}^{t})$ a one-parameter family as in the
statement. First let us assume conditions (1), (2), and (3) hold for this
family. Since $dK_{t}=0$ for all $t$, it follows from Corollary 2.4 that
$(\mathbf{J}_{1}^{t},\mathbf{J}_{2}^{t})$ are integrable generalized complex
structures. Furthermore, using that $K_{t}\in\Lambda^{1,1}_{J^{t}}$ for all
$t$, it follows from Lemma 2.2 and Proposition 3.3 that
$[\mathbf{J}_{1}^{t},\mathbf{J}_{2}^{t}]=0$ for all $t$. Since we have assumed
the positivity of $\left<-\mathbf{J}_{1}\mathbf{J}_{2}\cdot,\cdot\right>$ in
condition (3), it follows that $(\mathbf{J}_{1}^{t},\mathbf{J}_{2}^{t})$ is
generalized Kähler for all $t$.

Conversely, suppose $(\mathbf{J}_{1}^{t},\mathbf{J}_{2}^{t})$ defines a
generalized Kähler structure for all $t$. Condition (3) then holds by
definition, and condition (1) holds by Lemma 2.2 and Proposition 3.3. As the
structures $(\mathbf{J}_{1}^{t},\mathbf{J}_{2}^{t})$ are assumed integrable
for all times $t$, their Nijenhuis tensors vanish for all $t$, and thus it
follows from Proposition 2.3 that

$\displaystyle 0=$ $\displaystyle\
\pi_{0,1}^{\mathbf{J}_{i}}dK\left(\pi_{T}\pi_{1,0}^{\mathbf{J}_{i}}(\vec{x}),\pi_{T}\pi_{1,0}^{\mathbf{J}_{i}}(\vec{y}),\cdot\right)$

for all $\vec{x},\vec{y}\in T\oplus T^{*}$, all $t$, and $i=1,2$. It then
follows from Proposition 3.4 that $dK=0$, as required. ∎

With this characterization of canonical deformations in hand, we can now give
the definition of generalized Kähler _classes_ which emerges naturally from
Theorem 1.1.

######  Definition 3.5.

Let $M$ be a smooth manifold. An _exact canonical deformation_ is a one-
parameter family of generalized Kähler structures
$(\mathbf{J}^{t}_{1},\mathbf{J}^{t}_{2})$ such that, for all $t$,

$\displaystyle\frac{\partial}{\partial
t}\mathbf{J}_{i}^{t}=\Phi_{da_{t}}\mathbf{J}_{i}^{t},$

for some $a_{t}\in\Lambda^{1}$. Given
$\widetilde{\mathbf{J}}_{1},\widetilde{\mathbf{J}}_{2}$ another generalized
Kähler structure, we define an equivalence relation where

$\displaystyle(\widetilde{\mathbf{J}}_{1},\widetilde{\mathbf{J}}_{2})\sim_{\mbox{\tiny{exact}}}(\mathbf{J}_{1},\mathbf{J}_{2})$

if and only if there exists an exact canonical deformation
$(\mathbf{J}_{1}^{t},\mathbf{J}_{2}^{t})$, $t\in[0,1]$, such that
$(\mathbf{J}_{1}^{0},\mathbf{J}_{2}^{0})=(\mathbf{J}_{1},\mathbf{J}_{2})$,
$(\mathbf{J}_{1}^{1},{\mathbf{J}}_{2}^{1})=(\widetilde{\mathbf{J}}_{1},\widetilde{\mathbf{J}}_{2})$.
Furthermore, the _generalized Kähler class of
$(\mathbf{J}_{1},\mathbf{J}_{2})$_ is

$\displaystyle[(\mathbf{J}_{1},\mathbf{J}_{2})]=\\{(\widetilde{\mathbf{J}}_{1},\widetilde{\mathbf{J}}_{2})\mbox{
generalized K\"{a}hler }|\
(\widetilde{\mathbf{J}}_{1},\widetilde{\mathbf{J}}_{2})\sim_{\mbox{\tiny{exact}}}(\mathbf{J}_{1},\mathbf{J}_{2})\\}.$

###  3.5. Induced variations

In this section we derive the variation on the associated bihermitian data
induced by a canonical deformation through an analysis of the Gualtieri map.

######  Proposition 3.6.

Let $(\mathbf{J}_{1}^{t},\mathbf{J}_{2}^{t})$ be a canonical family, and let
$(g_{t},b_{t},I_{t},J_{t})$ denote the corresponding $1$-parameter family of
bihermitian data. Then

(3.3) $\displaystyle\begin{split}\dot{g}=-\tfrac{1}{2}[K,I],&\
\qquad\dot{b}=-\tfrac{1}{2}\\{K,I\\},\\\
\dot{\omega}_{I}=-\tfrac{1}{2}[K,I]I,&\
\qquad\dot{\omega}_{J}=-\tfrac{1}{2}\\{K,IJ\\},\\\ \dot{I}=0,&\
\qquad\dot{J}=\tfrac{1}{2}[I,J]g^{-1}K.\end{split}$

###### Proof.

We use the notation and computations of Proposition 3.3. In particular, we
recall that

$\displaystyle
e^{-b}\dot{\mathbf{J}}_{1/2}e^{b}=\frac{1}{4}\begin{pmatrix}-(\omega_{I}^{-1}\mp\omega_{J}^{-1})K(I\pm
J)&(\omega_{I}^{-1}\mp\omega_{J}^{-1})K(\omega_{I}^{-1}\mp\omega_{J}^{-1})\\\\[4.0pt]
4K-(I^{*}\pm J^{*})K(I\pm J)&(I^{*}\pm
J^{*})K(\omega_{I}^{-1}\mp\omega_{J}^{-1})\end{pmatrix}.$

On the other hand, writing expression 3.3 as
$\mathbf{J}_{1/2}=\tfrac{1}{2}e^{b}\Upsilon_{1/2}e^{-b}$ and differentiating,
using the fact that $e^{b}\dot{e}^{b}=\dot{e}^{b}=\dot{e}^{b}e^{b}$, yields

(3.4) $\displaystyle\begin{split}e^{-b}\dot{\mathbf{J}}_{1/2}e^{b}=&\
\tfrac{1}{2}e^{-b}\left(\dot{e}^{b}\Upsilon_{1/2}e^{-b}+e^{b}\dot{\Upsilon}_{1/2}e^{-b}-e^{b}\Upsilon_{1/2}\dot{e}^{b}\right)\\\
=&\
\tfrac{1}{2}[\dot{e}^{b},\Upsilon_{1/2}]+\tfrac{1}{2}\dot{\Upsilon}_{1/2}\\\
=&\
\tfrac{1}{2}\begin{pmatrix}(\omega_{I}^{-1}\mp\omega_{J}^{-1})\dot{b}+(\dot{I}\pm\dot{J})&-(\dot{\omega}_{I}^{-1}\mp\dot{\omega}_{J}^{-1})\\\\[6.0pt]
\\{\dot{b},I\pm
J\\}+(\dot{\omega}_{I}\mp\dot{\omega}_{J})&-\dot{b}(\omega_{I}^{-1}\mp\omega_{J}^{-1})-(\dot{I}^{*}\pm\dot{J}^{*})\end{pmatrix},\end{split}$

Then equating the appropriate expressions coming from above shows

$\displaystyle e^{-b}(\dot{\mathbf{J}}_{1}+\dot{\mathbf{J}}_{2})e^{b}=$
$\displaystyle\
\begin{pmatrix}\dot{I}+\omega_{I}^{-1}\dot{b}&\omega_{I}^{-1}\dot{\omega}_{I}\omega_{I}^{-1}\\\\[4.0pt]
\dot{\omega}_{I}+\\{\dot{b},I\\}&-(\dot{I}^{*}+\dot{b}\omega_{I}^{-1})\end{pmatrix}$
$\displaystyle=$ $\displaystyle\
\tfrac{1}{2}\begin{pmatrix}\omega_{J}^{-1}KJ-\omega_{I}^{-1}KI&\omega_{I}^{-1}K\omega_{I}^{-1}+\omega_{J}^{-1}K\omega_{J}^{-1}\\\\[4.0pt]
4K-(I^{*}KI+J^{*}KJ)&I^{*}K\omega_{I}^{-1}-J^{*}K\omega_{J}^{-1}\end{pmatrix},$
$\displaystyle e^{-b}(\dot{\mathbf{J}}_{1}-\dot{\mathbf{J}}_{2})e^{b}=$
$\displaystyle\
\begin{pmatrix}\dot{J}-\omega_{J}^{-1}\dot{b}&-\omega_{J}^{-1}\dot{\omega}_{J}\omega_{J}^{-1}\\\\[4.0pt]
\\{\dot{b},J\\}-\dot{\omega}_{J}&\dot{b}\omega_{J}^{-1}-\dot{J}^{*}\end{pmatrix}$
$\displaystyle=$ $\displaystyle\
\tfrac{1}{2}\begin{pmatrix}\omega_{J}^{-1}KI-\omega_{I}^{-1}KJ&-(\omega_{I}^{-1}K\omega_{J}^{-1}+\omega_{J}^{-1}K\omega_{I}^{-1})\\\\[4.0pt]
-(I^{*}KJ+J^{*}KI)&J^{*}K\omega_{I}^{-1}-I^{*}K\omega_{J}^{-1}\end{pmatrix}.$

Turning to the top right entries of each expression shows

$\displaystyle\dot{\omega}_{I}$
$\displaystyle=\tfrac{1}{2}(K+\omega_{I}\omega_{J}^{-1}K\omega_{J}^{-1}\omega_{I})=\tfrac{1}{2}(K+I^{*}J^{*}KJI)=\tfrac{1}{2}(K+I^{*}KI)=-\tfrac{1}{2}[K,I]I,$
$\displaystyle\dot{\omega}_{J}$
$\displaystyle=\tfrac{1}{2}(\omega_{J}\omega_{I}^{-1}K+K\omega_{I}^{-1}\omega_{J})=-\tfrac{1}{2}(J^{*}I^{*}K+KIJ)=-\tfrac{1}{2}\\{K,IJ\\}.$

For the remaining data we differentiate the generalized metric
$\mathbf{G}_{t}=-(\mathbf{J}_{1})_{t}(\mathbf{J}_{2})_{t}=\begin{pmatrix}A_{t}&g^{-1}_{t}\\\
\delta_{t}&A_{t}^{*}\end{pmatrix}$, where $g_{t}$ is the associated metric and
$b_{t}=-g_{t}A_{t}$. It follows that

$e^{-b}\dot{\mathbf{G}}e^{b}=\begin{pmatrix}\dot{A}-g^{-1}\dot{g}g^{-1}b&-g^{-1}\dot{g}g^{-1}\\\
-b\dot{A}+\dot{A}^{*}b+\dot{\delta}-bg^{-1}\dot{g}g^{-1}&\dot{A}^{*}-bg^{-1}\dot{g}g^{-1}\end{pmatrix}$

and in particular we will only be interested in the first row. Focusing on the
top row, we furthermore compute

$\displaystyle e^{-b}\dot{\mathbf{G}}e^{b}$
$\displaystyle=-(e^{-b}\dot{\mathbf{J}}_{1}e^{b})(e^{-b}\mathbf{J}_{2}e^{b})-(e^{-b}\mathbf{J}_{1}e^{b})(e^{-b}\dot{\mathbf{J}}_{2}e^{b})$
$\displaystyle=-\tfrac{1}{8}\begin{pmatrix}-(\omega_{I}^{-1}-\omega_{J}^{-1})K(I+J)&(\omega_{I}^{-1}-\omega_{J}^{-1})K(\omega_{I}^{-1}-\omega_{J}^{-1})\\\\[4.0pt]
4K-(I^{*}+J^{*})K(I+J)&(I^{*}+J^{*})K(\omega_{I}^{-1}-\omega_{J}^{-1})\end{pmatrix}\begin{pmatrix}I-J&-(\omega_{I}^{-1}+\omega_{J}^{-1})\\\
\omega_{I}+\omega_{J}&-(I^{*}-J^{*})\end{pmatrix}$
$\displaystyle-\tfrac{1}{8}\begin{pmatrix}I+J&-(\omega_{I}^{-1}-\omega_{J}^{-1})\\\
\omega_{I}-\omega_{J}&-(I^{*}+J^{*})\end{pmatrix}\begin{pmatrix}-(\omega_{I}^{-1}+\omega_{J}^{-1})K(I-J)&(\omega_{I}^{-1}+\omega_{J}^{-1})K(\omega_{I}^{-1}+\omega_{J}^{-1})\\\\[4.0pt]
4K-(I^{*}-J^{*})K(I-J)&(I^{*}-J^{*})K(\omega_{I}^{-1}+\omega_{J}^{-1})\end{pmatrix}$
$\displaystyle=\tfrac{1}{2}\begin{pmatrix}g^{-1}K(I-J)+(\omega_{I}^{-1}-\omega_{J}^{-1})K&-(\omega_{I}^{-1}-\omega_{J}^{-1})Kg^{-1}-g^{-1}K(\omega_{I}^{-1}+\omega_{J}^{-1})\\\
*&*\end{pmatrix}$
$\displaystyle=\tfrac{1}{2}\begin{pmatrix}g^{-1}\\{K,I\\}&-(\omega_{I}^{-1}-\omega_{J}^{-1})Kg^{-1}-g^{-1}K(\omega_{I}^{-1}+\omega_{J}^{-1})\\\
*&*\end{pmatrix}.$

Thus

$\displaystyle\dot{g}=$ $\displaystyle\
\tfrac{1}{2}g(\omega_{I}^{-1}-\omega_{J}^{-1})K+\tfrac{1}{2}K(\omega_{I}^{-1}+\omega_{J}^{-1})g$
$\displaystyle=$ $\displaystyle\
\tfrac{1}{2}(I^{*}K-KI)-\tfrac{1}{2}(KJ+J^{*}K)$ $\displaystyle=$
$\displaystyle\ -\tfrac{1}{2}[K,I],$

where we have used that $K\in\Lambda^{1,1}_{J}$. Then differentiating the
formulas $I/J=-\omega_{I/J}^{-1}g$ gives

$\displaystyle\dot{I}$
$\displaystyle=\omega_{I}^{-1}\dot{\omega}_{I}\omega_{I}^{-1}g-\omega_{I}^{-1}\dot{g}$
$\displaystyle=-\tfrac{1}{2}\omega_{I}^{-1}[K,I]I\omega_{I}^{-1}g+\tfrac{1}{2}\omega_{I}^{-1}[K,I]=0,$
$\displaystyle\dot{J}$
$\displaystyle=\omega_{J}^{-1}\dot{\omega}_{J}\omega_{J}^{-1}-\omega_{J}^{-1}\dot{g}$
$\displaystyle=-\tfrac{1}{2}\omega_{J}^{-1}\\{K,IJ\\}\omega_{J}^{-1}+\tfrac{1}{2}\omega_{J}^{-1}[K,I]$
$\displaystyle=-\tfrac{1}{2}\omega_{J}^{-1}KI-\tfrac{1}{2}g^{-1}I^{*}KJ+\tfrac{1}{2}\omega_{J}^{-1}KI-\tfrac{1}{2}\omega_{J}^{-1}I^{*}K$
$\displaystyle=\tfrac{1}{2}g^{-1}I^{*}J^{*}K-\tfrac{1}{2}g^{-1}J^{*}I^{*}K=\tfrac{1}{2}[I,J]g^{-1}K.$

Lastly, differentiating the formula $b=-gA$ yields

$\displaystyle\dot{b}$ $\displaystyle=-\dot{g}A-g\dot{A}$
$\displaystyle=-\dot{g}A-g(g^{-1}\dot{g}g^{-1}b+\tfrac{1}{2}g^{-1}\\{K,I\\})$
$\displaystyle=-\tfrac{1}{2}\\{K,I\\}.$

∎

Using Proposition 3.6 we give the proof of Corollary 1.3.

###### Proof of Corollary 1.3.

By Proposition 3.3 we know that $\dot{I}=0$, and moreover it follows that

$\displaystyle\dot{\sigma}$
$\displaystyle=\tfrac{1}{2}\left(I[I,J]g^{-1}K-[I,J]g^{-1}KI\right)g^{-1}+\tfrac{1}{2}[I,J]g^{-1}[K,I]g^{-1}$
$\displaystyle=\tfrac{1}{2}I[I,J]g^{-1}Kg^{-1}-\tfrac{1}{2}[I,J]g^{-1}I^{*}Kg^{-1}$
$\displaystyle=\tfrac{1}{2}\left(I[I,J]+[I,J]I\right)g^{-1}Kg^{-1}$
$\displaystyle=0.$

∎

###  3.6. Examples

######  Example 3.7.

Given $(M^{2n},g,J)$ a Kähler manifold, we interpret this as a generalized
Kähler structure by setting $I=J$, and $b=0$. Suppose
$(g_{t},b_{t},I_{t},J_{t})$ is a canonical family with this initial condition.
Initially we have $\sigma=0$, thus $\sigma_{t}\equiv 0$ from Corollary 1.3. It
follows that $[I,J]\equiv 0$ for all times, so from the equations of
Proposition 3.6 it follows that that $\dot{J}\equiv 0$ for all times, and so
$J_{t}\equiv J=0$. In turn it follows easily that

$\displaystyle\dot{\omega}_{I}=\dot{\omega}_{J}=K,$

in other words, the complex structures stay fixed and the Kähler forms change
by $K$. By the $\sqrt{-1}\partial\overline{\partial}$-Lemma, an exact
canonical deformation satisfies $K=da=dJdu$ for some $u\in C^{\infty}(M)$.
Using the construction above and the normalization that
$\sqrt{-1}\partial\overline{\partial}=dId$ we verify item (1) of Proposition
1.2, noting that positivity of the Kähler forms
$(\omega_{I})_{t}=\omega_{I}+tdJdu$ is equivalent to the positivity condition
(3) in Theorem 1.1.

######  Example 3.8.

Suppose $(M^{2n},g,b,I,J)$ is a generalized Kähler structure and $[I,J]\equiv
0$. These manifolds are characterized by a holomorphic splitting of the
tangent bundle and simple examples are given by quotients of products, for
instance on Hopf surfaces. For further background on these structures see
([2]). Setting $Q=-IJ$ we see that $Q^{2}=1$ and so we can split
$T=T_{+}\oplus T_{-}$ in terms of the $\pm 1$-eigenbundles of $Q$. Let
$J_{+}=\left.I\right|_{T_{+}}$ and $J_{-}=\left.I\right|_{T_{-}}$. Since on
$T_{\pm}$ we have $I=\pm J$, it follows $\left.J\right|_{T_{+}}=J_{+}$ and
$\left.J\right|_{T_{-}}=-J_{-}$. Similarly, we define
$\omega_{\pm}=\left.\omega_{I}\right|_{T_{\pm}}$. Note that the Kähler form
indeed block diagonalizes along $T_{\pm}$ since

$\displaystyle\omega_{I}(X_{+},IY_{-})=g(IX_{+},IY_{-})=g(JX_{+},JY_{-})=g(-IX_{+},IY_{-})=-\omega_{I}(X_{+},IY_{-})=0.$

Now suppose $(g_{t},b_{t},I_{t},J_{t})$ is a canonical family with this
initial condition. Arguing as above, since $\sigma=0$ for the given structure,
it follows that $[I,J]\equiv 0$ for all times $t$, and hence $\dot{J}=0$. Thus
along the variation we preserve the splitting induced by $Q$, and we can
decompose $K=K_{++}+K_{+-}+K_{-+}+K_{--}$. Tracing through the formulas in
Proposition 3.6 yields

$\displaystyle\dot{g}=$ $\displaystyle\ \begin{pmatrix}J_{+}^{*}K_{++}&0\\\
0&J_{-}^{*}K_{--}\end{pmatrix},\qquad\dot{b}=\begin{pmatrix}0&-J_{+}^{*}K_{-+}\\\
-J_{-}^{*}K_{+-}&0\end{pmatrix},$ $\displaystyle\dot{\omega}_{I}=$
$\displaystyle\ \begin{pmatrix}K_{++}&0\\\
0&K_{--}\end{pmatrix},\qquad\dot{\omega}_{J}=\begin{pmatrix}-K_{++}&0\\\
0&K_{--}\end{pmatrix}.$

A special case of this occurs when $K=dJdu$, yielding

$\displaystyle\dot{\omega}_{I}=$ $\displaystyle\
\sqrt{-1}\left(\partial_{+}\overline{\partial}_{+}-\partial_{-}\overline{\partial}_{-}\right)u,$

where
$d=\partial_{+}+\partial_{-}+\overline{\partial}_{+}+\overline{\partial}_{-}$
is the fourfold splitting of $d$ induced by $Q$. This construction verifies
item (2) of Proposition 1.2, again noting that positivity of

$\displaystyle(\omega_{I})_{t}=\omega_{I}+t\sqrt{-1}\left(\partial_{+}\overline{\partial}_{+}-\partial_{-}\overline{\partial}_{-}\right)u$

is equivalent to the positivity condition of item (3) in Theorem 1.1.

######  Example 3.9.

Suppose $(M^{2n},g,b,I,J)$ is a generalized Kähler structure where the
associated Poisson tensor $\sigma$ is nondegenerate. In this setting the
endomorphism $[I,J]$ is invertible, and this in turn implies that $I\pm J$ are
invertible. We define the 2-forms $F_{\pm}=-2g(I\pm J)^{-1}$. Moreover let
$\Omega=\sigma^{-1}$. It turns out that the three symplectic forms
$F_{\pm},\Omega$ completely determine the generalized Kähler structure in this
case. Direct computations show that the generalized complex structures can be
expressed as

$\displaystyle\mathbf{J}_{1}=e^{-4\Omega}\begin{pmatrix}0&-F_{-}^{-1}\\\
F_{-}&0\end{pmatrix}e^{4\Omega},\quad\mathbf{J}_{2}=\begin{pmatrix}0&-F_{+}^{-1}\\\
F_{+}&0\end{pmatrix},$

Let $K$ be an infinitessmal deformation of the generalized Kahler pair
$(\mathbf{J}_{1},\mathbf{J}_{2})$. Then with respect to the data
$(F_{+},F_{-},\Omega)$, direct computations show that

$\displaystyle\dot{F}_{+}=\dot{F}_{-}=K,\qquad\dot{\Omega}=0.$

As in the above examples we can choose $K_{t}=dJ_{t}du_{t}$. We claim that for
such a variation,

$\displaystyle\dot{J}=L_{\sigma du}J.$

To show this we compute, using that $\sigma=\Omega^{-1}$ and $d\Omega=0$,

$\displaystyle\left(L_{\sigma du}J\right)\Omega=$ $\displaystyle\ L_{\sigma
du}(J\Omega)-JL_{\sigma du}\Omega=dJdu=K.$

Comparing against Proposition 3.6 we see that

$\displaystyle\dot{J}=\sigma K=L_{\sigma du}J,$

as required. It follows that $J_{t}=\phi_{t}^{*}J_{0}$, where $\phi$ is the
one-parameter family of $\Omega$-Hamiltonian diffeomorphisms driven by
$u_{t}$. As discussed in [3], the positivity of
$-\operatorname{Im}\pi_{\Lambda_{I}^{1,1}}\Omega_{J}$ is equivalent to the
positivity condition (3) of Theorem 1.1. This verifies item (3) of Proposition
1.2.

###### Proof of Proposition 1.2.

The three deformations claimed in the proposition are described in the
examples above. ∎

##  4\. Generalized Kähler-Ricci flow as canonical deformation

In this section we establish Theorem 1.4, namely that solutions to the
generalized Kähler-Ricci flow are canonical deformations, driven by the Bismut
Ricci curvature. This generalizes and unifies various instances of this
phenomenon which have previously been observed. In particular, it is well
known that Kähler-Ricci flow moves within the Kähler cone against a fixed
complex structure, and so is a canonical deformation as in Example 3.7. Next,
comparing Example 3.8 against the curvature identities of ([12]), we see that
the generalized Kähler-Ricci flow in the case $[I,J]=0$ is a canonical
deformation driven by $\rho_{I}$. Also, we can compare the discussion in
Example 3.9 with ([3]) to see the phenomenon holds in the nondegenerate case.
The key point in establishing the general case is to show that the evolution
of $J$ is indeed determined by $K=\rho_{I}$. This requires a delicate
curvature identity we build up below.

######  Lemma 4.1.

([11]) Let $(M^{2n},g,I)$ be a pluriclosed structure. Then

(4.1) $\displaystyle\begin{split}\rho_{I}(X,Y)=&\
-\operatorname{Rc}^{B}(X,IY)+\nabla^{B}_{X}\theta(IY)\\\
\rho^{1,1}_{I}(\cdot,I\cdot)=&\
\operatorname{Rc}^{g}-\tfrac{1}{4}H^{2}-\tfrac{1}{2}L_{\theta^{\sharp}}g,\\\
\rho_{I}^{2,0+0,2}(X,Y)=&\
\tfrac{1}{2}\left(d^{*}H(IX,Y)+d^{\nabla}\theta(IX,Y)\right)\\\ =&\
\left(dI\theta\right)^{2,0+0,2}(X,Y).\end{split}$

######  Proposition 4.2.

Let $(M^{2n},g,I,J)$ be a generalized Kähler structure. Then

$\displaystyle
g\left(\left(L_{\theta_{J}^{\sharp}-\theta_{I}^{\sharp}}J\right)X,Y\right)=$
$\displaystyle\ \rho_{I}([I,J]X,Y).$

###### Proof.

We will use that for a Hermitian manifold $(M^{2n},g,J)$ one has

(4.2) $\displaystyle g((L_{X}J)Y,Z)=$ $\displaystyle\
g((D_{X}J)Y-D_{JY}X+JD_{Y}X,Z).$

First note, using that $\rho_{I}\in\Lambda^{1,1}_{J}$,

$\displaystyle\rho_{I}([I,J]X,JY)$ $\displaystyle+\rho_{I}([I,J]JY,X)$
$\displaystyle=$ $\displaystyle\ \rho_{I}(IJX-JIX,JY)-\rho_{I}(X,IJJY-JIJY)$
$\displaystyle=$ $\displaystyle\
\rho_{I}(IJX,JY)-\rho_{I}(IX,Y)+\rho_{I}(X,IY)-\rho_{I}(JX,IJY)$
$\displaystyle=$ $\displaystyle\
-\rho_{I}(JY,IJX)+\rho_{I}(Y,IX)+\rho_{I}(X,IY)-\rho_{I}(JX,IJY).$

Applying Lemma 4.1 we further compute

$\displaystyle-\rho_{I}(JY,IJX)$
$\displaystyle+\rho_{I}(Y,IX)+\rho_{I}(X,IY)-\rho_{I}(JX,IJY)$
$\displaystyle=$ $\displaystyle\
-\operatorname{Rc}^{I}(JY,JX)+\nabla^{I}_{JY}\theta^{I}(JX)+\operatorname{Rc}^{I}(Y,X)-\nabla^{I}_{Y}\theta^{I}(X)$
$\displaystyle\
+\operatorname{Rc}^{I}(X,Y)-\nabla^{I}_{X}\theta^{I}(Y)-\operatorname{Rc}^{I}(JX,JY)+\nabla^{I}_{JX}\theta^{I}(JY)$
$\displaystyle=$ $\displaystyle\
-\operatorname{Rc}^{J}(JY,JX)-\operatorname{Rc}^{J}(JX,JY)+\operatorname{Rc}^{J}(X,Y)+\operatorname{Rc}^{J}(Y,X)$
$\displaystyle\
+\nabla^{I}_{JY}\theta^{I}(JX)-\nabla^{I}_{Y}\theta^{I}(X)-\nabla^{I}_{X}\theta^{I}(Y)+\nabla^{I}_{JX}\theta^{I}(JY)$
$\displaystyle=$ $\displaystyle\
\rho_{J}(JY,X)+\rho_{J}(JX,Y)+\rho_{J}(X,JY)+\rho_{J}(Y,JX)$ $\displaystyle\
-\nabla^{J}_{JY}\theta^{J}(JX)-\nabla^{J}_{JX}\theta^{J}(JY)+\nabla^{J}_{X}\theta^{J}(Y)+\nabla^{J}_{Y}\theta^{J}(X)$
$\displaystyle\
+\nabla^{I}_{JY}\theta^{I}(JX)-\nabla^{I}_{Y}\theta^{I}(X)-\nabla^{I}_{X}\theta^{I}(Y)+\nabla^{I}_{JX}\theta^{I}(JY)$
$\displaystyle=$ $\displaystyle\
D_{X}(\theta^{J}-\theta^{I})(Y)+D_{Y}(\theta^{J}-\theta^{I})(X)-D_{JX}\left(\theta^{J}-\theta^{I}\right)(JY)-D_{JY}\left(\theta^{J}-\theta^{I}\right)(JX)$
$\displaystyle=$ $\displaystyle\
g\left(\left(L_{\theta_{J}^{\sharp}-\theta_{I}^{\sharp}}J\right)X,JY\right)+g\left(\left(L_{\theta_{J}^{\sharp}-\theta_{I}^{\sharp}}J\right)JY,X\right).$

where the last line follows by comparing (4.2).

To address the skew symmetric piece we first compute, again using that
$\rho_{I}\in\Lambda^{1,1}_{J}$,

$\displaystyle\rho_{I}$
$\displaystyle\left([I,J]X,Y\right)-\rho_{I}\left([I,J]Y,X\right)$
$\displaystyle=$ $\displaystyle\ \rho_{I}((IJ-JI)X,Y)+\rho_{I}(X,(IJ-JI)Y)$
$\displaystyle=$ $\displaystyle\
\rho_{I}(IJX,Y)+\rho_{I}(IX,JY)+\rho_{I}(X,IJY)+\rho_{I}(JX,IY)$
$\displaystyle=$ $\displaystyle\
2\left(\left(\rho_{I}\right)^{2,0}(JX,IY)+\left(\rho_{I}\right)^{2,0}(IX,JY)\right).$

Now we compute using Lemma 4.1 and equation (4.2),

$\displaystyle 2$
$\displaystyle\left(\left(\rho_{I}\right)^{2,0}(JX,IY)+\left(\rho_{I}\right)^{2,0}(IX,JY)\right)$
$\displaystyle=$ $\displaystyle\
2\left(\left(-\rho_{I}\right)^{2,0}(IY,JX)+\left(\rho_{I}\right)^{2,0}(IX,JY)\right)$
$\displaystyle=$ $\displaystyle\
d^{*}H_{I}(Y,JX)-d^{*}H_{I}(X,JY)+d^{\nabla^{I}}\theta^{I}(Y,JX)-d^{\nabla^{I}}\theta^{I}(X,JY)$
$\displaystyle=$ $\displaystyle\
-d^{*}H_{J}(Y,JX)+d^{*}H_{J}(X,JY)+d^{\nabla^{I}}\theta^{I}(Y,JX)-d^{\nabla^{I}}\theta^{I}(X,JY)$
$\displaystyle=$ $\displaystyle\
2(\rho_{J})^{2,0}(X,Y)-d^{\nabla^{J}}\theta^{J}(JX,Y)-2(\rho_{J})^{2,0}(Y,X)+d^{\nabla^{J}}\theta^{J}(JY,X)$
$\displaystyle\
+d^{\nabla^{I}}\theta^{I}(Y,JX)-d^{\nabla^{I}}\theta^{I}(X,JY)$
$\displaystyle=$ $\displaystyle\
d^{\nabla^{J}}\theta^{J}(JX,Y)-d^{\nabla^{J}}\theta^{J}(JY,X)+d^{\nabla^{I}}\theta^{I}(Y,JX)-d^{\nabla^{I}}\theta^{I}(X,JY)$
$\displaystyle=$ $\displaystyle\
g\left(\left(L_{\theta_{J}^{\sharp}-\theta_{I}^{\sharp}}J\right)X,Y\right)-g\left(\left(L_{\theta_{J}^{\sharp}-\theta_{I}^{\sharp}}J\right)Y,X\right),$

as required. ∎

###### Proof of Theorem 1.4.

We first assume that equations (1.2) hold. By Proposition 3.6 with
$K=\rho_{I}$ one notes that $I$ is fixed and one easily derives the evolution
equation $\dot{\omega}_{I}=-(\rho_{I})^{1,1}$. By Proposition 4.2 it follows
that

$\displaystyle\dot{J}=\tfrac{1}{2}[I,J]g^{-1}\rho_{I}=L_{\tfrac{1}{2}\left(\theta_{J}^{\sharp}-\theta_{I}^{\sharp}\right)}J.$

as required. Thus we have verified the evolution equations of (1.1). Assuming
equations (1.1) and imposing $\dot{b}=-\\{\rho_{B}^{I},I\\}$, we can apply the
computations in Proposition 3.6 and use Proposition 4.2 to establish equations
(1.2). ∎

## References

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[a,1]Sipaz Sharma 11footnotetext: For the HotQCD Collaboration.

# Charm fluctuations in (2+1)-flavor QCD at high temperature

###### Abstract

Using the high statistics datasets of the HotQCD Collaboration, generated with
the HISQ (2+1)-flavor action for light and strange quarks, and treating the
charm sector in the quenched approximation, we analyze the second and fourth
order cumulants of charm fluctuations and the correlations of charm with
lighter conserved flavor quantum numbers. We can make use of a factor 100
larger statistics on ${N_{\tau}=8}$ lattices which never have been used in
studies of charm fluctuations.

Analyzing correlations of charm fluctuations with baryon number and electric
charge fluctuations we can project onto charmed baryon and meson correlations
and compare results with quark model extended hadron resonance gas model
calculations. We aim at a precise determination of the dissociation
temperature of charmed hadrons and will probe the sensitivity of the
fluctuations observables to the presence of multiple-charmed baryons.

##  1 Introduction

Charmed hadrons have been proposed as an important probe in understanding
properties as well as signatures of the deconfined medium consisting of quark-
gluon plasma formed in the heavy-ion collision experiments ever since the
seminal work of Matsui and Satz [1].

It is now well established that the strong interaction matter at vanishing
baryon chemical potential undergoes restoration of the spontaneously broken
chiral symmetry via a crossover transition since the small yet non-vanishing
up and down quark masses result also in the explicit breaking of the
${SU(2)_{L}\times SU(2)_{R}}$ chiral symmetry group. This chiral crossover
transition occurs at a pseudo-critical temperature, ${T_{pc}}$ which
calculated using Lattice QCD formalism is ${T_{pc}=156.5\pm 1.5}$ MeV [2].
Interestingly, as per observations made in heavy-ion collision experiments,
deconfinement of light-quark hadrons also occurs at ${T_{pc}}$, and thus the
relevant degrees of freedom change from partonic to hadronic in going from
high temperature phase to temperatures below ${T_{pc}}$.

One of the open questions heavy-ion experiments aim to answer is whether open-
charm states melt beyond ${T_{pc}}$, or at ${T_{pc}}$ like other light-quark
hadrons. Statistical Hadronization Model (SHM) is one of theoretical tools to
model experimentally measured hadron abundances, and has been extended to
SHMc: which incorporates multiple-charmed states based on the observation that
the production of charmed-states is strongly Boltzmann suppressed given the
massive charm quark mass in relation to the relevant temperature scale, and is
instead a result of the initial hard collisions [3]. Another approach to this
problem is using lattice calculations of hadronic correlation functions
[4],[5] to extract details of the charm hadron spectrum; or by comparing
various charm cumulants calculated on the lattice with the Hadron Resonance
Gas (HRG) model predictions [6], which are valid only in the low temperature
phase, and hence considering the deviation of lattice results from HRG
calculations a signal of the open-charm sector melting [7]. Moreover,
existence of the not-yet-discovered open charm states can also be predicted by
the comparison of lattice results with HRG calculations [7]; and signals of
exotic charm states such as tetraquarks can shed light on how quarks arrange
themselves inside the bound states [8].

##  2 Hadron Resonance Gas Model

###  2.1 Partial Pressure from the Open-Charm Sector

HRG describes a non-interacting gas of hadron resonances, and therefore is
valid in the hadronic phase below ${T_{pc}}$. It has been successful in
describing particle abundance ratios measured in heavy-ion experiments. Grand-
canonical partition function in the charm sector provided by Hadron resonance
gas model given by
${\text{ln}\mathcal{Z}^{HRG}_{C}(T,V,\overrightarrow{\mu})}$ is dependent upon
temperature ${T}$, volume ${V}$, and chemical potentials
${\overrightarrow{\mu}=(\mu_{B},\mu_{Q},\mu_{S},\mu_{C})}$ corresponding to
various conserved quantum numbers such as baryon number ${B}$, electric charge
${Q}$, strangeness ${S}$, charm ${C}$ respectively. Partial pressure ${P_{C}}$
from the open-charm sector is related to ${\text{ln}\mathcal{Z}^{HRG}_{C}}$ as
follows,

${\dfrac{P_{C}(T,V,\overrightarrow{\mu})}{T^{4}}=\dfrac{\text{ln}\mathcal{Z}^{HRG}_{C}(T,V,\overrightarrow{\mu})}{VT^{3}}}\text{
.}$ (1)

Left-hand-side of Eq.1 can be further decomposed into separate contributions
to partial open-charm pressure coming from charmed meson (C-mesons) and
charmed baryon (C-baryons) sectors, denoted by ${M_{C}}$ and ${B_{C}}$
respectively:

${P_{C}(T,\overrightarrow{\mu})/T^{4}=M_{C}(T,\overrightarrow{\mu})+B_{C}(T,\overrightarrow{\mu})}\text{
.}$ (2)

Based on the calculations presented in References [6], [7], ${M_{C}}$ and
${B_{C}}$ take following forms, In above equations, at a given temperature
${T}$, summation is over all C-mesons/baryons with masses given by ${m_{i}}$;
degeneracy factors of states with equal mass and same quantum numbers are
represented by ${g_{i}}$; to work with dimensionless variables,
${\hat{\mu}_{X}=\mu/T}$, ${\forall X\in\\{B,Q,S,C\\}}$; ${K_{2}(x)}$ is a
modified Bessel function, which for large argument can be approximated by
${K_{2}(x)}\sim\sqrt{\pi/2x}\;e^{-x}\;[1+\mathbb{O}(x^{-1})]$[6]:
consequently, if a charmed state under consideration is much heavier than the
relevant temperature scale, such that ${m_{i}\gg T}$, then the contribution to
${P_{C}}$ from that particular state will be exponentially suppressed, e.g.,
singly-charmed ${\Lambda}_{c}^{+}$ baryon has a PDG mass of about $2286$ MeV,
whereas doubly-charmed ${\Xi_{cc}^{++}}$ baryon’s mass as tabulated in PDG
records is about $3621$ MeV, therefore at ${T_{pc}}$, contribution to
${B_{C}}$ from ${\Xi_{cc}^{++}}$ will be suppressed by a factor of $10^{-3}$
in relation to ${\Lambda}_{c}^{+}$ contribution. In other words, in order to
distinguish signals of multiple-charm states from ${C=1}$ sector, a precision
study based on orders of magnitude larger statistics than that has been used
in the present study would be required.

###  2.2 Generalized Susceptibilities of the Conserved Charges

To project out the relevant degrees of freedom in the charm sector, one
calculates the generalized susceptibilities, ${\chi^{BQSC}_{klmn}}$ of the
conserved charges by taking appropriate derivatives of total pressure ${P}$ –
which contains contribution from ${P_{C}}$ – with respect to the chemical
potentials of the quantum number combinations one is interested in:

${\chi^{BQSC}_{klmn}=\dfrac{\partial^{(k+l+m+n)}\;\;[P\;(\hat{\mu}_{B},\hat{\mu}_{Q},\hat{\mu}_{S},\hat{\mu}_{C})\;/T^{4}]}{\partial\hat{\mu}^{k}_{B}\;\;\partial\hat{\mu}^{l}_{Q}\;\;\partial\hat{\mu}^{m}_{S}\;\;\partial\hat{\mu}^{n}_{C}}}\bigg{|}_{\overrightarrow{\mu}=0}\text{.}$
(3)

In Equation 3, as long as ${n\neq 0}$ and ${(k+l+m+n)\in\text{even}}$, ${P}$
can be replaced with ${P_{C}}$, since derivative w.r.t. ${\hat{\mu}_{C}}$ will
always project onto the open-charm states. In the following, if the subscript
corresponding to a conserved charge is zero in the L.H.S. of Eq. 3, then both
the corresponding superscript as well the zero subscript will be suppressed.

Based on following observations, it is possible to construct some useful
combinations of various generalized susceptibilities which can indirectly shed
light on the properties of the open-charm sector: $\bullet$ Baryon-charm
correlations, ${\chi^{BC}_{mn}}$ will receive contributions from singly,
doubly and triply-charmed states carrying baryon number denoted by
${B_{C,1}}$, ${B_{C,2}}$ and ${B_{C,3}}$ respectively, but as argued earlier,
dominant contribution to the partial pressure will come only from $C=1$
sector, therefore,

${\chi^{BC}_{mn}=B_{C,1}+2^{n}B_{C,2}+3^{n}B_{C,3}\simeq B_{C,1}}\text{.}$ (4)

$\bullet$ Since baryons carry ${B=1}$, ratios such as
${\chi^{BC}_{mn}/\chi^{BC}_{kl}}$, will always be unity in HRG phase
irrespective of the details of the baryon mass spectrum,
${\forall\\{(m+n),(k+l)\\}\in\text{even}}$. $\bullet$ Ratios like
${\chi^{BC}_{1n}/\chi^{BC}_{1l}}$ should be unity throughout the entire
temperature range, ${\forall\\{n,l\\}\in\text{odd}}$.

##  3 Simulation Details

###  3.1 Lattice Setup

We have used approximately one-third of the available (2+1)-flavor HISQ
configurations generated by HotQCD collaboration – tabulated in Tab. I of Ref.
[9] – for physical strange to light quark mass ratio, ${m_{s}/m_{l}}=27$ at a
temporal lattice extent, ${N_{\tau}}=8$. To set the temperature scale, $f_{K}$
scale setting from Ref. [9] is used. Charm-quark sector is treated in the
quenched approximation. Details of the charm mass tuning and its
parametrization will be discussed in the next subsection 3.2.

Partition function of QCD with two light, one strange and one charm quark
flavors with ${m_{l}}$, ${m_{s}}$ and ${m_{c}}$ being their respective masses
is given by:

${\mathcal{Z}=\displaystyle\int\mathcal{D}[U]\\{\text{det
}D(m_{l})\\}^{2/4}\\{\text{det }D(m_{s})\\}^{1/4}\\{\text{det
}D(m_{c})\\}^{1/4}e^{-S_{g}}}\text{.}$ (5)

Here, ${U}$ are the ${SU(3)}$ gauge links; ${S_{g}}$ is the gluonic part of
the action; ${D(m_{i})}$ for ${i\in(l,s,c)}$ represent the fermionic matrices;
determinants of the fermionic matrices are raised to powers of $1/4$ because
we are dealing with staggered quarks. As pointed out in Eq. 1, partition
function of Eq. 5 can be related to the total pressure, which can be further
used to calculate the generalized susceptibilities in the $BQSC$ basis,
presented in Eq. 3. Calculation of $\chi^{BQSC}_{klmn}$ involves derivatives
of pressure, and on lattice this is achieved by stochastic estimation of
various traces – consisting of inversions and derivatives of the fermionic
matrices – using random noise method. In particular, $500$ random vectors have
been used to calculate various traces per configuration, expect for Tr
${\big{(}D^{-1}\frac{\partial D}{\partial\mu}\big{)}}$ – which turned out be
particularly noisy and therefore, $2000$ random vectors have gone into its
calculation. At present, we have gone upto fourth order in calculating various
charm cumulants.

Figure 1: Charm mass tuning – by considering spin-averaged charmonium mass,
$(3m_{J/\psi}+m_{\eta})/4$ as the line of constant physics (LCP) – for
thirteen $\beta$ values using various trial $am_{c}$ values is shown [left].
RG inspired fit using Eq. 6 to the intersections of LCP with linear
interpolations of each beta value, $am_{c}^{lcp}(\beta)$ is shown [right];
various fit coefficients are also quoted in the figure.

###  3.2 Charm Mass Tuning on the Lattice

For each of the thirteen beta values, $\beta\in[6.285,7.825]$, more than three
trial values of the bare charm quark mass, $am_{c}$ have gone into the
calculation of the spin-averaged charmonium masses given by
${(3m_{J/\psi}+m_{\eta_{c\bar{c}}})/4}$, which are represented by different
colors as well markers for each beta value in Fig. 1 [left]. $f_{K}$ scale
setting from Ref. [9] has been used to convert various spin-averaged
charmonium masses in lattice units to GeV. Linearly interpolated solid lines
for each beta intersect the black-horizontal Line of constant physics (LCP)
defined by the PDG ${(3m_{J/\psi}+m_{\eta_{c\bar{c}}})/4}=3.06865$ GeV at
$am_{c}^{lcp}(\beta)$ in Fig. 1 [left]. These intersections – olive-stars –
are then fitted to the following Renormalization Group (RG) inspired ansatz in
Fig. 1 [right],

${am_{c}^{lcp}(\beta)=\bigg{(}\dfrac{20b_{0}}{\beta}\bigg{)}^{\frac{4}{9}}c_{0m}f(\beta)\;\bigg{[}\;\dfrac{1+c_{2m}(\frac{10}{\beta})f^{2}(\beta)+c_{3m}(\frac{10}{\beta})f^{3}(\beta)}{1+d_{2m}(\frac{10}{\beta})f^{2}(\beta)+d_{3m}(\frac{10}{\beta})f^{3}(\beta)}\;\bigg{]}}\text{.}$
(6)

Here, $f(\beta)$ is the 2-loop QCD beta function,

$f(\beta)=\bigg{(}\dfrac{10b_{0}}{\beta}\bigg{)}^{-b_{1}/2b_{0}^{2}}\text{ exp
}(-\beta/(20b_{0}))\text{,}$ (7)

with $b_{0}$ and $b_{1}$ being the perturbative expansion coefficients of the
QCD $\beta$-function for number of colors, $N=3$ and number of flavors,
$N_{f}=3$ ,

$\displaystyle\begin{aligned}
b_{0}&=\dfrac{1}{(4\pi)^{2}}\bigg{(}\dfrac{11}{3}N-\dfrac{2}{3}N_{f}\bigg{)}\text{
,}\\\
b_{1}&=\dfrac{1}{(4\pi)^{4}}\bigg{(}\dfrac{34}{3}N^{2}-\dfrac{10}{3}NN_{f}-\dfrac{N^{2}-1}{N}N_{f}\bigg{)}\text{
.}\end{aligned}$ (8)

Fig. 1 [right] quotes the value of the fit coefficients in Eq. 6 as well the
$\chi^{2}/dof$ which indicates the quality of the fit.

##  4 Results and Conclusions

Figure 2: Ratios of various $BC$ correlations [left]. Ratios of $BC$, $BS$,
$BQ$ correlations [right].

As pointed out in Subsec.2.2, ratios of baryon-charm correlations, e.g.,
$\chi^{BC}_{13}/\chi^{BC}_{22}$ – which comprise of different orders of
derivative w.r.t $\hat{\mu}_{B}$, such that the number of derivatives w.r.t.
$\hat{\mu}_{C}$ are adjusted based on the availability of the highest order
cumulant – provide an indication about the onset of the melting of open-charm
baryon states by deviating from their low temperature HRG value – which is
unity. Fig 2 [left] shows updated $\chi^{BC}_{13}/\chi^{BC}_{22}$ values with
red-square markers for $T\in(144.95,171.02)$ MeV, in addition to that Fig 2
[left] is augmented with high temperature data, $T\in(156.8,330.2)$ MeV –
brown-downward-triangles – from the 2014 HotQCD analysis [7], which took into
account an order of magnitude smaller statistics at $T_{pc}$, and this fact
reflects in the errors of the data points. Combination of low-temperature and
high-temperature data sets, as well as the shrinkage of yellow band
representing $T_{pc}$ based on [2], enables discerning the deviation of
$\chi^{BC}_{13}/\chi^{BC}_{22}$ from unity right after $T_{pc}$, which in turn
hints at the change of the carriers of the charm degrees-of-freedom at
$T_{pc}$.

Given the massive charm quark, ratios of baryon-charm correlations containing
one derivative w.r.t. $\hat{\mu}_{B}$ should stay unity for the entire
temperature range, and blue-circles representing updated
$\chi^{BC}_{11}/\chi^{BC}_{13}$ values as well as the teal-stars depicting
2014 HotQCD results in Fig 2 [left] are a clear testament of this fact.

Fig. 2 [right] depicts the reciprocal of $\chi^{BC}_{13}/\chi^{BC}_{22}$ for
both updated and 2014 HotQCD data, therefore above discussion is valid. Ratios
of the baryon-electric-charge as well as the baryon-strangeness correlations
separated by even $\hat{\mu}_{B}$ derivatives should be unity in the HRG
phase, but should show deviation from unity when the relevant of degrees of
freedom are no longer baryons. In Fig. 2 [right], unlike red-squares which
take into account one-third of the available statistics, green-triangles
representing $\chi^{BQ}_{31}/\chi^{BQ}_{11}$ and blue-circles representing
$\chi^{BS}_{31}/\chi^{BS}_{11}$ with relatively smaller errors are calculated
by taking into account all the available (2+1)-flavor HISQ configurations
generated by the HotQCD collaboration. Although, the electrically-charged-
baryon sector as well as the strange-baryon sector start deviating from the
HRG expectation slightly before $T_{pc}$, the charmed-baryon sector shows
clear deviation from unity right at $T_{pc}$.

Figure 3: Plots depicting ratios of baryonic to mesonic contribution to the
partial charm pressure in charm sector [left], electrically-charged charm
sector [middle], strange-charm sector [right].

Quantities plotted in Fig. 2 are independent of the details of the hadron mass
spectrum, whereas ratios such as,
$\chi^{BC}_{13}/(\chi^{C}_{4}-\chi^{BC}_{13})$ – which effectively translates
to the ratio of the contribution to the partial charm pressure from charmed-
baryonic and charmed-mesonic sector – are sensitive to the details of the
hadron mass spectrum. In Fig. 3 [left], dashed-teal curve representing HRG
calculation based on the states tabulated in the PDG record clearly misses the
numerically calculated lattice data for
$\chi^{BC}_{13}/(\chi^{C}_{4}-\chi^{BC}_{13})$, whereas upon including states
predicted via Quark-Model calculations [10], [11] – represented by dashed-
magenta curve – lattice data upto $T=180$ MeV can be very well described by
the HRG calculations, beyond that the data starts to approach its non-
interacting quark gas limit. Similar scenario persists for electrically-
charged charm sector tapped into using
$\chi^{BQC}_{112}/(\chi^{QC}_{13}-\chi^{BQC}_{112})$ in Fig. 3 [middle], where
deviation from QM-HRG starts at $T_{pc}$. Errors of the strange-charm sector
projected out using $-\chi^{BSC}_{112}/(\chi^{SC}_{13}-\chi^{BSC}_{112})$ in
Fig. 3 [right] are not very much under control yet for the updated data and
for the 2014 HotQCD data, due to lower statistics might be getting
underestimated, therefore, at present it is not possible to make a precise
statement about the deviation of lattice data from the QM-HRG calculations.
Nonetheless, depiction of PDG-HRG curve missing the lattice data is quite
vivid. In conclusion, all sub-figures of Fig. 3 reiterate the possibility of
the existence of not-yet-discovered baryonic states in the charm sector as was
argued in [7].

## Acknowledgments

This work was supported by The Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) - Project numbers 315477589-TRR 211.

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