---
title: Constant Scalar Curvature Kähler Metric
url: https://www.emergentmind.com/topics/constant-scalar-curvature-kahler-metric
type: topic
---

# Constant Scalar Curvature Kähler Metric

A constant scalar curvature Kähler (cscK) metric is a Kähler metric whose scalar curvature is constant on a compact or noncompact complex manifold. If $(M^{2n},J)$ is a complex manifold, a Hermitian metric $g$ satisfies
\[
g(JX,JY)=g(X,Y),
\]
and its associated real $(1,1)$-form is
\[
\omega(X,Y)=g(JX,Y).
\]
The metric is Kähler when $d\omega=0$. In a fixed Kähler class, every Kähler form can be written
\[
\omega_\varphi=\omega_0+\sqrt{-1}\,\partial\bar\partial\varphi>0,
\]
and the cscK equation is
\[
S(\omega_\varphi)=\overline S,
\]
where $\overline S$ is determined by the cohomology class. The cscK condition is a fourth-order nonlinear elliptic equation in the potential $\varphi$, and occupies a central position at the intersection of complex differential geometry, symplectic moment-map theory, nonlinear PDE, algebraic geometry, and moduli theory.

## 1. Differential-geometric formulation

The Kähler form locally admits a real-valued potential:
\[
\omega=\sqrt{-1}\,\partial\bar\partial\phi.
\]
The Ricci form is
\[
\rho=-\sqrt{-1}\,\partial\bar\partial\log\det(g_{j\bar k}),
\]
and the scalar curvature is its contraction with the Kähler form:
\[
S(\omega)=\Lambda_\omega\rho
\]
up to convention-dependent factors. Equivalently,
\[
S(\omega)
=-g^{i\bar j}\partial_i\partial_{\bar j}
\log\det(g_{k\bar\ell}).
\]

For a compact Kähler manifold, the average scalar curvature is topological:
\[
\overline S
=
n\frac{\displaystyle\int_M c_1(M)\wedge[\omega]^{n-1}}
{\displaystyle\int_M[\omega]^n},
\]
with normalization depending on the convention for the Ricci form. Thus a cscK metric in a fixed Kähler class necessarily has the unique constant prescribed by $c_1(M)$ and $[\omega]$.

The equation
\[
S(\omega_\varphi)=\overline S
\]
is fourth order because $\omega_\varphi$ itself contains second derivatives of $\varphi$, while scalar curvature differentiates the logarithm of its determinant twice more. A useful equivalent formulation introduces the logarithmic volume ratio
\[
V_\varphi=\log\frac{\omega_\varphi^n}{\omega^n}.
\]
Since
\[
\operatorname{Ric}(\omega_\varphi)
=
\operatorname{Ric}(\omega)
-\sqrt{-1}\,\partial\bar\partial V_\varphi,
\]
the scalar curvature equation becomes
\[
\Delta_\varphi V_\varphi
=
g_\varphi^{i\bar j}R_{i\bar j}-\overline S.
\]
This reduction to a second-order Laplace equation coupled to a complex Monge–Ampère equation is important in regularity theory [1705.01236].

A cscK metric is an extremal Kähler metric. Extremal metrics are critical points of Calabi’s functional
\[
g\longmapsto\int_M S_g^2\,d\mu_g
\]
and are characterized by the holomorphicity of the $(1,0)$-part of $\operatorname{grad}S_g$. The cscK condition is the special case $\operatorname{grad}S_g=0$; the converse does not hold in general.

The linearization of scalar curvature at a cscK metric is the Lichnerowicz operator:
\[
L_\omega u
=
-\mathcal D_\omega^*\mathcal D_\omega u
=
-\frac12\Delta_\omega^2u
-\left\langle
\operatorname{Ric}(\omega),\sqrt{-1}\partial\bar\partial u
\right\rangle_\omega,
\]
with normalization-dependent factors. Its kernel is generated by real holomorphy potentials, so holomorphic automorphisms are the fundamental source of nonuniqueness and analytic degeneracy.

## 2. Variational structure and regularity

The Mabuchi K-energy $\mathcal K$ is defined, up to an additive constant, by
\[
\frac{d}{dt}\mathcal K(\varphi_t)
=
-\int_M\dot\varphi_t
\bigl(S(\omega_{\varphi_t})-\overline S\bigr)
\frac{\omega_{\varphi_t}^n}{n!}.
\]
Its critical points are precisely cscK metrics. The Mabuchi functional is convex along geodesics in the space of Kähler potentials, and a cscK metric is a minimizer.

For finite-energy potentials, the relevant metric geometry is often the $d_1$-geometry. The $L^1$ Finsler norm is
\[
\|\dot\varphi\|_{\varphi,1}
=
\frac1V\int_M|\dot\varphi|\,\omega_\varphi^n,
\]
and the completion of the smooth space is the finite-energy space $\mathcal E^1$. The Mabuchi functional extends as a lower semicontinuous convex functional on $\mathcal E^1$.

Under the assumption that the identity component of the holomorphic automorphism group is trivial,
\[
\operatorname{Aut}_0(M,J)=0,
\]
properness of the Mabuchi energy with respect to $d_1$ is equivalent to existence of a cscK metric:
\[
\mathcal K(\varphi)\geq A\,d_1(0,\varphi)-B,
\qquad A>0.
\]
In this setting, failure of cscK existence is equivalent to the existence of a locally finite-energy geodesic ray along which the Mabuchi energy is non-increasing. Positive asymptotic slopes along all such rays define geodesic stability [1801.00656].

Weak regularity is governed by the Kähler structure. If $\omega_\varphi$ is an $L^\infty$ Kähler metric satisfying
\[
\varepsilon\omega\leq\omega_\varphi\leq\Lambda\omega
\]
and is a distributional weak solution of the cscK equation, then it is smooth. The argument proceeds by solving the associated Laplace equation in divergence form, establishing $W^{2,2}$ regularity, obtaining Hölder regularity for $V_\varphi$, applying complex Monge–Ampère regularity to $\varphi$, and bootstrapping by Schauder estimates. The Kähler identity identifying divergence and nondivergence formulations remains valid with merely $L^\infty$ coefficients, a feature absent for general uniformly elliptic equations with measurable coefficients [1705.01236].

The same method proves that uniformly positive $L^\infty$ K-energy minimizers are smooth cscK metrics. It does not resolve the fully degenerate $C^{1,\bar 1}$ case in which the associated Kähler current may fail to be strictly positive.

## 3. Stability and the Yau–Tian–Donaldson framework

For a polarized projective manifold $(X,L)$, a test configuration is an equivariant degeneration
\[
\pi:\mathcal X\to\mathbb A^1
\]
whose nonzero fibers are isomorphic to $(X,L^r)$. If
\[
h(k)=a_0k^n+a_1k^{n-1}+O(k^{n-2})
\]
is the Hilbert polynomial and
\[
w(k)=b_0k^{n+1}+b_1k^n+O(k^{n-1})
\]
is the weight polynomial, the Donaldson–Futaki invariant is
\[
\operatorname{DF}(\mathcal X,\mathcal L)
=
\frac{b_0a_1-b_1a_0}{a_0}
\]
up to normalization conventions.

K-semistability requires nonnegativity of this invariant for every test configuration; K-polystability requires equality only for product configurations; K-stability requires strict positivity for nontrivial configurations. A negative Donaldson–Futaki invariant is a destabilizing test configuration and obstructs cscK existence.

The Yau–Tian–Donaldson philosophy predicts
\[
(X,L)\text{ admits a cscK metric}
\quad\Longleftrightarrow\quad
(X,L)\text{ is K-polystable}.
\]
A stronger formulation identifies uniform K-stability with coercivity of the Mabuchi functional. The minimum norm of a test configuration is
\[
\|\mathcal X\|_m
=
\sum_j\bigl(b_{0,j}-\lambda_j a_{0,j}\bigr),
\]
and uniform K-stability means
\[
\operatorname{DF}(\mathcal X,\mathcal L)
\geq
\epsilon\|\mathcal X\|_m.
\]
The minimum norm vanishes precisely for almost trivial test configurations, equivalently when the $L^2$-norm vanishes [1412.0648].

Several implications are established in important settings. Existence of a twisted cscK metric implies twisted K-semistability for semipositive twisting and uniform twisted K-stability for ample twisting. The twisted equation is
\[
S(\omega)-\Lambda_\omega\alpha=C_\alpha,
\]
and its algebraic counterpart adds a twisting contribution to the Donaldson–Futaki invariant. The full equivalence between twisted cscK existence and uniform twisted K-stability remains conjectural in general.

A direct continuity-path approach considers
\[
t\bigl(S(\omega_{\varphi_t})-\overline S\bigr)
=
(1-t)\bigl(\operatorname{tr}_{\omega_{\varphi_t}}\chi-\overline\chi\bigr),
\qquad t\in[0,1],
\]
where $\chi$ is a fixed positive closed $(1,1)$-form. At $t=0$ this is the $J_\chi$-equation, while at $t=1$ it is the cscK equation. Positivity of $\chi$ gives strict ellipticity for $t<1$, and openness follows from an implicit-function argument. Global closedness up to $t=1$ requires uniform a priori estimates and is not supplied by the continuity path alone [1506.06423].

Microscopic stability thresholds provide another sufficient criterion. If $\gamma(L)$ denotes the limiting integrability threshold of negative powers of Slater determinants, then conditions of the form
\[
\gamma(L)>n\mu(L)-(n-1)s(L)
\]
together with ampleness of $K_X+\gamma(L)L$ imply coercivity of the Mabuchi energy and hence cscK existence. This probabilistic criterion is related to, but not identified with, the delta-invariant; one has
\[
\delta(L)\geq\gamma(L),
\]
while equality remains open in general [2410.22090].

## 4. Existence, deformation, and examples

Existence depends strongly on the Kähler class and on the automorphism group. In an algebraic family of polarized manifolds, the cscK locus is Zariski-open under Property $R$ and finite automorphism groups. Property $R$ requires uniform diameter and Ricci bounds for the cscK metrics in the family:
\[
\operatorname{diam}(X_s,\omega_s)\leq C,
\qquad
\|\operatorname{Ric}(\omega_s)\|_{L^\infty}\leq C.
\]
For Fano manifolds with normalized Kähler–Einstein metrics, the Ricci bound is automatic and the diameter bound follows from Myers’ theorem, yielding unconditional openness of the Kähler–Einstein locus [1503.05174].

The rational-surface case exhibits sharp dependence on polarization. The only projective rational surfaces admitting cscK metrics in every Kähler class are
\[
\mathbb P^2
\qquad\text{and}\qquad
\mathbb P^1\times\mathbb P^1.
\]
For Hirzebruch surfaces $\mathbb F_n$ with $n\geq1$, the connected automorphism group is non-reductive, so the Matsushima–Lichnerowicz obstruction excludes cscK metrics in every Kähler class. Ross–Thomas slope test configurations provide explicit destabilizations. Many blow-ups and all rational surfaces of Picard rank $3$ are totally unstable [1712.04857].

Projectivized vector bundles provide a contrasting phenomenon. Let
\[
\pi:\mathbb P(E)\to B
\]
and consider adiabatic classes
\[
\Omega_k=c_1(\mathcal O_{\mathbb P(E)}(1))+k\pi^*c_1(L).
\]
For a sufficiently smooth, simple, slope-semistable bundle $E$ over a cscK base with discrete automorphism group, cscK existence on $\mathbb P(E)$ for sufficiently large $k$ is equivalent to adiabatic slope stability. The relevant degenerations arise from subsheaves
\[
S\subset E
\]
and degenerate $E$ to $S\oplus E/S$. In this setting, adiabatic slope stability, cscK existence, and ordinary K-stability coincide for sufficiently large $k [2406.08284].

Strict slope semistability of $E$ does not prevent cscK existence on $\mathbb P(E)$. A nontrivial simple extension can fail to admit a Hermite–Einstein metric while its projectivization admits cscK metrics in adiabatic classes. This separates cscK geometry of a projectivized bundle from Hermite–Einstein geometry of the original bundle.

## 5. Noncompact, conical, and Poincaré-type cscK metrics

The cscK problem also has noncompact variants. If $D\subset X$ is a smooth divisor and $X\setminus D$ is equipped with a complete metric, the behavior near $D$ becomes part of the geometric data.

For a cscK cone metric of angle $2\pi\beta$, the local model near
\[
D=\{z_1=0\}
\]
is
\[
\omega_{\beta,\mathrm{model}}
=
\sqrt{-1}\left(
\beta^2|z_1|^{2\beta-2}\,dz_1\wedge d\bar z_1
+
\sum_{j=2}^n dz_j\wedge d\bar z_j
\right).
\]
The logarithmic first Chern class is
\[
c_1(X,D)=c_1(X)-(1-\beta)c_1(\mathcal O(D)),
\]
and the scalar-curvature constant is determined by this class. Existence of a cscK cone metric implies log K-polystability and uniform log K-stability relative to the automorphism group preserving $D$:
\[
\operatorname{DF}(\mathcal X,\mathcal D,\mathcal L,\beta)
\geq
\epsilon J^{\mathrm{NA}}_G(\mathcal X,\mathcal L).
\]
The converse logarithmic Yau–Tian–Donaldson statement remains open in general [2110.02518].

For generic smooth divisors in sufficiently high multiples of an ample line bundle, cscK cone metrics exist for every fixed angle $2\pi\beta$ with $0<\beta<1$. Conversely, deformation to the normal cone gives explicit instability criteria when the scalar curvature of the divisor is sufficiently small relative to the angle.

Poincaré-type cscK metrics correspond to the limiting cusp geometry. Near a smooth divisor, the normal model is
\[
\frac{\sqrt{-1}\,dz_n\wedge d\bar z_n}
{|z_n|^2\log^2|z_n|^2}
+
\sum_{i=1}^{n-1}\sqrt{-1}\,dz_i\wedge d\bar z_i.
\]
The divisor lies at infinite distance, while the cusp end has finite volume. A Poincaré cscK metric has an asymptotic expansion
\[
\omega
=
p^*\omega_D
+
a\,\frac{\sqrt{-1}\,dz_n\wedge d\bar z_n}
{|z_n|^2\log^2|z_n|^2}
+
O\bigl(|\log|z_n||^{-\eta}\bigr),
\]
where $\omega_D$ is a cscK metric on $D$. If $D$ has no nontrivial holomorphic vector fields, Poincaré-type cscK metrics in a fixed class are unique up to holomorphic transformations preserving $D$. In the general case, uniqueness is conjecturally controlled by the asymptotic cscK metric on $D$ [2410.14130].

There are also complete scalar-flat Kähler constructions on affine complements. If $D\in|L_X|$ is smooth and admits a positive cscK metric, a complete scalar-flat Kähler metric can be constructed on $X\setminus D$ under dimension, positivity, very-ampleness, and weighted invertibility conditions. The cscK geometry of $D$ improves the scalar-curvature decay of the natural asymptotically conical background, enabling a weighted fixed-point argument [1910.12317].

A related gluing construction removes finitely many points from a compact cscK manifold. The resulting manifold admits a complete cscK metric with Poincaré cusp ends. The Kähler class on the corresponding blow-up is
\[
\pi^*[\omega_X]-\epsilon^2\sum_i[E_i],
\]
and the local model is supplied by a scalar-flat Hwang–Singer metric with both asymptotically Euclidean and Poincaré-cusp behavior [2405.11952]. Higher-codimension weighted orbifold resolutions can likewise inherit cscK metrics by gluing scalar-flat ALE models to a compact cscK orbifold, provided the singularities have the prescribed type and there are no holomorphic-vector-field obstructions [2411.02823].

## 6. Moduli, generalized equations, and current directions

The uniqueness of cscK metrics modulo automorphisms permits a moduli theory. The moduli space of smooth polarized manifolds admitting cscK metrics is constructed as a Hausdorff complex space by gluing local analytic GIT quotients. Closed complexified automorphism orbits correspond to cscK fibers, while unique cscK degenerations provide the separatedness mechanism. The moduli space carries a natural Weil–Petersson-type Kähler form
\[
\eta
=
\int_{X/B}
\left(
n\mu\,\omega_X^{n+1}
-
(n+1)\rho\wedge\omega_X^n
\right),
\]
where $\rho$ is the curvature form of the relative anticanonical bundle. On deformation spaces, this form is the $L^2$-pairing of harmonic Kodaira–Spencer classes. In the projective case, it represents the first Chern class of a CM $\mathbb Q$-line bundle [1810.02576].

The $\mu$-scalar-curvature formalism extends cscK geometry by introducing a holomorphic vector field $\xi$ with Hamiltonian potential $\theta_\xi$. The $\mu$-scalar curvature is
\[
s_\xi^\lambda(\omega)
=
s_\xi(\omega)-\lambda\theta_\xi,
\]
and a constant $\mu_\xi^\lambda$-scalar curvature metric satisfies
\[
s_\xi^\lambda(\omega)=\text{constant}.
\]
The specialization $\xi=0$ recovers the ordinary cscK equation. Kähler–Ricci solitons occur for nonzero $\xi$ and $\lambda>0$, while the limit $\lambda\to-\infty$ formally approaches the extremal Kähler equation. A corresponding $\mu$-K-energy is convex along invariant geodesics, and a provisional $\mu K$-stability theory is formulated through non-Archimedean energy slopes [1902.00664].

Coupled continuity equations provide another second-order route toward cscK metrics. For a Kähler form $\omega(t)$ and closed $(1,1)$-form $\alpha(t)$,
\[
\frac{\omega-\hat\omega}{t}
=
-\operatorname{Ric}(\omega)+\lambda\omega+\alpha,
\]
\[
\frac{\alpha-\hat\alpha}{t}
=
\kappa\Delta_\omega\alpha.
\]
Stationary pairs are exactly
\[
\omega\text{ cscK},
\qquad
\alpha\text{ }\omega\text{-harmonic}.
\]
Under a uniform metric-equivalence estimate
\[
C^{-1}\hat\omega\leq\omega(t)\leq C\hat\omega,
\]
higher-order estimates and smooth convergence follow. When $c_1(X)<0$ and $\alpha=0$, the system reduces to a continuity equation for the negative Kähler–Einstein metric [2601.07677].

For elliptic surfaces, semi-flat cscK currents combine flat metrics on elliptic fibers with hyperbolic geometry on the base. The universal model
\[
\eta
=
\sqrt{-1}\partial\bar\partial
\left(
-\log v^2+\frac{y^2}{v}
\right)
\]
has fiber volume $1$ and scalar curvature $-3$. Its modular invariance allows it to descend to elliptic surfaces, where singular fibers produce controlled current singularities. Under suitable monodromy conditions, the semi-flat cscK current is unique among invariant metrics [2509.08669].

The principal unresolved issues concern global existence without auxiliary hypotheses, uniform a priori estimates along continuity paths, the full equivalence between cscK existence and uniform K-stability, treatment of continuous automorphism groups, regularity of degenerate finite-energy minimizers, compactification by singular objects, and uniqueness in noncompact settings with nontrivial asymptotic automorphism data. Together, these problems define the continuing development of cscK geometry as a theory of canonical metrics governed simultaneously by scalar curvature, energy coercivity, degeneration, and moduli.

Source: https://www.emergentmind.com/topics/constant-scalar-curvature-kahler-metric