Positive holomorphic sectional curvature on rational surfaces

Abstract: In 1975, Hitchin proved that any compact complex surface admitting a Kähler metric with positive holomorphic sectional curvature $HSC>0$ is rational. Conversely, he constructed such metrics on all Hirzebruch surfaces $\mathbb{F}_k$, as a first step towards characterizing rational surfaces by the existence of a Kähler metric with suitable curvature positivity. In this paper, we prove that every projective manifold $X$ obtained from a projective toric manifold by a finite sequence of blow-ups at points admits a Kähler metric with $HSC>0$. This statement applies to all rational surfaces and therefore completes Hitchin's result, resolving the complex surface case of a problem of Yau listed in "Open Problems in Geometry". The proof has two main ingredients. First, we prove that the toric Kähler metric on a projective toric manifold arising from Delzant's construction has $HSC>0$. Second, via a one-parameter degeneration, we construct, for any such $X$, a smooth projective family $π:\mathcal X\to\mathbb C$ such that $\mathcal X_t\simeq X$ for $t\ne0$, while $\mathcal X_0$ is a projective toric manifold.

• The paper resolves Yau’s problem by proving that every rational surface, formed via toric blow-ups, admits a Kähler metric with positive holomorphic sectional curvature.
• It employs toric geometry, symplectic reduction, and deformation techniques to perform explicit curvature calculations ensuring strict positivity.
• The work completes Hitchin’s program by fully classifying compact complex surfaces through their curvature properties, establishing a definitive curvature–rationality correspondence.

Positive Holomorphic Sectional Curvature on Rational Surfaces

Background and Motivation

The problem of characterizing rational surfaces via curvature conditions has occupied a central position in complex differential geometry since Hitchin's seminal work (1975), which established that a compact Kähler surface admitting a metric with positive holomorphic sectional curvature (HSC$>0$) must be rational. Hitchin further constructed explicit Kähler metrics with HSC$>0$ for all Hirzebruch surfaces $\mathbb{F}_k$. The outstanding question, posed by Yau and elaborated in his open problems list, concerns the converse: must every rational surface admit a Kähler metric with HSC$>0$, and do blow-ups of projective manifolds with HSC$>0$ preserve this property?

This paper resolves the surface case of Yau's problem by proving that every rational surface—specifically, every surface obtained from a projective toric manifold by a finite sequence of point blow-ups—admits a Kähler metric with HSC$>0$ (2606.23333). This result completes Hitchin's characterization, closes the gap left by previous constructions (notably, those for Hirzebruch surfaces and projectivizations of vector bundles), and fully settles the curvature-rationality correspondence for complex dimension two.

Main Results

Existence of Positive Holomorphic Sectional Curvature Metrics

The principal theorem asserts that any projective manifold obtained from a projective toric manifold by finitely many point blow-ups supports a Kähler metric with HSC$>0$. As every rational surface is obtained by blowing up $\mathbb{P}^2$ or a Hirzebruch surface at points, this result ensures the existence of such metrics on all rational surfaces. The paper further establishes the following strong classification: a compact complex surface is rational if and only if it admits a Kähler metric with HSC$>0$.

A corollary refines the classification for semi-positive holomorphic sectional curvature, showing that a compact Kähler surface with HSC$\geq0$ is either rational, a finite étale quotient of a complex torus, or a projectively unitary flat $>0$0-bundle over an elliptic curve.

Construction and Deformation Techniques

The proof utilizes tools from toric geometry, symplectic reduction, and deformation theory. The first ingredient is a computation showing that the canonical toric Kähler metric derived via Delzant's symplectic reduction has HSC$>0$1. The second is a deformation argument: for any ordered cluster of blow-up points (not necessarily torus-fixed), a generic one-parameter subgroup of the torus is used to construct a smooth projective family whose central fiber is a toric manifold. Kodaira-Spencer local stability then implies the existence of HSC$>0$2 metrics on nearby fibers, i.e., the target blown-up rational surface.

The technical heart of the argument is an explicit curvature calculation, using Gauss equations and O'Neill's formula for Kähler reductions. Compactness of the moment level set is shown to force strict positivity of the holomorphic sectional curvature.

Theoretical Implications

This work precisely closes the curvature–rationality loop for compact complex surfaces. The rational surfaces are now fully characterized in purely differential-geometric terms via HSC$>0$3 metrics. The methods are robust, leveraging both algebraic and symplectic aspects of toric geometry, and are applicable to a broad class of surfaces due to the abundance of smooth projective toric manifolds.

The classification of surfaces with HSC$>0$4—effectively a birational classification—is now complete. All examples constructed via toric blow-ups or projectivization of vector bundles over rational bases are rational; this raises further questions about whether higher-dimensional analogues exhibit similar rigidity.

Future Directions and Open Problems

While the surface case is settled, the classification of threefolds and higher-dimensional varieties with HSC$>0$5 remains unresolved. Rational connectedness, demonstrated for HSC$>0$6 in dimension three and higher, is not equivalent to rationality beyond dimension two. The precise position of HSC$>0$7 among rationality, stable rationality, unirationality, and rational connectedness is an outstanding issue.

Beyond dimension two, blow-ups along smooth curves (rather than points) present technical obstacles, as existing deformation techniques (Kleiman’s iterated blow-up spaces) do not directly generalize. The extension of these methods to higher-dimensional birational operations requires significant further development.

The relationship between HSC$>0$8 and Ric$>0$9 is clarified in dimension two, but in higher dimensions remains poorly understood. A deeper classification of HSC$\mathbb{F}_k$0 manifolds may address open questions related to curvature hierarchies and their algebraic counterparts (Fano, uniruled, rationally connected, etc.).

Conclusion

This paper decisively settles the characterization of rational surfaces by the existence of Kähler metrics with positive holomorphic sectional curvature, completing Hitchin’s program and fully resolving Yau’s surface case problem. The results are both definitive and technically robust, relying on explicit curvature calculations and deformation techniques rooted in toric geometry. Extensions to higher dimensions remain open, with implications for the broader landscape of curvature and rationality in complex geometry.