---
title: Positive Scalar Curvature Kähler Surfaces
url: https://www.emergentmind.com/topics/positive-scalar-curvature-kahler-surfaces
type: topic
---

# Positive Scalar Curvature Kähler Surfaces

A positive scalar curvature Kähler surface is a compact complex surface equipped with an integrable complex structure and a Kähler metric whose scalar curvature is strictly positive at each point. The study of such surfaces connects complex differential geometry, algebraic surface classification, GIT stability (K-polystability), and comparison/systolic geometry. This article presents the technical landscape governing positive scalar curvature Kähler surfaces, including their classification, moduli, rigidity, stability-theoretic obstructions and key geometrical inequalities.

## 1. Classification and Existence Criteria

The classification of compact Kähler surfaces with positive scalar curvature (PSC) is exhaustively understood as a consequence of the Enriques–Kodaira classification, algebraic criteria, and recent analytic advances. A compact complex surface $X$ admits a Kähler metric $\omega$ with $S(\omega)>0$ if and only if its minimal model is either:

- The projective plane $\mathbb{P}^2$,
- A Hirzebruch surface $F_e$ (rational ruled, $\mathbb{P}^1$-bundle over $\mathbb{P}^1$),
- A non-rational ruled surface $\mathbb{P}^1 \to X_0 \to B$ over a base curve $B$ of genus $g(B)\ge1$.

Crucially, blowing up any of these at finitely many points preserves the existence of a PSC Kähler metric, as established in [2405.12189]. This resolves a conjecture of LeBrun and extends Hitchin’s result for $n\ge3$ to complex surfaces ($n=2$), completing the classification in terms of Kodaira dimension: a compact Kähler surface admits $S(\omega)>0$ if and only if $\kappa(X)=-\infty$.

## 2. Constant Scalar Curvature Kähler Metrics and K-polystability

The existence of constant scalar curvature Kähler (cscK) metrics on a polarized projective surface $(X,L)$ is tightly linked to the notion of K-polystability, following the Yau–Tian–Donaldson framework. For rational surfaces, only two admit a cscK metric in every Kähler class: $\mathbb{P}^2$ and $\mathbb{P}^1\times\mathbb{P}^1$. These are termed projective rational strong Calabi dream surfaces [1712.04857]. The proof relies on:

- Reduction to minimal models and application of destabilizing slope (Ross–Thomas) test configurations,
- Computation of the Donaldson–Futaki invariant 
  $$
  DF_S(\lambda) = \frac{2}{3}\nu(L)[-3\lambda^2(L\cdot Z)+\lambda^3(Z^2)] + \lambda^2(2-2g(Z))+2\lambda(L\cdot Z),
  $$
  where $\nu(L) = -K_S \cdot L / L^2$,
- Showing that for $F_n$ with $n\ge1$ (Hirzebruch surfaces other than $\mathbb{P}^1\times\mathbb{P}^1$) or blow-ups, there exist polarizations for which $DF<0$, obstructing cscK metrics in any class.

Thus, outside of $\mathbb{P}^2$ and $\mathbb{P}^1\times\mathbb{P}^1$, rational surfaces always possess destabilizing test configurations for some polarization. The automorphism group obstruction (Matsushima–Lichnerowicz) further precludes cscK existence when $\mathrm{Aut}^0(X)$ is non-reductive, as for $F_n$, $n>0$ [1712.04857].

## 3. Systolic Inequalities and Rigidity Phenomena

Positive scalar curvature on Kähler surfaces imposes sharp geometric inequalities, particularly for the area of minimal 2-cycles—captured by the systolic invariant. For any closed PSC Kähler surface $(X,\omega)$,
$$
\min_X S(\omega)\cdot \mathrm{sys}_2(\omega)\le 12\pi,
$$
with equality if and only if $X \cong \mathbb{P}^2$ with the Fubini–Study metric [2601.02901]. This estimate is attained and rigid only in the cscK case on $\mathbb{P}^2$; for Hirzebruch surfaces, the optimal constant is $8\pi$, and for non-rational ruled surfaces, $4\pi$. These bounds are stable under blow-up.

The proof applies a calibration by the Kähler form, optimization over the Kähler cone, and, in the non-rational ruled case, an adaptation of Stern’s level-set method to the holomorphic fibration context [2601.02901, 2510.13353]. The holomorphic 2-systole of a Kähler class $[\omega]$ is defined as
$$
\mathrm{sys}_2([\omega]) = \inf\{[\omega]\cdot[C] \mid C\subset X\ \text{effective},\ [C]\neq 0\},
$$
and the scale-invariant functional
$$
\mathcal{J}_X([\omega]) = \mathrm{sys}_2([\omega])\cdot\hat S([\omega]),\quad \hat S([\omega])=\frac{4\pi c_1(X)\cdot [\omega]}{[\omega]^2}.
$$

The full table of optimal constants is:

| Minimal Model | PSC Systolic Bound | Rigidity Case                   |
|:-------------:|:------------------:|:-------------------------------:|
| $\mathbb{P}^2$ | $12\pi$           | Fubini–Study metric             |
| $F_e$         | $8\pi$             | Product structure, $e=0$        |
| Non-rational  | $4\pi$             | Product of $S^2$ and flat torus |

## 4. Analytic Techniques and Compactness Results

Sequences of PSC Kähler surfaces exhibit strong compactness properties under global geometric bounds. If a family of Kähler surfaces $(M_i,J_i,g_i)$ has:
- $\mathrm{Vol}(M_i,g_i)\ge V_0$,
- $\mathrm{diam}(M_i,g_i)\le D$,
- $|\mathrm{Ric}_{g_i}|\le \Lambda$,
- Signature $\tau(M_i)\ge\tau_0$,

then after extraction, the sequence converges in the Gromov–Hausdorff sense to an orbifold Kähler surface of constant scalar curvature, singular only at isolated conical points [1304.0853]. The convergence improves to smooth Cheeger–Gromov convergence if the $L^\infty$ norm of the anti-self-dual Weyl tensor is uniformly bounded.

The Einstein–Maxwell interpretation shows that PSC ($R-|F|^2>0$) provides lower Ricci bounds, and by Myers' theorem the universal cover is compact. Rigidity results include: if the $L^q$ norm of the curvature operator is sufficiently small, the metric is isometric to a round sphere; with nonnegative isotropic curvature, the only example is the standard Fubini–Study metric on $\mathbb{C}P^2$.

## 5. Stability and Kähler–Einstein/Extremal Metrics on Del Pezzo Surfaces

Del Pezzo surfaces—degree $d=K_S^2>0$, $-K_S$ ample—occupy a central position. The existence of Kähler–Einstein or cscK metrics on polarized del Pezzo surfaces is governed by K-polystability, specifically via the Donaldson–Futaki invariant and the alpha-invariant.

On low-degree del Pezzo surfaces ($d\le2$), a sufficient condition for the existence of cscK metrics in a class $c_1(L)$ is the nefness of $-K_S - \frac{2}{3}\mu(L) L$, with $\mu(L) = -K_S\cdot L/L^2$ [1606.04370]. For anticanonical polarization, every del Pezzo surface of degree $1$ or $2$ admits a positive cscK metric.

For higher-degree (degree $\ge4$), no non-anticanonical polarization satisfies the necessary alpha-invariant criterion, and extremal metrics become rare, with only the quadric surface (degree $8$, $d=8$) realizing equality.

The functional-theoretic landscape is enriched by Weyl curvature minimization: for a smooth compact 4-manifold $M$ supporting a positive scalar curvature Kähler–Einstein metric $g$, the conformal class $[g]$ minimizes $\int_M |W_g|^2\,d\mu_g$ among all conformal classes with positive Yamabe constant, with equality only for the Kähler–Einstein case [1310.0848].

## 6. Methodologies: Blowing Up, Gluing, and Operator Analysis

The analytic preservation of PSC under blow-up involves gluing methods. Starting from $(M,\omega)$ with $S(\omega)>0$, for the blow-up $\mathrm{Bl}_p M$ one constructs a metric in $[\pi^*\omega - \varepsilon^2 [E]]$ via cutoff and correction with a scalar-flat Burns–Simanca metric near the exceptional divisor [2405.12189]. Weighted Hölder spaces and bi-Laplacian invertibility with appropriate kernel cancellation allow the scalar curvature perturbation to be controlled explicitly, ensuring $S(\omega_\varepsilon)>0$ for small $\varepsilon$.

The calculation of stability invariants relies on intersection-theoretic expansions (Hilbert polynomial, slope $\mu(X,L)$, and the explicit formula for the Donaldson–Futaki invariant in terms of test configurations), and reduction to explicit algebraic data (sections, fibers, exceptional divisors) for concrete models.

## 7. Fundamental Geometric Formulas

Several structure formulas universal for positive scalar curvature Kähler surfaces:

- Hilbert polynomial (surface case):
  $$
  \mathrm{Hilb}(kL) = a_0 k^2 + a_1 k + O(1), \quad a_0 = \frac{1}{2}L^2, \quad a_1 = -\frac{1}{2}K_X\cdot L
  $$
- Slope:
  $$
  \mu(X,L) = -K_X \cdot L / L^2
  $$
- Donaldson–Futaki invariant (test configuration):
  $$
  DF(\mathcal{X},\mathcal{L}) = (2/3)\nu(L)[-3\lambda^2(L\cdot Z)+\lambda^3(Z^2)] + \lambda^2(2-2g(Z)) + 2\lambda(L\cdot Z)
  $$
  for surface slope configurations.
- Holomorphic 2-systole:
  $$
  \mathrm{sys}_2([\omega]) = \inf\{[\omega]\cdot[C] \mid C\subset X\ \text{effective},\ [C]\neq 0\}
  $$
- Average scalar curvature:
  $$
  \hat S([\omega]) = \frac{4\pi c_1(X)\cdot[\omega]}{[\omega]^2}
  $$
- Scale-invariant systolic functional:
  $$
  \mathcal{J}_X([\omega]) = \mathrm{sys}_2([\omega])\cdot \hat S([\omega])
  $$

These formulas are central in establishing classification, existence, and rigidity results across all positive scalar curvature Kähler surfaces.

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In sum, positive scalar curvature Kähler surfaces are fully classified up to blow-up, with their local and global geometry tightly controlled by the interplay of K-stability, algebraic data, and analytic gluing techniques. The sharp geometric inequalities, rigidity phenomena, and explicit invariants define a rigid landscape with precise moduli, distinguished by the projective plane and the quadric as unique models for maximal systolic ratio and universal stability [1712.04857, 2405.12189, 2601.02901, 1304.0853, 1310.0848, 1606.04370, 2510.13353].

Source: https://www.emergentmind.com/topics/positive-scalar-curvature-kahler-surfaces