Constant Scalar Curvature Kähler Metric

• Constant Scalar Curvature Kähler (cscK) metrics are Kähler metrics with a constant scalar curvature on a compact or noncompact complex manifold, crucial in symplectic moment-map theory and algebraic geometry.
• cscK metrics can be written as given by a fixed Kähler class $\omega_{\varphi} = \omega_{0} + \sqrt{-1}\, \partial \bar\[\partial \varphi > 0.$
• The second-order Laplace equation and complex Monge–Ampère equation are essential in proving the regularity of cscK metrics.

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: $g$0 The Ricci form is

$g$1

and the scalar curvature is its contraction with the Kähler form: $g$2 up to convention-dependent factors. Equivalently,

$g$3

For a compact Kähler manifold, the average scalar curvature is topological: $g$4 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 $g$5 and $g$6.

The equation

$g$7

is fourth order because $g$8 itself contains second derivatives of $g$9, while scalar curvature differentiates the logarithm of its determinant twice more. A useful equivalent formulation introduces the logarithmic volume ratio

$g(JX,JY)=g(X,Y),$0

Since

$g(JX,JY)=g(X,Y),$1

the scalar curvature equation becomes

$g(JX,JY)=g(X,Y),$2

This reduction to a second-order Laplace equation coupled to a complex Monge–Ampère equation is important in regularity theory (He et al., 2017).

A cscK metric is an extremal Kähler metric. Extremal metrics are critical points of Calabi’s functional

$g(JX,JY)=g(X,Y),$3

and are characterized by the holomorphicity of the $g(JX,JY)=g(X,Y),$4-part of $g(JX,JY)=g(X,Y),$5. The cscK condition is the special case $g(JX,JY)=g(X,Y),$6; the converse does not hold in general.

The linearization of scalar curvature at a cscK metric is the Lichnerowicz operator: $g(JX,JY)=g(X,Y),$7 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 $g(JX,JY)=g(X,Y),$8 is defined, up to an additive constant, by

$g(JX,JY)=g(X,Y),$9

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 $(1,1)$0-geometry. The $(1,1)$1 Finsler norm is

$(1,1)$2

and the completion of the smooth space is the finite-energy space $(1,1)$3. The Mabuchi functional extends as a lower semicontinuous convex functional on $(1,1)$4.

Under the assumption that the identity component of the holomorphic automorphism group is trivial,

$(1,1)$5

properness of the Mabuchi energy with respect to $(1,1)$6 is equivalent to existence of a cscK metric: $(1,1)$7 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 (Chen et al., 2018).

Weak regularity is governed by the Kähler structure. If $(1,1)$8 is an $(1,1)$9 Kähler metric satisfying

$\omega(X,Y)=g(JX,Y).$0

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 $\omega(X,Y)=g(JX,Y).$1 regularity, obtaining Hölder regularity for $\omega(X,Y)=g(JX,Y).$2, applying complex Monge–Ampère regularity to $\omega(X,Y)=g(JX,Y).$3, and bootstrapping by Schauder estimates. The Kähler identity identifying divergence and nondivergence formulations remains valid with merely $\omega(X,Y)=g(JX,Y).$4 coefficients, a feature absent for general uniformly elliptic equations with measurable coefficients (He et al., 2017).

The same method proves that uniformly positive $\omega(X,Y)=g(JX,Y).$5 K-energy minimizers are smooth cscK metrics. It does not resolve the fully degenerate $\omega(X,Y)=g(JX,Y).$6 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 $\omega(X,Y)=g(JX,Y).$7, a test configuration is an equivariant degeneration

$\omega(X,Y)=g(JX,Y).$8

whose nonzero fibers are isomorphic to $\omega(X,Y)=g(JX,Y).$9. If

$d\omega=0$0

is the Hilbert polynomial and

$d\omega=0$1

is the weight polynomial, the Donaldson–Futaki invariant is

$d\omega=0$2

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

$d\omega=0$3

A stronger formulation identifies uniform K-stability with coercivity of the Mabuchi functional. The minimum norm of a test configuration is

$d\omega=0$4

and uniform K-stability means

$d\omega=0$5

The minimum norm vanishes precisely for almost trivial test configurations, equivalently when the $d\omega=0$6-norm vanishes (Dervan, 2014).

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

$d\omega=0$7

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

$d\omega=0$8

where $d\omega=0$9 is a fixed positive closed $$\omega_\varphi=\omega_0+\sqrt{-1}\,\partial\bar\partial\varphi>0,$$0-form. At $$\omega_\varphi=\omega_0+\sqrt{-1}\,\partial\bar\partial\varphi>0,$$1 this is the $$\omega_\varphi=\omega_0+\sqrt{-1}\,\partial\bar\partial\varphi>0,$$2-equation, while at $$\omega_\varphi=\omega_0+\sqrt{-1}\,\partial\bar\partial\varphi>0,$$3 it is the cscK equation. Positivity of $$\omega_\varphi=\omega_0+\sqrt{-1}\,\partial\bar\partial\varphi>0,$$4 gives strict ellipticity for $$\omega_\varphi=\omega_0+\sqrt{-1}\,\partial\bar\partial\varphi>0,$$5, and openness follows from an implicit-function argument. Global closedness up to $$\omega_\varphi=\omega_0+\sqrt{-1}\,\partial\bar\partial\varphi>0,$$6 requires uniform a priori estimates and is not supplied by the continuity path alone (Chen, 2015).

Microscopic stability thresholds provide another sufficient criterion. If $$\omega_\varphi=\omega_0+\sqrt{-1}\,\partial\bar\partial\varphi>0,$$7 denotes the limiting integrability threshold of negative powers of Slater determinants, then conditions of the form

$$\omega_\varphi=\omega_0+\sqrt{-1}\,\partial\bar\partial\varphi>0,$$8

together with ampleness of $$\omega_\varphi=\omega_0+\sqrt{-1}\,\partial\bar\partial\varphi>0,$$9 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

$S(\omega_\varphi)=\overline S,$0

while equality remains open in general (Aoi, 2024).

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 $S(\omega_\varphi)=\overline S,$1 and finite automorphism groups. Property $S(\omega_\varphi)=\overline S,$2 requires uniform diameter and Ricci bounds for the cscK metrics in the family: $S(\omega_\varphi)=\overline S,$3 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 (Donaldson, 2015).

The rational-surface case exhibits sharp dependence on polarization. The only projective rational surfaces admitting cscK metrics in every Kähler class are

$S(\omega_\varphi)=\overline S,$4

For Hirzebruch surfaces $S(\omega_\varphi)=\overline S,$5 with $S(\omega_\varphi)=\overline S,$6, 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 $S(\omega_\varphi)=\overline S,$7 are totally unstable (Martinez-Garcia, 2017).

Projectivized vector bundles provide a contrasting phenomenon. Let

$S(\omega_\varphi)=\overline S,$8

and consider adiabatic classes

$S(\omega_\varphi)=\overline S,$9

For a sufficiently smooth, simple, slope-semistable bundle $\overline S$0 over a cscK base with discrete automorphism group, cscK existence on $\overline S$1 for sufficiently large $\overline S$2 is equivalent to adiabatic slope stability. The relevant degenerations arise from subsheaves

$\overline S$3

and degenerate $\overline S$4 to $\overline S$5. In this setting, adiabatic slope stability, cscK existence, and ordinary K-stability coincide for sufficiently large $$k (<a href="/papers/2406.08284" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Ortu et al., 2024</a>).</p> <p>Strict slope semistability of$$\overline S$6 does not prevent cscK existence on $\overline S$7. 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 $\overline S$8 is a smooth divisor and $\overline S$9 is equipped with a complete metric, the behavior near $\varphi$0 becomes part of the geometric data.

For a cscK cone metric of angle $\varphi$1, the local model near

$\varphi$2

is

$\varphi$3

The logarithmic first Chern class is

$\varphi$4

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 $\varphi$5: $\varphi$6 The converse logarithmic Yau–Tian–Donaldson statement remains open in general (Aoi et al., 2021).

For generic smooth divisors in sufficiently high multiples of an ample line bundle, cscK cone metrics exist for every fixed angle $\varphi$7 with $\varphi$8. 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

$\varphi$9

The divisor lies at infinite distance, while the cusp end has finite volume. A Poincaré cscK metric has an asymptotic expansion

$g$00

where $g$01 is a cscK metric on $g$02. If $g$03 has no nontrivial holomorphic vector fields, Poincaré-type cscK metrics in a fixed class are unique up to holomorphic transformations preserving $g$04. In the general case, uniqueness is conjecturally controlled by the asymptotic cscK metric on $g$05 (Xu et al., 2024).

There are also complete scalar-flat Kähler constructions on affine complements. If $g$06 is smooth and admits a positive cscK metric, a complete scalar-flat Kähler metric can be constructed on $g$07 under dimension, positivity, very-ampleness, and weighted invertibility conditions. The cscK geometry of $g$08 improves the scalar-curvature decay of the natural asymptotically conical background, enabling a weighted fixed-point argument (Aoi, 2019).

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

$g$09

and the local model is supplied by a scalar-flat Hwang–Singer metric with both asymptotically Euclidean and Poincaré-cusp behavior (Feng, 2024). 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 (Najafpour, 2024).

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

$g$10

where $g$11 is the curvature form of the relative anticanonical bundle. On deformation spaces, this form is the $g$12-pairing of harmonic Kodaira–Spencer classes. In the projective case, it represents the first Chern class of a CM $g$13-line bundle (Dervan et al., 2018).

The $g$14-scalar-curvature formalism extends cscK geometry by introducing a holomorphic vector field $g$15 with Hamiltonian potential $g$16. The $g$17-scalar curvature is

$g$18

and a constant $g$19-scalar curvature metric satisfies

$g$20

The specialization $g$21 recovers the ordinary cscK equation. Kähler–Ricci solitons occur for nonzero $g$22 and $g$23, while the limit $g$24 formally approaches the extremal Kähler equation. A corresponding $g$25-K-energy is convex along invariant geodesics, and a provisional $g$26-stability theory is formulated through non-Archimedean energy slopes (Inoue, 2019).

Coupled continuity equations provide another second-order route toward cscK metrics. For a Kähler form $g$27 and closed $g$28-form $g$29,

$g$30

$g$31

Stationary pairs are exactly

$g$32

Under a uniform metric-equivalence estimate

$g$33

higher-order estimates and smooth convergence follow. When $g$34 and $g$35, the system reduces to a continuity equation for the negative Kähler–Einstein metric (Shen et al., 12 Jan 2026).

For elliptic surfaces, semi-flat cscK currents combine flat metrics on elliptic fibers with hyperbolic geometry on the base. The universal model

$g$36

has fiber volume $g$37 and scalar curvature $g$38. 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 (Wang et al., 10 Sep 2025).

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.