Positive Scalar Curvature Kähler Surfaces

• Positive Scalar Curvature Kähler Surfaces are compact complex surfaces with integrable structures and Kähler metrics that exhibit strictly positive scalar curvature at every point.
• They are classified via the Enriques–Kodaira framework, including key models like the projective plane and Hirzebruch surfaces, with PSC preserved under finite blow-ups.
• Analytic and algebraic techniques such as gluing constructions and Donaldson–Futaki invariant computations provide insight into stability, rigidity phenomena, and sharp systolic inequalities.

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 (Brown, 2024). 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 (Martinez-Garcia, 2017). 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 $F_n$, $n>0$ (Martinez-Garcia, 2017).

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 (Sha, 6 Jan 2026). 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 (Sha, 6 Jan 2026, Sha, 15 Oct 2025). 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 (Shao, 2013). 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 $\mu(L) = -K_S\cdot L/L^2$ (Cheltsov et al., 2016). 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 (LeBrun, 2013).

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 (Brown, 2024). 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 (Martinez-Garcia, 2017, Brown, 2024, Sha, 6 Jan 2026, Shao, 2013, LeBrun, 2013, Cheltsov et al., 2016, Sha, 15 Oct 2025).