1 Introduction

Recall that an equivalence class of affine torsion-free connections on the tangent bundle of a smooth manifold \(N\) is called a (real) projective structure [11, 38, 39]. Two connections \(\nabla\) and \(\nabla^{\prime}\) are projectively equivalent if they share the same unparametrised geodesics. This condition is equivalent to \(\nabla\) and \(\nabla^{\prime}\) inducing the same parallel transport on the projectivised tangent bundle \(\mathbb{P}TN\).

It is a natural task to (locally) characterise the projective structures arising via the Levi-Civita connection of a (pseudo-)Riemannian metric. R. Liouville [25] made the crucial observation that the Riemannian metrics on a surface whose Levi-Civita connection belongs to a given projective class precisely correspond to nondegenerate solutions of a certain projectively invariant finite-type linear system of partial differential equations. In [4] Bryant, Eastwood and Dunajski used Liouville’s observation to solve the two-dimensional version of the Riemannian metrisability problem. In another direction it was shown in [29] that on a surface locally every affine torsion-free connection is projectively equivalent to a conformal connection (see also [28]). Local existence of a connection with skew-symmetric Ricci tensor in a given projective class was investigated in [36] (see also [23] for a connection to Veronese webs). Liouville’s result generalises to higher dimensions [30] and the corresponding finite-type differential system was prolonged to closed form in [14, 30]. Several necessary conditions for Riemann metrisability of a projective structure in dimensions larger than two were given in [33]. See also [7, 16] for the role of Einstein metrics in projective geometry.

Now let \(M\) be a complex manifold of complex dimension \(d>1\) with integrable almost complex structure map \(J\). Two affine torsion-free connections \(\nabla\) and \(\nabla^{\prime}\) on \(TM\) which preserve \(J\) are called complex projectively equivalent if they share the same generalised geodesics (for the notion of a curved complex projective structure on Riemann surfaces see [5]). A generalised geodesic is a smoothly immersed curve \(\gamma \subset M\) with the property that the \(2\)-plane spanned by \(\dot{\gamma}\) and \(J \dot\gamma\) is parallel along \(\gamma\). Complex projective geometry was introduced by Otsuki and Tashiro [35, 37]. Background on the history of complex projective geometry and its recently discovered connection to Hamiltonian \(2\)-forms (see [1] and references therein) may be found in [26].

In the complex setting it is natural to study the Kähler metrisability problem, i.e. try to (locally) characterise the complex projective structures which arise via the Levi-Civita connection of a (pseudo-)Kähler metric. Similar to the real case, the Kähler metrics whose Levi-Civita connection belongs to a given complex projective class precisely correspond to nondegenerate solutions of a certain complex projectively invariant finite-type linear system of partial differential equations [12, 26, 31].

In this note we prolong the relevant differential system to closed form in the surface case. In doing so we obtain necessary conditions for Kähler metrisability of a complex projective structure \([\nabla]\) on a complex surface and show in particular that the generic complex projective structure is not Kähler metrisable. Furthermore we show that the space of Kähler metrics compatible with a given complex projective structure is algebraically constrained by the complex projective Weyl curvature of \([\nabla]\). We also show that the (pseudo-)Kähler metrics defined on some domain in \(\mathbb{CP}^2\) which are compatible with the standard complex projective structure are in one-to-one correspondence with the hermitian forms on \(\mathbb{C}^3\) whose rank is at least two. A result whose real counterpart is a well-known classical fact. This note concerns itself with the complex 2-dimensional case, but there are obvious higher dimensional generalisations that can be treated with the same techniques.

The reader should be aware that the results presented here can also be obtained by using the elegant and powerful theory of Bernstein–Gelfand–Gelfand (BGG) sequences developed by Čap, Slovák and Souček [10] (see also the article of Calderbank and Diemer [6]). In particular, the prolongation computed here is an example of a prolongation connection of a first BGG equation in parabolic geometry and may be derived using the techniques developed in [18].

This note aims at providing an intermediate analysis between the abstract BGG machinery and pure local coordinate computations. This is achieved by carrying out the computations on the parabolic Cartan geometry of a complex projective surface.

Acknowledgments

The author was supported by Forschungsinstitut für Mathematik (FIM) at ETH Zürich and Schweizerischer Nationalfonds SNF via the postdoctoral fellowship PA00P2_142053. The author is grateful to Friedrich-Schiller-Universität Jena for financial support for several trips to Jena where a part of the writing for this article took place and to V. S. Matveev and S. Rosemann for introducing him to the subject of complex projective geometry through many stimulating discussions. The author also would like to thank R. L. Bryant for sharing with him his notes [3] which contain the proofs of the counterparts for real projective surfaces of Theorem 3.7 and Corollary 3.8. Furthermore, the author also would like to thank the referees for several valuable comments.