\( \widetilde{\mathcal F}_\zeta \) of (<ref>)
can be re-written as

\begin{equation}
f_k(\zeta) = \int_M \difLie_\zeta F_k \wedge \omega^{n-k+1}
\end{equation}

where, writing $Hc_k(\omega)$ for the harmonic part
of $c_k(\omega)$, we have

\begin{equation}
c_k(\omega) - Hc_k(\omega) =
\sqrt{-1} \partial \barpartial F_k
\end{equation}

with $ F_k \in \Omega^{k-1,k-1}(M)$, see
Futaki2005,Futaki2006 and
Bando2006 for details.
This $f_k$ is an obstruction
to the existence of $\omega \in \mathcal{K}$
for which the $k$-th Chern form $c_k(\omega)$ is
harmonic. Note that $f_1$ coincides with the cscK
obstruction in the previous subsection. This result
is due to Bando2006.
In Futaki2008, the perturbed version of
extremal Kähler metrics and the
Calabi-Lichnerowicz-Matsushima type decomposition theorem
has been studied. The Kempf–Ness functional can be defined
as in (<ref>), but not much has been
done so far in this perturbed case.

§.§ Deformation quantization

A deformation quantization is a formal associative
deformation of a Poisson algebra
$(\sFunctionSpace(M), \poissonDot)$
into the space $\sFunctionSpace(M)[[\nu]]$ of
formal power series in $\nu$ with a composition
law $\ast$, called the star product,
with the following property. The constant function
$1$ is a unit and if
we write for $f,\ g  \in \sFunctionSpace(M)$

\begin{eqnarray*}%\label{star}
f \ast g = \sum_{r=0}^\infty C_r(f,g) \nu^r,
\end{eqnarray*}

then $\ast$ is required to satisfy

\begin{eqnarray*}
C_0(f,g) = f g, \qquad C_1(f,g) - C_1(g,f)  = \{f,g\},
\end{eqnarray*}

where the $C_r$'s are required to be bidifferential operators.

For symplectic manifolds, the existence of star products
was shown by DeWildeLecomte1983, Fedosov1994,
and OmoriMaedaYoshioka1991. For general Poisson
manifolds, the existence of star products was shown by
Kontsevitch2003. In this subsection, we are concerned
with the star product constructed by Fedosov.

A star product on a compact symplectic manifold
$(M,\omega)$ of dimension $2n$
is called closed (in the sense of
ConnesFlatoSternheimer1992) if

\begin{eqnarray*}%\label{star3}
\int_M F\ast H\ \mu_\omega = \int_M H \ast F\ \mu_\omega
\end{eqnarray*}

for all $F,\ H \in \sFunctionSpace(M)[[\nu]]$.

A symplectic connection $\nabla$ on a compact
symplectic manifold $(M,\omega)$ is a torsion free
affine connection such that $\nabla\omega = 0$.
There always exists a symplectic connection on
any symplectic manifold. Unlike the Levi-Civita
connection on a Riemannian manifold, a symplectic
connection is not unique on a symplectic manifold.
Given two symplectic connections $\nabla$ and $\nabla^\prime$, let

\begin{equation}
S(X,Y) \defeq \nabla_X Y - \nabla^\prime_X Y.
\end{equation}

Then $\omega(S(X,Y),Z)$ is totally symmetric in $X$, $Y$, $Z$.
Conversely, if $\nabla$ is a symplectic connection
and $\omega(S(X,Y),Z)$ is totally symmetric, then
$\nabla^\prime \defeq \nabla + S$ is a
symplectic connection.

Thus, on a symplectic manifold $(M,\omega)$, the
space of symplectic connections, denoted by
$\mathcal E (M,\omega)$, is an affine space modeled
on the set of all smooth sections $\Gamma(S^3(T^\ast M))$
of symmetric covariant 3-tensors. Hence, we may identify
$\mathcal E (M,\omega)$ with

\begin{equation}
\mathcal E (M,\omega) \cong \nabla + \Gamma(S^3(T^\ast M)).
\end{equation}

On $\mathcal E (M,\omega)$ there is a natural symplectic
structure $\Omega^{\mathcal E}$
whose value at $\nabla$ is given by

\begin{equation}\label{symp str}
\Omega^{\mathcal E}_\nabla(\underline{A},\underline{B}) =
\int_M \omega^{i_1j_1}\omega^{i_2j_2}\omega^{i_3j_3}
\underline{A}_{i_1i_2i_3}\,\underline{B}_{j_1j_2j_3}\
\mu_\omega \nonumber
\end{equation}

for $\underline{A},\ \underline{B} \in
T_\nabla \mathcal E(M,\omega) \cong \Gamma(S^3(T^\ast M))$,
see CahenGutt2005.

$\Omega^{\mathcal E}_\nabla$ is independent of $\nabla$
we may omit $\nabla$ and write $\Omega^{\mathcal E}$.
There is a natural action of the group
$\DiffGroup(M,\omega)$ of symplectomorphisms

on $\mathcal E(M,\omega)$, which is given for a
symplectomorphism $\varphi$ by

\begin{equation}
(\varphi(\nabla))_X Y =
\varphi_\ast \bigl(\nabla_{\varphi_\ast^{-1} X}
(\varphi_\ast^{-1} Y)\bigr)
\end{equation}

for any $\nabla \in \mathcal E(M,\omega)$ and any
smooth vector fields  $X$ and $Y$ on $M$.
This action preserves the symplectic structure
$\Omega^{\mathcal E}$ on $\mathcal E(M,\omega)$.
In particular, the group $\HamDiffGroup(M,\omega)$
of Hamiltonian diffeomorphisms acts on
$\mathcal E(M,\omega)$ as symplectomorphisms.

Let $X_f$ be the Hamiltonian vector field on
$M$ defined by a smooth function $f$ on $M$.
Then the induced infinitesimal action of
$- X_f$ on $\mathcal E(M,\omega)$ is computed as

\begin{equation}\label{tangent}
\begin{split}
(\difLie_{X_f}\nabla)_Y Z
&= [X_f,\nabla_Y Z] - \nabla_{[X_f, Y]} Z - \nabla_Y [X_f,Z] \\
&= R^\nabla(X_f,Y)Z + (\nabla\nabla X_f )(Y,Z).
\end{split}
\end{equation}

For $\nabla \in \mathcal E(M,\omega)$, we define
the Cahen–Gutt momentum map $J(\nabla)$ by

\begin{equation}  \label{CGmoment}
J(\nabla) = \nabla_p\nabla_q \mathrm{Ric}(\nabla)^{pq}
\frac12 \mathrm{Ric}(\nabla)_{pq}\mathrm{Ric}(\nabla)^{pq}
+ \frac14 \mathrm{R}(\nabla,\omega)_{pqrs}
\mathrm{R}(\nabla,\omega)^{pqrs},
\end{equation}

\begin{equation}\label{curvature}
\mathrm{R}(\nabla,\omega)(X,Y,Z,W) = \omega (R(X,Y)Z, W)
\end{equation}

\begin{equation}
\mathrm{Ric}(X,Y) = - \tr ( Z \mapsto R(X,Z)Y).
\end{equation}

The function $J$ on $\mathcal E(M,\omega)$ given by (<ref>) is an equivariant momentum map
for the action of $\HamDiffGroup(M,\omega)$.
Moreover, the action of the group of symplectomorphisms
on $\mathcal E(M,\omega)$ has a non-equivariant momentum map.

Thus, the zeros of the momentum map are the symplectic
connections $\nabla$ for which $J(\nabla)$ is constant.
An important observation
is that if the Fedosov star product is closed, then
the Cahen–Gutt momentum $J(\nabla)$ is constant, see

Now we assume that $M$ is a compact Kähler manifold
and that $\omega$ is a fixed symplectic form. We set,
as before, $\mathcal I$ to be the set of all integrable
complex structures $j$ such that $(M,\omega, j)$ is a Kähler
manifold. Let $lv : \mathcal I \to \mathcal E(M,\omega)$
be the Levi-Civita map sending $j$ to the Levi-Civita
connection $\nabla^j$ of the Kähler manifold $(M,\omega,j)$.
Then $lv^\ast \Omega^{\mathcal E}$ gives a new symplectic
structure on $\mathcal I$ if it is non-degenerate.
A condition for non-degeneracy, under suitable assumptions
on the Ricci curvature, is given in
[Proposition 17]Fuente-Gravy2016. In the following,
we assume that $lv^\ast \Omega^{\mathcal E}$ is symplectic.
Then, by <ref>, we can apply
<ref> and recover the following result.

Let $(M, \omega)$ be a compact Kähler manifold. Then

\begin{equation}
\widetilde{\mathcal F}_\zeta (\omega, 1) = - \int_M (J(\nabla^j) - J(\nabla^j)_0)\,\varrho_\zeta\ \mu_\omega, \qquad j=\chi(\omega,1) \label{obst3}
\end{equation}

is independent of the choice of $\omega \in \mathcal K$. In particular, if $ \widetilde{\mathcal F}_\zeta (\omega, 1) $
is not identically zero as a function of $\zeta$, then there is no closed Fedosov star-product for
$\omega \in \mathcal K$.

The last statement follows from <ref>.

In FutakiOno2018 it has been shown that
for a real smooth function $f$,
$\difLie_{X_f}\nabla^j = 0$ if and only if
$\difLie_{X_f}j = 0$. In this case, $X_f$ is a holomorphic
Killing vector field. Note also that the momentum map $J(\nabla)$
defined in (<ref>) is written in the Kähler case as

\begin{equation*}
J(\nabla) = \Delta S - \frac12 |\mathrm{Ric}|^2
+ \frac14 |\mathrm{R}|^2.
\end{equation*}

If a vector field $Y$ generates a Hamiltonian isometric
$S^1$-action, , \( Y \contr \omega=-\dif v_Y \)
for the Hamiltonian function $v_Y$, then
adapting the cohomology formula of Odaka2013
and Wang2012a to our context, we obtain
the version of the Donaldson-Futaki invariant
$\mathrm{Fut}$ for a test configuration
$(\mathcal M, \mathcal L)$

\begin{eqnarray}
\label{intersection1}\\
&&\quad = \frac{-2}{n+1}\kappa(M,L)\cup c_1(\mathcal{L})^{n+1}
+2n\left( c_2(\mathcal{M})- \frac{1}{2}c_1^2\Bigl(
\mathcal{K}^{-1}_{\mathcal{M}/\bfC\bfP^1}\Bigr)
\right)\cup c_1(\mathcal{L})^{n-1},\nonumber
\end{eqnarray}

where $\kappa(M,L)$ is the average of the Cahen–Gutt momentum

\begin{equation}\label{intersection2}
\kappa(M,L) \defeq
\right)(M)\cup c_1(L)^{n-2}}{c_1(L)^n},
\end{equation}

and the cup products \( \cup \) are taken on the total
space of $\mathcal M$, which has dimension $n+1$ in (<ref>), and on $M$, which has dimension
$n$ in (<ref>), see FutakiFuenteGravy2019.
This suggests that one could define $K$-stability related
to the study of a Kähler metric with constant Cahen–Gutt
momentum, at least if one can restrict to smooth test
configurations. The Kempf–Ness functional can be also
defined as in (<ref>), but nothing has been
done so far in the literature in this Cahen–Gutt
momentum map case.

Replacing the role of the scalar curvature by $J(\nabla)$,
many results in Kähler geometry can be
carried over to the geometry of the Cahen–Gutt momentum
map, see, e.g., Fuente-Gravy2016,FutakiOno2018.
For example, one can define a Cahen–Gutt version of extremal
Kähler metrics and prove the same structure theorem as
the Calabi extremal Kähler metrics, see
[Theorem 4.7]FutakiOno2018 and
[Theorem 6.9]DiezRatiuSymplecticConnections.

§.§ Z-critical Kähler metrics

In this subsection, we connect Dervan's
work Dervan2023 to our general results of <ref>.
Let $(M,L)$ be a smooth polarized variety of dimension $n$,
with $L$ an ample line bundle. Let $\mathcal K$ be the set
of Kähler forms in the class $c_1(L)$.
Suppose we are given
$\rho = (\rho_0, \dotsc, \rho_n) \in \mathbb C^{n+1}$
with $\rho_n = \sqrt{-1}$, and
$\Theta \in \oplus_{j=0}^n H^{j,j}(M,\mathbb C)$
of the form $\Theta = 1 + \Theta'$ where
$\Theta' \in \oplus_{j>0} H^{j,j}(M,\mathbb C)$.

We define the
central charge $Z(M,L) : \mathbb N \to \mathbb C$ by

\begin{equation}
Z_k(M,L) =
\sum_{\ell=1}^n \rho_\ell k^\ell
\int_M L^\ell \cup \sum_{j=0}^n a_j (-K_M)^j \cup \Theta
\end{equation}

where \( a_j \in \C \) with $a_0=a_1=1$.
Here, \( K_M \) is the canonical bundle of \( M \)
and we use the convention $K_M^0 = 1$.
We define the phase $\varphi_k (M,L)$ to be
the argument of $Z_k(M,L)$. Hereafter we often omit $k$
and write $Z(M,L)$ and $\varphi(M,L)$, or simply $Z$ and $\varphi$.

We fix a representative $\theta \in \Theta$.
For $\omega \in \mathcal K$, we define

\begin{equation}
\tilde Z(\omega) =
\sum_{\ell=1}^n \rho_\ell k^\ell
\sum_{j=0}^n a_j\tilde Z^{\ell,j} \in
\sFunctionSpace(M,\mathbb C)\end{equation}

\begin{equation}
\tilde Z^{\ell,j} =
\frac{\omega^\ell \wedge \mathrm{Ric}\, \omega^j
\wedge \theta}{\omega^n}
- \frac{j}{\ell +1} \Delta
\left( \frac{\omega^{\ell + 1} \wedge \mathrm{Ric}\,
\omega^{j-1} \wedge \theta}{\omega^n}  \right).
\end{equation}

Here $\mathrm{Ric}\, \omega$ denotes the Ricci form
of $\omega$ and $\Delta$ the Laplacian.
Note that

\begin{equation}
\int_M \tilde Z(\omega) \omega^n = Z(M,L).
\end{equation}

We say that $\omega \in \mathcal K$ is a $Z$-critical Kähler metric if

\begin{equation}
\Im\bigl(e^{-\I\varphi(M,L)}\tilde Z(\omega)\bigr) = 0
\quad \text{and} \quad
\mathrm{Re}\bigl(e^{-\I\varphi(M,L)}\tilde Z(\omega)\bigr) > 0.
\end{equation}

Notice that the first condition is equivalent to
$\arg \tilde Z(\omega) = \arg Z(M,L) \mod \pi$.

Fix $\omega \in \mathcal K$ and consider it as
a symplectic form. As before, let  $\mathcal I$ be
the set of all $\omega$-compatible complex structures
$j$ such that $(M,\omega, j)$ is a Kähler
manifold. Dervan constructs a Kähler structure
$\Omega_\epsilon$, $\epsilon = \frac 1 k$,  on
$\mathcal I$  when $k$ is large, , for the
large volume limit. Unlike the $L^2$-inner product
in the previous subsections, the construction of
$\Omega_\epsilon$ is more nonlinear and we will
not reproduce it here, see
[Equation (3.11)]Dervan2023. Dervan
shows that $\Im (e^{-i\varphi_\epsilon}\tilde Z)$
is a momentum map for the action of the group of
Hamiltonian diffeomorphisms. The Futaki invariant
of <ref> (and <ref>)
in this case is given by

\begin{equation}
\widetilde{\mathcal F}_\zeta (\omega, 1) =
- \int_M (\Im (e^{-\I\varphi_\epsilon}\tilde Z)
- \Im (e^{-\I\varphi_\epsilon}\tilde Z)_0)\,
\varrho_\zeta\ \mu_\omega.
\label{obst4}
\end{equation}

The vanishing of $\widetilde{\mathcal F}_\zeta (\omega, 1)$
is a necessary condition for the existence
of $Z$-critical Kähler metrics; in this way, we
recover [Corollary 3.7]Dervan2023.
A special form of $\widetilde{\mathcal{F}}_\zeta (\omega, 1)$
has been considered by Leung1998a.
The Kempf–Ness functional can be defined again as
in (<ref>).

Instead of treating only $K_M$, the higher Chern classes
can also be treated, see [Section 4]Dervan2023
and DervanHallam2023.

§ GAUGE THEORY

Let $E \to M$ be a $\sFunctionSpace$ complex vector bundle
over a compact Kähler manifold $(M,\omega)$,
$h_0$ a fixed Hermitian metric of $E$, and $\mathcal G$
the group of unitary gauge transformations of $E$ with
respect to $h_0$. Let $\mathcal A(h_0)$ be the space of
$h_0$-unitary connections $A$ of $E$
such that the curvature $F_A$ is type $(1,1)$.
Thus, $F_A^{0,2} = \barpartial_A\circ \barpartial_A = 0$
and $A$ defines a holomorphic structure on $E$,
see [Theorem 2.1.53]DonaldsonKronheimer1997.
Here we denote by $\dif_A$ the covariant derivative with
respect to $A \in \mathcal A(h_0)$;
$\barpartial_A$ is the type $(0,1)$-part of $\dif_A$.
Then $\mathcal G$ acts on $\mathcal A(h_0)$ by
$\dif_{g(A)} = g\circ \dif_A \circ g^{-1}$ so that
$g(A) = A - \dif_A g \cdot g^{-1}$ for \( g \in \mathcal{G} \).
Here we used the same letter $A$ to denote the connection
form with respect to a chosen unitary frame.

We also denote by $\mathcal G^c$ the space of complex gauge
transformations of $E$; thus $\mathcal G^c$ is the
complexification of $\mathcal G$.

We take the Klein pair $(\mathfrak{g}^c, \mathfrak{g})$, where
$\mathfrak{g}$ is the Lie algebra of $\mathcal{G}$ and
$\mathfrak{g}^c$ is the complexification of $\mathfrak{g}$.
In this case, $\mathfrak{g}^c$ integrates to
the complex gauge group $\mathcal{G}^c$ of $E$.
We take as the Cartan bundle $\SectionSpaceAbb{P}$ modeled
on the Klein pair $(\mathfrak{g}^c, \mathfrak{g})$ with
group $\mathcal{G}$ the principal bundle $\SectionSpaceAbb{P}
=\mathcal{G}^c \to \mathcal{G}\backslash \mathcal{G}^c$
with the left $\mathcal{G}$-action. With the right Maurer–Cartan
form $\theta$ on \( \SectionSpaceAbb{G}^c \), the pair
$(\SectionSpaceAbb{P},\theta)$ is a Cartan geometry modeled
on the Klein pair $(\mathfrak{g}^c, \mathfrak{g})$.
Alternatively, one can consider the space of pairs $(h, g)$,
where $h$ is a Hermitian metric on $E$ and $g \in \mathcal G^c$,
satisfying $h(x,y) = h_0(gx,gy)$. The left action of $\mathcal{G}$
on $\SectionSpaceAbb{P}$ is given by  $k\cdot (h,g) = (h,kg)$.
Then $\SectionSpaceAbb{P}$ is a principal $\mathcal{G}$-bundle
over the homogenous space $\mathcal{G}\backslash \mathcal{G}^c$
and $\SectionSpaceAbb{P}$ is diffeomorphic to $\mathcal{G}^c$.
Note the formal similarity of the latter definition of
$\SectionSpaceAbb{P}$ to the Cartan bundle in Kähler
geometry introduced in <ref>.
Since any two Hermitian metrics are related by a complex
gauge transformation, the base
\( \mathcal G\backslash \mathcal G^c \) is identified
with the space of Hermitian metrics on \( E \).
In particular, \( \mathcal G\backslash \mathcal G^c \)
is simply connected.

Fix $A \in \mathcal A(h_0)$. There is a smooth map
$\chi : \SectionSpaceAbb{P} \to \mathcal A(h_0)$ induced
by the action of $\mathcal G^c$ on $\mathcal A(h_0)$.
To describe this action, let $A = A^{1,0} + A^{0,1}$ be
the decomposition of $A \in \mathcal A(h_0)$ into its
$(1,0)$ and $(0,1)$-components so that $A^{1,0} =
- (A^{0,1})^\ast$, where ${}^\ast$ denotes the transpose-conjugate
with respect to $h_0$. For $g \in \mathcal G^c$, define the
partial connection by $\barpartial_{g(A)}\defeq
g\circ \barpartial_A \circ g^{-1}$ or, equivalently, by

\begin{equation}
g(A)^{0,1} = A^{0,1} - \barpartial_A^\ast g \cdot g^{-1}.
\end{equation}

In these formulas, $\circ$ is the composition of operators
and $\cdot$ denotes matrix multiplication.
A partial connection defines a unique unitary connection
$g(A)$ by

\begin{equation}
g(A)^{1,0} \defeq -(g(A)^{0,1})^\ast
= A^{1,0} + (\barpartial_A^\ast g \cdot g^{-1})^\ast,
\end{equation}

see [Lemma 2.1.54]DonaldsonKronheimer1997.
If $g$ is a unitary gauge transformation in $\mathcal G$,
then $\dif_{g(A)} = g\circ \dif_A \circ g^{-1}$ or, equivalently,

\begin{equation}
g(A) = A + g\dif_A(g^{-1}) = A - \dif_A g\cdot g^{-1}.
\end{equation}

Thus, the $\mathcal{G}^c$-action on $\mathcal A(h_0)$
is a natural extension of the $\mathcal{G}$-action.

Since a unitary connection is uniquely determined by a
partial connection, the tangent space $T_A\mathcal A(h_0)$
can be identified with $\Omega^{0,1}(\End(E))$ and there
is an almost complex structure on $\mathcal A(h_0)$ defined
by the multiplication by $\I$ on $\Omega^{0,1}(\End(E))$.
It is easy to see that the Cartan bundle
\( (\mathcal{P}, \theta) \) provides a model for the
complex orbit through \( A \) in the sense of
<ref>. Note that $\theta$ is torsion free.

§.§ Hermitian Yang–Mills connections

Keeping the same notations, let $h_0$ be a Hermitian metric
of a holomorphic vector bundle $E \to M$ over a compact
\(2n\)-dimensional Kähler manifold $(M,\omega)$ and
$\mathcal A(h_0)$ the space of $h_0$-unitary connections.
There is a symplectic form $\Omega$ on $\mathcal A(h_0)$
defined by

\begin{equation}
\Omega(\xi,\eta) = - \int_M \tr(\xi\wedge\eta) \,\omega^{n-1}
\end{equation}

for $ \xi , \eta \in \TBundle_A \mathcal A(h_0)=\Omega^{0,1}(\End(E))$.
It is well known
and easy to show that

\begin{equation}
J(A) = F_A\wedge \omega^{n-1} -  \lambda \, \id\, \omega^n
\end{equation}

is a momentum map on the symplectic manifold
$\bigl(\mathcal A(h_0), \Omega\bigr)$ for the action
of the unitary gauge transformations $\mathcal{G}$ for
any purely imaginary constant $\lambda$.
Thus, by <ref>, the Futaki
functional $\widetilde{\mathcal{F}}$ on the stabilizer
\( (\mathfrak{g}^c)_A \) of $A= \chi(h_0,1)$ of
the action of $\mathfrak{g}^c$, defined by

\begin{equation}\label{obst5}
\widetilde{\mathcal F}(\zeta) =
- \Im \int_M \tr\bigl(\zeta (F_A \wedge \omega^{n-1}
-  \lambda \, \id \, \omega^n)\bigr), \qquad
\zeta \in (\mathfrak g^c)_A
\end{equation}

is independent of the Hermitian metric $h_0$. Note that
the unitary connection $A$ determines a holomorphic
structure $\barpartial_A$ on $E$ and thus the infinitesimal
complex gauge transformation $\zeta$ in the
stabilizer of $A$ is just a holomorphic endomorphism of
$E$ with respect to $\barpartial_A$.

Choosing $\lambda$ to be the topological invariant determined

\begin{equation}
-2\pi \I r\lambda = c_1(E)[\omega]^{n-1}
\end{equation}

with $r=\mathrm{rank}(E)$, the Chern connection $A$ of
a Hermitian metric $h$ is called a Hermitian
Yang–Mills connection (or $h$ is called a
Hermitian-Einstein metric) if its curvature
$F_A$ satisfies

\begin{equation}
F_A\, \omega^{n-1} =  \lambda \, \id \, \omega^n.
\end{equation}

Thus $\widetilde{\mathcal F}$ is an obstruction to the
existence of Hermitian Yang–Mills connections.
However, it is well known that if a Hermitian Yang–Mills
connection exists on an irreducible holomorphic
vector bundle $E$, then $E$ is simple, meaning that
any holomorphic section $\zeta$ of the endomorphism
bundle of $E$ is a multiple of the identity. Therefore,
when trying to prove existence of Hermitian Yang–Mills
solutions, it is necessary to assume that $E$ is simple
(and even slope stable). With this natural assumption,
the obstruction (<ref>) always vanishes.

Note, in passing, that we have the following criterion
for the vanishing of the Futaki character which follows from
Kodaira–Serre duality.

The Futaki character (<ref>) vanishes if and only
if $F_A \omega^{n-1} - \lambda \omega^n$ is trivial in
$H^{n,n}(M) = H^{2n}(M, \C)$.

The Kempf–Ness functional defined in <ref>
recovers in this case the Donaldson functional, seen as a functional
on the space of Hermitian metrics.
The Donaldson functional has been effectively used by
Donaldson1985 for the existence problem of Hermitian
Yang–Mills connections on algebraic surfaces
and was the precursor of the Mabuchi K-energy.
See UhlenbeckYau1986 for the higher dimensional result.

We remark that the above arguments still work even if
the Hermitian metric is indefinite.
GarciaFernandezMolina2023 used (<ref>)
for indefinite metrics as an obstruction to the existence
of solutions of the Hull–Strominger system. See also

§.§ Z-critical connections

In this subsection, we apply our results of <ref>
to the setting of DervanMcCarthySektnan2020.
Let $E \to M$ be a holomorphic vector bundle over a compact
Kähler manifold $(M,\omega)$.
A polynomial central charge $Z_k(E)$ of $E$ is

\begin{equation}
Z_k(E) = \int_M \sum_{d=0}^n\, \rho_d \,k^d
\,[\omega]^d \cup \chernClass(E)\cup U
\end{equation}

where $(\rho_0, \rho_1, \dotsc, \rho_n) \in
(\mathbb C^\ast)^{n+1}$ with $\Im \rho_n > 0$,
$\Re\rho_{n-1} <0$, and $U = 1 + N$ with
$N \in H^{>0}(M,\mathbb R)$ are given data.
Here, \( \chernClass(E) \in H^*(M) \) denotes  the Chern
character of \( E \). For each $E$ with $Z_k(E) \ne 0$,
we define the phase

\begin{equation}
\varphi_k(E) \defeq \arg Z_k(E).
\end{equation}

We often omit $k$, and write \( Z(E) \) and
\( \varphi(E) \) when no confusion is likely to occur.
Let $\tilde U \in \Omega^\ast(M, \mathbb R)$ be a
closed form representing $U \in H^\ast(M,\mathbb R)$.
For a Hermitian metric $h$ of $E$ with Chern connection
$A$, we denote by $\widetilde{\chernClass}(A) =
\exp\left(\frac{i}{2\pi}\,F_A\right)$ the endomorphism-valued
differential form whose trace represents the
Chern character $\chernClass(E)$.
We define an endomorphism-valued
$(n,n)$-form $\tilde Z_k(A)$ by

\begin{equation}
\tilde Z_k(A) = \left[\sum_{d=0}^n \rho_d\,k^d\,
\omega^d\wedge\widetilde{\chernClass}(A)\wedge
\tilde{U}\right]_{n,n},
\end{equation}

where on the right-hand side above we only select the
$(n,n)$-form component of the differential
form in the brackets with various mixed degrees.  Obviously,

\begin{equation}\label{average1}
0 = \Im\bigl(e^{-\I\varphi_k(E)}\tilde Z_k(E)\bigr)
= \int_M\,\tr \, \Im
\bigl(e^{-\I\varphi_k(E)}\tilde Z_k(A)\bigr).
\end{equation}

We say that $A$ is a $Z$-critical (or $Z_k$-critical)
connection if

\begin{equation}
\Im\bigl(e^{-\I\varphi_k(E)}\tilde Z_k(A)\bigr) = 0,
\end{equation}

where the imaginary part means $-\I$-times the $h$-skew
Hermitian part.

Fix a Hermitian metric $h_0$ and let $\mathcal A(h_0)$ be
the space of all $h_0$-unitary connections.
Define a Hermitian pairing on $\TBundle_A \mathcal A(h_0) =
\Omega^{0,1}(\End(E))$ by

\begin{equation}
\langle \xi,\eta \rangle_A =
-\I \int_M \tr \,
\left[ \Im( e^{-\I\varphi(E)}\,\tilde Z^\prime(A)
\wedge \xi \wedge \eta^\ast)\right]_{\mathrm{sym}}
\end{equation}

where $\tilde Z^\prime(A)$ denotes the derivative of
$\tilde Z(A)$ with respect to $\frac{\I}{2\pi}\,F_A$,
considered as a variable, and where
$\mathrm{sym}$ denotes the graded symmetric product
of endomorphism valued forms

\begin{equation}
[B_1 \wedge \cdots \wedge B_j]_{\mathrm{sym}}
= \frac 1{j!} \sum_{\sigma \in S_j}
B_{\sigma(1)}\wedge \cdots \wedge B_{\sigma(j)};
\end{equation}

here, gradsgn means the sign resulting from the
permutation of the differential forms $B_i$'s.
Then the imaginary part

\begin{equation}
\Omega_A(\xi,\eta) = \Im \langle \xi,\eta \rangle_A
\end{equation}

is a symplectic form on $\mathcal A(h_0)$ provided
$ \langle \cdot \, ,\,\cdot \rangle_A $ is positive
definite. When this latter condition is satisfied,
$A$ is called a subsolution.
When $A$ is a subsolution, then the PDE for $A$ being
a $Z$-critical connection becomes elliptic,
see [Lemma 2.36 and Proposition 2.43]DervanMcCarthySektnan2020.
By [Theorem 2.45]DervanMcCarthySektnan2020, the map

\begin{equation}
A \mapsto 2\pi \I \Im \bigl(e^{-\I\varphi(E)}\tilde Z(A)\bigr)
\end{equation}

is a momentum map for the $\mathcal G$-action on
$\bigl(\mathcal A(h_0),\Omega\bigr)$. Thus, by
<ref>, the Futaki functional
$\widetilde{\mathcal F}$ on $H^0(M,\End(E))$, defined
in this case by

\begin{equation}\label{obst6}
\widetilde{\mathcal F}(\zeta) \defeq
2\pi \int_M \tr\Bigl(\zeta \, \Im \bigl(
e^{-\I\varphi(E)}\tilde Z(A)\bigr)\Bigr),
\end{equation}

is independent of the Hermitian metric $h_0$.
Clearly, the non-vanishing of $\widetilde{\mathcal F}$
is an obstruction to the existence of a Hermitian metric
$h$ whose Chern connection $A$ is a $Z$-critical connection.
By <ref>, we can also obtain the Kempf–Ness functional just as
the Donaldson functional, CollinsYau2021.

A holomorphic vector bundle $E \to (M,\omega)$ with
$Z_k(E) \ne 0$ for all $k \gg 0$ is said to be
asymptotically $Z$-stable if for all proper non-zero
subsheaves $F \subset E$ there exists some $k_0$ such that
$ \varphi_k(F) < \varphi_k(E)$ for all $k \ge k_0$. By
[Lemma 2.18]DervanMcCarthySektnan2020,
any asymptotically $Z$-stable
bundle is simple. Thus, by (<ref>),
$\widetilde{\mathcal F}$ vanishes on any asymptotically
$Z$-stable bundle.

In the special case where \( E \) is a line bundle \( L \),
the central charge is

\begin{equation}
Z_k(L) = \I^{n+1}\int_M e^{-\I k[\omega]} \, \chernClass(L).
\end{equation}

Moreover, in this case, a $Z$-critical connection
is a solution of

\begin{equation}
\Im\bigl(e^{-\I\varphi}\tilde Z_k(A)\bigr) =
\frac1{n!} \Im
\left( e^{-\I\varphi}\left( \omega -
\frac{F_A}{2\pi} \right) \right) = 0.
\end{equation}

This equation is called the LYZ equation after
LeungYauZaslow2000, which was formulated as the mirror
to the special Lagrangian equation; its solutions
are called the deformed Yang–Mills connections.
This subject has been well-studied and there are
many interesting papers, but most relevant to
our setting are CollinsYau2021,CollinsShi2022.
Since the endomorphism bundle of a line bundle is trivial,
its holomorphic sections are constant functions. Thus,
the obstruction (<ref>) vanishes. The Kempf–Ness
functional is explicitly described using the Calabi-Yau
functional and is studied in terms of geodesics in

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

journaltitleMath. Ann.
titleA Lie group structure for pseudodifferential operators

journaltitlePhilos. Trans. Roy. Soc. London Ser. A
titleThe Yang-Mills equations over Riemann surfaces

journaltitleC. R. Acad. Sci. Paris
titleEquations du type Monge-Ampère sur les variétés kähleriennes compactes

journaltitleKodai Math. J.
titleAn obstruction for Chern class forms to be harmonic

journaltitleJ. Amer. Math. Soc.
titleConvexity of the $K$-energy on the space of Kähler metrics and uniqueness of extremal metrics

journaltitleJ. Amer. Math. Soc.
titleA variational approach to the Yau-Tian-Donaldson conjecture

Societe Mathematique de France

journaltitleAnn. Sci. Éc. Norm. Supér. (4)
titleRegularity of weak minimizers of the $K$-energy and applications to properness and $K$-stability

World Scientific Publishing Co. Pte Ltd

noteIncludes bibliographical references and index
subtitleWith applications to maximum principles and Lie groups
titleAn Introduction to the Geometrical Analysis of Vector Fields

Cellule MathDoc/CEDRAM

journaltitleAnn. Inst. Fourier
titleUniform $K$-stability, Duistermaat-Heckman measures and singularities of pairs

Academia Press

booktitleLiber Amicorum Richard Delanghe: Een Veelzijdig Wiskundige
groupsSymplectic connection
titleMoment map for the space of symplectic connections

Springer, Berlin

booktitleDifferential Geometry and Complex Analysis
groupsSymplectic connection
titleExtremal Kähler metrics II
53C55 (58E30)

journaltitleJ. Amer. Math. Soc.
titleKähler-Einstein metric on Fano manifolds. III: limits with cone angle approaches $2\pi$ and completion of the main proof

American Mathematical Society (AMS)

journaltitleJournal of the American Mathematical Society
shorttitleKähler-Einstein metrics on Fano manifolds. I
titleKähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities

journaltitleGeom. Topol.
titleKähler-Ricci flow, Kähler-Einstein metric, and K-stability

Paris: Société Mathématique de France

booktitleGéométrie différentielle, physique mathématique, mathématiques et société (I). Volume en l'honneur de Jean Pierre Bourguignon
groupsFutaki Character
titleTest configuration and geodesic rays

Somerville, MA: International Press

annotationComment: Survey article
booktitleDifferential Geometry, Calabi-Yau Theory, and General Relativity. Part 2. Lectures and Articles Celebrating the 70th Birthday of Shing-Tung Yau, Harvard University, Cambridge, Ma, USA, May 2019
groupsFutaki Character
notearXiv:2004.04831 [math] type: article
titleStability and the deformed Hermitian-Yang-Mills equation

groupsFutaki Character
journaltitleAnnals of PDE
shorttitleMoment Maps, Nonlinear PDE and Stability in Mirror Symmetry, I
titleMoment maps, nonlinear PDE and stability in mirror symmetry, I: Geodesics
Mirror symmetry,Stability,Deformed Hermitian-Yang-Mills,Special Lagrangian,Calabi-Yau,paper

groupsDeformation quantization
journaltitleLett. Math. Phys.
titleClosed star products and cyclic cohomology

* Entry 'ConnesFlatoSternheimer1992' (d:/Research/A_Central/Latex/bibtex/bib/Literature.bib): Invalid format '2019-03-09TZ' of date field 'urldate' - ignoring

journaltitleGeom. Funct. Anal.
titleKähler-Einstein metrics along the smooth continuity method

journaltitleLett. Math. Phys.
titleExistence of star-products and of formal deformations of the Poisson Lie Algebra of arbitrary symplectic manifolds

journaltitleForum Math. Sigma
titleStability conditions for polarised varieties
pagesPaper No. e10457pp

titleZ-critical connections and Bridgeland stability conditions

groupsInfinite-dimensional symplectic geometry
titleThe universal structure of moment maps in complex geometry

groupsMy publications
titleNorm-squared of the momentum map in infinite dimensions with applications to Kähler geometry and symplectic connections

journaltitleProc. London Math. Soc. (3)
titleAnti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles

World Sci. Publ., River Edge, NJ

booktitleFields Medallists' Lectures
seriesWorld Sci. Ser. 20th Century Math.
titleRemarks on gauge theory, complex geometry and 4-manifold topology
57R57 (57N13 58D27 58G05)

American Mathematical Society

booktitleNorthern California symplectic geometry seminar
groupsFutaki Character
titleSymmetric spaces, Kähler geometry and Hamiltonian dynamics

Oxford University Press

groupsYang Mills (heat) flow, Math (Riemannian)
seriesOxford Mathematical Monographs
titleThe Geometry of Four-Manifolds

journaltitleJ. Differ. Geom.
titleScalar curvature and stability of toric varieties

journaltitleJ. Differ. Geom.
titleA simple geometrical construction of deformation quantization

journaltitleSugaku Expositions
noteTranslation of Sûgaku 42 (1990), no. 3, 231–243
titleModuli space of polarized algebraic manifolds and Kähler metrics

journaltitleInvent. Math.
titleAn obstruction to the existence of Einstein Kähler metrics

Springer-Verlag, Berlin

seriesLecture Notes in Mathematics
titleKähler-Einstein Metrics and Integral Invariants
53C25 (14J40 32J99 53C55 53C57 58E15 58G30),paper

journaltitleProc. 9-th Internat. Conf. on Differential Geometry and its Applications, Prague, 2004
titleStability, integral invariants and canonical Kahler metrics

annotationComment: 20 pages, to appear in Kodai Mathematical Journal
groupsSymplectic connection
journaltitleKodai Math. J.
notearXiv: math/0603706 version: 2
titleHarmonic total Chern forms and stability
atics - Differential Geometry,Mathematics - Symplectic Geometry,53C55,53C21,paper

International Press of Boston

groupsSymplectic connection
issn1527-5256, 1540-2347
journaltitleJ. Symplect. Geom.
titleHolomorphic vector fields and perturbed extremal Kähler metrics

Int. Press, Boston, MA

booktitleProceedings of the International Consortium of Chinese Mathematicians 2018
groupsSymplectic connection
titleDeformation quantization and Kähler geometry with moment map

journaltitleMath. Ann.
titleBilinear forms and extremal Kähler vector fields associated with Kähler classes

journaltitleJ. Differential Geom.
titleInvariant polynomials of the automorphism group of a compact complex manifold
Complex Lie groups,group actions on complex spaces,paper

journaltitleKodai Math. J.
titleCharacters of automorphism groups associated with Kähler classes and functionals with cocycle conditions

annotationComment: The curvature condition is added in the main result, and other modifications are made
groupsSymplectic connection
journaltitleJ. Symplectic Geom.
titleCahen-Gutt moment map, closed Fedosov star product and structure of the automorphism group
Mathematics - Differential Geometry,32Q15,46L65,53C55,paper

titleFutaki Invariants and Yau's Conjecture on the Hull-Strominger system

journaltitleTrans. Amer. Math. Soc.
titleHolomorphic string algebroids

noteTo appear in J. Differential Geom.
titleGauge theory for string algebroids

groupsSymplectic connection
titleCalabi’s Extremal Kähler Metrics: An Elementary Introduction
ive.org/web/20210108083652/https://cims.nyu.edu/ rodion/lib/P.

Cham: Springer

groupsGIT, Symplectic connection
isbn978-3-030-89299-9; 978-3-030-89300-2
seriesLect. Notes Math.
titleThe Moment-Weight Inequality and the Hilbert-Mumford Criterion. GIT from the Differential Geometric Viewpoint
.ethz.ch/ salamon/PREPRINTS/momentweight.pdf
Hessian,momentum map squared,paper

pubstateTo appear
titleInfinite-dimensional Lie Groups. General Theory and Main Examples

journaltitleMath. Ann.
titleCartan decomposition of the moment map

booktitleConference Held at Katata, September 3–8, 1979. Proceedings of the 6th International Taniguchi Symposium
howpublishedSendai/Japan: Tohoku University, Department of Mathematics. v, 23 p. (1979).
titleNon-linear problems in geometry

Springer, Berlin

booktitleAlgebraic Geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978)
seriesLecture Notes in Math.
titleThe length of vectors in representation spaces
14L30 (20G05)

annotationComment: 62 pages. Revised version, accepted in Bull. AMS
groupsFutaki Character
issn0273-0979, 1088-9485
journaltitleBulletin of the American Mathematical Society
titleGeometric hydrodynamics and infinite-dimensional Newton's equations

journaltitleProc. Japan Acad. Ser. A Math. Sci.
titleCurvature and stability of vector bundles
53C55 (32L99),paper

librarySK 370 K75 (CB, ITP)
titleFoundations of Differential Geometry

journaltitleLetters in Math. Phys.
titleDeformation quantization of Poisson manifolds

American Mathematical Society

seriesMathematical Surveys and Monographs
titleThe Convenient Setting of Global Analysis
Infinite dimensional manifolds,paper

groupsSymplectic connection
journaltitleAnn. Global Anal. Geom.
titleInfinite dimensional moment map geometry and closed Fedosov’s star products

groupsSymplectic connection
journaltitleJ. Symplectic Geom.
titleFutaki invariant for Fedosov's star products

journaltitleComm. Anal. Geom.
titleBando Futaki invariants and Kähler Einstein metric

journaltitleAdv. Theor. Math. Phys.
titleFrom special Lagrangian to Hermitian-Yang-Mills via Fourier-Mukai transform

journaltitleInvent. Math.
titleG-uniform stability and Kähler-Einstein metrics on Fano varieties

journaltitleAnn. of Math.
titleFinite generation for valuations computing stability thresholds and applications to K-stability

groupsFutaki Character
journaltitleCommun. Math. Phys.
titleSome geometric calculations on Wasserstein space
Sectional Curvature,Symplectic Manifold,Poisson Structure,Geometric Calculation,Parallel Transport,paper

journaltitleTohoku Math. J.
titleK-energy maps integrating Futaki invariants

journaltitleOsaka J. Math.
titleSome symplectic geometry on compact Kähler manifolds. I

groupsFutaki Character
titleGeometry of infinite dimensional Cartan Developments

annotationComment: Major revision of the first version. 61 pages, 10 figures
groupsFutaki Character
journaltitleJ. Geom. Mech.
titleGeometry of matrix decompositions seen through optimal transport and information geometry
Mathematics - Differential Geometry,Mathematics - Numerical Analysis,15A23,53C21,58B20,15A18,49M99,65F15,65F40,paper

Springer Japan

journaltitleJapan. J. Math.
titleTowards a Lie theory of locally convex groups

journaltitleOsaka J. Math.
titleA generalization of the Ross-Thomas slope theory

journaltitleAdv. in Math.
titleWeyl manifolds and deformation quantization

Taylor & Francis

groupsFutaki Character
journaltitleCommun. Part. Diff. Eq.
shorttitleThe Geometry of Dissipative Evolution Equations
titleThe geometry of dissipative evolution equations: the porous medium equation
AMS code: 35K55; 76S05; 58B20; 58F39,paper

journaltitleJ. Symplectic Geom.
titleTest configurations for K-stability and geodesic rays

titleDifferential Geometry and Mathematical Physics. Part II. Fibre Bundles, Topology and Gauge Fields

groupsFutaki Character
journaltitleAmer. J. Math.
titleComplex Monge-Ampère and symplectic manifolds

New York

Berlin: Springer

groupsFutaki Character
seriesGrad. Texts Math.
subtitleCartan's Generalization of Klein's Erlangen Program
titleDifferential Geometry

journaltitleComm. Anal. Geom.
titleThe K-energy of hypersurfaces and stability

journaltitleInvent. Math.
titleKähler-Einstein metrics with positive scalar curvature

annotationComment: 47 pages
journaltitleCommun. Pure Appl. Math.
notearXiv:1211.4669 [math] type: article
titleK-Stability and Kähler-Einstein metrics
Mathematics - Differential Geometry,Mathematics - Algebraic Geometry,Mathematics - Analysis of PDEs,53 Differential geometry,14 Algebraic geometry,paper

Wiley Subscription Services, Inc., A Wiley Company

groupsAnalytic questions
journaltitleCommun. Pure Appl. Math.
titleOn the existence of Hermitian-Yang-Mills connections in stable vector bundles

International Press of Boston

groupsSymplectic connection
journaltitleComm. Anal. Geom.
titleMoment map, Futaki invariant and stability of projective manifolds
Hessian,momentum map squared,paper

* Entry 'Wang2004' (d:/Research/A_Central/Latex/bibtex/bib/Literature.bib): Invalid format 'CURRENT_TIMESTAMP' of date field 'urldate' - ignoring

journaltitleMath. Res. Lett.
titleHeight and GIT weight
Geometric invariant theory,paper

journaltitleComm. Pure Appl. Math.
titleOn the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation I

journaltitleProc. Symp. Pure Math.
titleOpen problems in geometry

librarySK 370 K75 (CB, ITP)
titleFoundations of Differential Geometry

journaltitleC. R. Acad. Sci. Paris
titleEquations du type Monge-Ampère sur les variétés kähleriennes compactes

journaltitleComm. Pure Appl. Math.
titleOn the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation I

booktitleConference Held at Katata, September 3–8, 1979. Proceedings of the 6th International Taniguchi Symposium
howpublishedSendai/Japan: Tohoku University, Department of Mathematics. v, 23 p. (1979).
titleNon-linear problems in geometry

Springer, Berlin

booktitleAlgebraic Geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978)
seriesLecture Notes in Math.
titleThe length of vectors in representation spaces
14L30 (20G05)

journaltitleProc. Japan Acad. Ser. A Math. Sci.
titleCurvature and stability of vector bundles
53C55 (32L99),paper

journaltitlePhilos. Trans. Roy. Soc. London Ser. A
titleThe Yang-Mills equations over Riemann surfaces

journaltitleLett. Math. Phys.
titleExistence of star-products and of formal deformations of the Poisson Lie Algebra of arbitrary symplectic manifolds

journaltitleInvent. Math.
titleAn obstruction to the existence of Einstein Kähler metrics

Springer, Berlin

booktitleDifferential Geometry and Complex Analysis
groupsSymplectic connection
titleExtremal Kähler metrics II
53C55 (58E30)

journaltitleProc. London Math. Soc. (3)
titleAnti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles

journaltitleJ. Differential Geom.
titleInvariant polynomials of the automorphism group of a compact complex manifold
Complex Lie groups,group actions on complex spaces,paper

journaltitleMath. Ann.
titleA Lie group structure for pseudodifferential operators

Wiley Subscription Services, Inc., A Wiley Company

groupsAnalytic questions
journaltitleCommun. Pure Appl. Math.
titleOn the existence of Hermitian-Yang-Mills connections in stable vector bundles

journaltitleTohoku Math. J.
titleK-energy maps integrating Futaki invariants

journaltitleOsaka J. Math.
titleSome symplectic geometry on compact Kähler manifolds. I

Springer-Verlag, Berlin

seriesLecture Notes in Mathematics
titleKähler-Einstein Metrics and Integral Invariants
53C25 (14J40 32J99 53C55 53C57 58E15 58G30),paper

journaltitleAdv. in Math.
titleWeyl manifolds and deformation quantization

groupsDeformation quantization
journaltitleLett. Math. Phys.
titleClosed star products and cyclic cohomology

* Entry 'ConnesFlatoSternheimer1992' (d:/Research/A_Central/Latex/bibtex/bib/Literature.bib): Invalid format '2019-03-09TZ' of date field 'urldate' - ignoring

journaltitleSugaku Expositions
noteTranslation of Sûgaku 42 (1990), no. 3, 231–243
titleModuli space of polarized algebraic manifolds and Kähler metrics

groupsFutaki Character
journaltitleAmer. J. Math.
titleComplex Monge-Ampère and symplectic manifolds

journaltitleProc. Symp. Pure Math.
titleOpen problems in geometry

journaltitleJ. Differ. Geom.
titleA simple geometrical construction of deformation quantization

journaltitleComm. Anal. Geom.
titleThe K-energy of hypersurfaces and stability

journaltitleMath. Ann.
titleBilinear forms and extremal Kähler vector fields associated with Kähler classes

World Sci. Publ., River Edge, NJ

booktitleFields Medallists' Lectures
seriesWorld Sci. Ser. 20th Century Math.
titleRemarks on gauge theory, complex geometry and 4-manifold topology
57R57 (57N13 58D27 58G05)

Oxford University Press

groupsYang Mills (heat) flow, Math (Riemannian)
seriesOxford Mathematical Monographs
titleThe Geometry of Four-Manifolds

American Mathematical Society

seriesMathematical Surveys and Monographs
titleThe Convenient Setting of Global Analysis
Infinite dimensional manifolds,paper

New York

Berlin: Springer

groupsFutaki Character
seriesGrad. Texts Math.
subtitleCartan's Generalization of Klein's Erlangen Program
titleDifferential Geometry

journaltitleInvent. Math.
titleKähler-Einstein metrics with positive scalar curvature

journaltitleComm. Anal. Geom.
titleBando Futaki invariants and Kähler Einstein metric

American Mathematical Society

booktitleNorthern California symplectic geometry seminar
groupsFutaki Character
titleSymmetric spaces, Kähler geometry and Hamiltonian dynamics

journaltitleAdv. Theor. Math. Phys.
titleFrom special Lagrangian to Hermitian-Yang-Mills via Fourier-Mukai transform

journaltitleKodai Math. J.
titleCharacters of automorphism groups associated with Kähler classes and functionals with cocycle conditions

Taylor & Francis

groupsFutaki Character
journaltitleCommun. Part. Diff. Eq.
shorttitleThe Geometry of Dissipative Evolution Equations
titleThe geometry of dissipative evolution equations: the porous medium equation
AMS code: 35K55; 76S05; 58B20; 58F39,paper

journaltitleJ. Differ. Geom.
titleScalar curvature and stability of toric varieties

journaltitleLetters in Math. Phys.
titleDeformation quantization of Poisson manifolds

International Press of Boston

groupsSymplectic connection
journaltitleComm. Anal. Geom.
titleMoment map, Futaki invariant and stability of projective manifolds
Hessian,momentum map squared,paper

* Entry 'Wang2004' (d:/Research/A_Central/Latex/bibtex/bib/Literature.bib): Invalid format 'CURRENT_TIMESTAMP' of date field 'urldate' - ignoring

Academia Press

booktitleLiber Amicorum Richard Delanghe: Een Veelzijdig Wiskundige
groupsSymplectic connection
titleMoment map for the space of symplectic connections

journaltitleProc. 9-th Internat. Conf. on Differential Geometry and its Applications, Prague, 2004
titleStability, integral invariants and canonical Kahler metrics

journaltitleKodai Math. J.
titleAn obstruction for Chern class forms to be harmonic

annotationComment: 20 pages, to appear in Kodai Mathematical Journal
groupsSymplectic connection
journaltitleKodai Math. J.
notearXiv: math/0603706 version: 2
titleHarmonic total Chern forms and stability
atics - Differential Geometry,Mathematics - Symplectic Geometry,53C55,53C21,paper

Springer Japan

journaltitleJapan. J. Math.
titleTowards a Lie theory of locally convex groups

journaltitleMath. Ann.
titleCartan decomposition of the moment map

journaltitleJ. Symplectic Geom.
titleTest configurations for K-stability and geodesic rays

Paris: Société Mathématique de France

booktitleGéométrie différentielle, physique mathématique, mathématiques et société (I). Volume en l'honneur de Jean Pierre Bourguignon
groupsFutaki Character
titleTest configuration and geodesic rays

International Press of Boston

groupsSymplectic connection
issn1527-5256, 1540-2347
journaltitleJ. Symplect. Geom.
titleHolomorphic vector fields and perturbed extremal Kähler metrics

groupsFutaki Character
journaltitleCommun. Math. Phys.
titleSome geometric calculations on Wasserstein space
Sectional Curvature,Symplectic Manifold,Poisson Structure,Geometric Calculation,Parallel Transport,paper

journaltitleMath. Res. Lett.
titleHeight and GIT weight
Geometric invariant theory,paper

journaltitleOsaka J. Math.
titleA generalization of the Ross-Thomas slope theory

journaltitleJ. Amer. Math. Soc.
titleKähler-Einstein metric on Fano manifolds. III: limits with cone angle approaches $2\pi$ and completion of the main proof

American Mathematical Society (AMS)

journaltitleJournal of the American Mathematical Society
shorttitleKähler-Einstein metrics on Fano manifolds. I
titleKähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities

groupsSymplectic connection
journaltitleAnn. Global Anal. Geom.
titleInfinite dimensional moment map geometry and closed Fedosov’s star products

annotationComment: 47 pages
journaltitleCommun. Pure Appl. Math.
notearXiv:1211.4669 [math] type: article
titleK-Stability and Kähler-Einstein metrics
Mathematics - Differential Geometry,Mathematics - Algebraic Geometry,Mathematics - Analysis of PDEs,53 Differential geometry,14 Algebraic geometry,paper

journaltitleGeom. Funct. Anal.
titleKähler-Einstein metrics along the smooth continuity method

journaltitleJ. Amer. Math. Soc.
titleConvexity of the $K$-energy on the space of Kähler metrics and uniqueness of extremal metrics

Cellule MathDoc/CEDRAM

journaltitleAnn. Inst. Fourier
titleUniform $K$-stability, Duistermaat-Heckman measures and singularities of pairs

groupsSymplectic connection
titleCalabi’s Extremal Kähler Metrics: An Elementary Introduction
ive.org/web/20210108083652/https://cims.nyu.edu/ rodion/lib/P.

annotationComment: Major revision of the first version. 61 pages, 10 figures
groupsFutaki Character
journaltitleJ. Geom. Mech.
titleGeometry of matrix decompositions seen through optimal transport and information geometry
Mathematics - Differential Geometry,Mathematics - Numerical Analysis,15A23,53C21,58B20,15A18,49M99,65F15,65F40,paper

titleDifferential Geometry and Mathematical Physics. Part II. Fibre Bundles, Topology and Gauge Fields

journaltitleGeom. Topol.
titleKähler-Ricci flow, Kähler-Einstein metric, and K-stability

annotationComment: The curvature condition is added in the main result, and other modifications are made
groupsSymplectic connection
journaltitleJ. Symplectic Geom.
titleCahen-Gutt moment map, closed Fedosov star product and structure of the automorphism group
Mathematics - Differential Geometry,32Q15,46L65,53C55,paper

Cham: Springer

groupsGIT, Symplectic connection
isbn978-3-030-89299-9; 978-3-030-89300-2
seriesLect. Notes Math.
titleThe Moment-Weight Inequality and the Hilbert-Mumford Criterion. GIT from the Differential Geometric Viewpoint
.ethz.ch/ salamon/PREPRINTS/momentweight.pdf
Hessian,momentum map squared,paper

World Scientific Publishing Co. Pte Ltd

noteIncludes bibliographical references and index
subtitleWith applications to maximum principles and Lie groups
titleAn Introduction to the Geometrical Analysis of Vector Fields

groupsSymplectic connection
journaltitleJ. Symplectic Geom.
titleFutaki invariant for Fedosov's star products

Societe Mathematique de France

journaltitleAnn. Sci. Éc. Norm. Supér. (4)
titleRegularity of weak minimizers of the $K$-energy and applications to properness and $K$-stability

titleZ-critical connections and Bridgeland stability conditions

Int. Press, Boston, MA

booktitleProceedings of the International Consortium of Chinese Mathematicians 2018
groupsSymplectic connection
titleDeformation quantization and Kähler geometry with moment map

journaltitleTrans. Amer. Math. Soc.
titleHolomorphic string algebroids

journaltitleJ. Amer. Math. Soc.
titleA variational approach to the Yau-Tian-Donaldson conjecture

groupsFutaki Character
journaltitleAnnals of PDE
shorttitleMoment Maps, Nonlinear PDE and Stability in Mirror Symmetry, I
titleMoment maps, nonlinear PDE and stability in mirror symmetry, I: Geodesics
Mirror symmetry,Stability,Deformed Hermitian-Yang-Mills,Special Lagrangian,Calabi-Yau,paper

annotationComment: 62 pages. Revised version, accepted in Bull. AMS
groupsFutaki Character
issn0273-0979, 1088-9485
journaltitleBulletin of the American Mathematical Society
titleGeometric hydrodynamics and infinite-dimensional Newton's equations

Somerville, MA: International Press

annotationComment: Survey article
booktitleDifferential Geometry, Calabi-Yau Theory, and General Relativity. Part 2. Lectures and Articles Celebrating the 70th Birthday of Shing-Tung Yau, Harvard University, Cambridge, Ma, USA, May 2019
groupsFutaki Character
notearXiv:2004.04831 [math] type: article
titleStability and the deformed Hermitian-Yang-Mills equation

journaltitleInvent. Math.
titleG-uniform stability and Kähler-Einstein metrics on Fano varieties

journaltitleAnn. of Math.
titleFinite generation for valuations computing stability thresholds and applications to K-stability

journaltitleForum Math. Sigma
titleStability conditions for polarised varieties
pagesPaper No. e10457pp

groupsInfinite-dimensional symplectic geometry
titleThe universal structure of moment maps in complex geometry

titleFutaki Invariants and Yau's Conjecture on the Hull-Strominger system

groupsMy publications
titleNorm-squared of the momentum map in infinite dimensions with applications to Kähler geometry and symplectic connections

noteTo appear in J. Differential Geom.
titleGauge theory for string algebroids

groupsFutaki Character
titleGeometry of infinite dimensional Cartan Developments

pubstateTo appear
titleInfinite-dimensional Lie Groups. General Theory and Main Examples