Conformally Einstein anti-self-dual spaces and their generalisations

Abstract: We study the existence of Einstein scales and compatible metrics in Grassmannian geometry. As a special case, this includes the conformal-to-Einstein problem for anti-self-dual (ASD) conformal structures. In the ASD setting, given a genericity condition algebraic in the Weyl curvature, we obtain necessary and sufficient conditions for the existence of a local Einstein scale that refine those previously appearing in the literature. We also prove that in Riemannian signature, a non-Ricci-flat ASD-Kähler metric is locally conformally Einstein if and only if it has a holomorphic Killing vector field with anti-self-dual derivative proportional to the Ricci spinor. Next, we consider some generalisations of the ASD conformal-to-Einstein problem for $(2n,2)$-Grassmannian structures. When $n > 1$, given an analogous genericity condition, we obtain necessary and sufficient conditions for the existence of a local Einstein scale. Using tractor calculus, we obtain algebraic obstructions to the existence of compatible metrics and show that the unique submaximally symmetric model of the geometry has the submaximal number of linearly independent compatible metrics.

• The paper derives a necessary-and-sufficient third-order obstruction for generic conformally Einstein ASD structures, reducing the test to four independent equations when the curvature invariants I and J are nonzero.
• It characterises non-Ricci-flat ASD-Kähler metrics that are locally conformally Einstein through a Kähler-preserving, or holomorphic in Riemannian signature, Killing vector whose anti-self-dual derivative is proportional to the Ricci spinor.
• For torsion-free (2n,2)-Grassmannian structures, it establishes generic Einstein-scale criteria, tractor-based metrisability obstructions, and the sharp compatible-metric bound 2n²−3n+4, attained by the Kruglikov–The model.

This paper develops a unified treatment of the conformal-to-Einstein problem for anti-self-dual (ASD) conformal structures and its generalisation to $(2n,2)$-Grassmannian geometry. The author, working in the framework of complex parabolic geometry, obtains necessary and sufficient conditions for the existence of local Einstein scales under algebraic genericity assumptions on the invariant curvature, proves a new characterisation of conformally Einstein ASD-Kähler metrics via holomorphic Killing vectors, and establishes sharp bounds on the number of compatible metrics for torsion-free Grassmannian structures (TFGs).

Background: Grassmannian structures and their calculus

The central object is a $(2n,2)$-almost-Grassmannian structure on a complex $4n$-manifold $X$: an isomorphism $TX \cong S \otimes S'$ with $\operatorname{rank} S = 2n$, $\operatorname{rank} S' = 2$. This generalises the flat model $F_2(\mathbb{C}^{2n+2})$ and, for $n=1$, recovers a holomorphic conformal structure. The structure admits distinguished $\alpha$- and $(2n,2)$0-planes; right-flatness (integrability of $(2n,2)$1-planes to $(2n,2)$2-surfaces) is equivalent to anti-self-duality of the Weyl tensor when $(2n,2)$3, while for $(2n,2)$4 it is equivalent to vanishing of part of the intrinsic torsion.

A key technical device is the pair of Weyl connections induced by scales $(2n,2)$5 and $(2n,2)$6, whose existence and uniqueness follow from an analogue of the fundamental theorem of Riemannian geometry. For a TFG (right-flat and torsion-free), the curvature decomposes into three irreducible pieces: $(2n,2)$7 (the analogue of the trace-free Ricci tensor), $(2n,2)$8 (generalising the Ricci scalar), and the invariant $(2n,2)$9 (generalising the ASD Weyl tensor). The Schouten tensor $4n$0 transforms by the standard conformal formula, so all four-dimensional spinorial machinery carries over verbatim. Compatible metrics are those of the form $4n$1 for which some Weyl connections induce the Levi-Civita connection; for $4n$2 these are automatically Einstein and constitute complexified analogues of quaternion-Kähler metrics.

Einstein scales for generic ASD conformal structures

Adapting the methods of Kozameh–Newman–Tod and Szekeres, the paper derives conditions for the existence of a scale $4n$3 solving the conformal-to-Einstein equation $4n$4. The necessary condition is that in some scale there exists a closed one-form $4n$5 satisfying

$4n$6

which expresses that the divergence of the Weyl tensor vanishes in the Einstein scale. Two results make this condition sufficient under genericity hypotheses:

• Closedness from algebraic solvability: if $4n$7, any one-form $4n$8 satisfying the equation algebraically is automatically closed. The proof combines the vanishing Bach tensor of ASD conformal structures with Calderbank's theorem that the Faraday curvature of an Einstein-Weyl structure on an ASD manifold is anti-self-dual.
• Sufficiency: given such a closed solution, the rescaled Schouten tensor becomes trace-free, i.e., the metric is Einstein.

Consequently, when $4n$9 and $X$0, the conformal class contains an Einstein metric if and only if a single explicit invariant vanishes:

$X$1

After choosing a basis this yields eight scalar equations, of which only four are independent — a refinement over the rank-fifteen bundle obstruction of Gover–Nurowski, and requiring only third derivatives of the metric components rather than fourth. Without the hypothesis $X$2, necessary and sufficient conditions are given by the closedness of the canonically defined one-form together with the Einstein-Weyl relation, at fourth order. A companion theorem gives necessary and sufficient conditions for a Ricci-flat scale: the Einstein condition plus the scalar constraint $X$3.

Two computed examples illustrate the theory. First, Araneda's ASD subfamily of Plebański-Demiański metrics satisfies the obstruction identically, with explicit Einstein scale $X$4 — consistent with the ambi-Kähler criterion of Apostolov–Calderbank–Gauduchon. Second, an ASD metric constructed from a monopole over Lorentzian Nil Einstein-Weyl space provides a counterexample showing that algebraic solutions need not be closed outside Riemannian signature: for $X$5 the unique algebraic solution has non-vanishing exterior derivative unless $X$6 is constant, demonstrating that the Riemannian assumption in Gover–Nagy's theorem is essential.

Conformally Einstein ASD-Kähler metrics

For scalar-flat Kähler metrics whose Kähler form is self-dual, the trace-free Ricci tensor factorises as $X$7 with $X$8 the Ricci spinor. The paper first re-proves, extending to indefinite signature, the result of Dunajski–Tod: an ASD-Einstein metric with non-zero cosmological constant is conformal to an ASD-Kähler metric on an open dense set if and only if it admits a Killing vector that is not everywhere null. The proof shows the Killing vector is Hamiltonian and remains Killing in the Kähler scale.

The main new result is the converse direction. Under the non-degeneracy condition $X$9, an ASD-Kähler metric is conformal to an Einstein metric if and only if it possesses a Kähler-preserving Killing vector whose anti-self-dual derivative is proportional to the Ricci spinor:

$TX \cong S \otimes S'$0

The proof extracts from this equation a function $TX \cong S \otimes S'$1 with $TX \cong S \otimes S'$2, which then solves the conformal-to-Einstein equation wherever it does not vanish. In Riemannian signature, "Kähler-preserving" means holomorphic, yielding the clean statement: a Riemannian ASD-Kähler metric that is not Ricci-flat is locally conformally Einstein if and only if it admits a holomorphic Killing vector field with anti-self-dual derivative proportional to the Ricci spinor. Together with Derdziński's treatment of the non-ASD case, this completes the local characterisation of conformally Einstein four-dimensional Riemannian Kähler metrics. The paper notes a global caveat: by the Hitchin-Thorpe inequality, no compact examples exist with a global conformal factor. The Plebański-Demiański example is revisited, where the relevant Killing vector is $TX \cong S \otimes S'$3.

Metrisability and tractor obstructions

For TFGs with $TX \cong S \otimes S'$4, compatible metrics are shown to correspond to non-degenerate solutions of the overdetermined metrisability equation

$TX \cong S \otimes S'$5

a first BGG operator previously studied via parabolic geometry machinery; here it is rederived elementarily. Invariant prolongation identifies solutions with parallel sections of the bundle $TX \cong S \otimes S'$6, where $TX \cong S \otimes S'$7 is the standard tractor bundle (the analogue of the local twistor bundle), equipped with the normalised tractor connection induced by the Cartan connection.

The tractor curvature yields algebraic obstructions: if a compatible metric exists, then for all $TX \cong S \otimes S'$8

$TX \cong S \otimes S'$9

By the Razmyslov-Procesi theorem these stop being independent beyond $\operatorname{rank} S = 2n$0. An explicit family $\operatorname{rank} S = 2n$1 ($\operatorname{rank} S = 2n$2) exhibits a non-vanishing cubic obstruction for $\operatorname{rank} S = 2n$3, confirming the obstructions are non-trivial. Conversely, an example with vanishing obstructions but no non-degenerate solution shows they are not sufficient — a limitation stated plainly by the author.

Submaximally symmetric models

The Kruglikov-The model, defined by $\operatorname{rank} S = 2n$4, realises the symmetry gap for almost-Grassmannian structures: it has the maximal non-flat symmetry count $\operatorname{rank} S = 2n$5, and by The's uniqueness result it is the unique such model. Its invariant curvature is $\operatorname{rank} S = 2n$6. The paper proves two quantitative statements:

Structure	Number of linearly independent compatible metrics
Flat model $\operatorname{rank} S = 2n$7	$\operatorname{rank} S = 2n$8
Non-flat TFG (upper bound)	$\operatorname{rank} S = 2n$9
Kruglikov-The model (lower bound)	$\operatorname{rank} S' = 2$0

The upper bound is obtained by a rank-nullity analysis of the pointwise solution space to the parallel-section conditions $\operatorname{rank} S' = 2$1 and $\operatorname{rank} S' = 2$2, bounding each projection slot separately. Since the lower bound matches, the Kruglikov-The model saturates the submaximal count of compatible metrics — a direct analogue, for metrisability, of its known saturation of the symmetry gap. Among its compatible metrics, those with $\operatorname{rank} S' = 2$3 are hyper-Kähler (products of ASD Ricci-flat pp-waves with flat factors), while generically the family has non-zero scalar curvature.

Einstein scales in Grassmannian geometry

An Einstein scale on a TFG is a pair of Weyl connections with $\operatorname{rank} S' = 2$4. The paper clarifies the relationship to compatible metrics: a compatible metric induces an Einstein scale, and conversely, using the vacuum Bianchi identity $\operatorname{rank} S' = 2$5 proved in the appendix, a non-degenerate $\operatorname{rank} S' = 2$6 in an Einstein scale determines a compatible metric $\operatorname{rank} S' = 2$7.

A Grassmannian analogue of Bach-flatness is derived from the tractor Bianchi identity:

$\operatorname{rank} S' = 2$8

Under the $\operatorname{rank} S' = 2$9-non-degeneracy condition (injectivity of $F_2(\mathbb{C}^{2n+2})$0, reducing to $F_2(\mathbb{C}^{2n+2})$1 when $F_2(\mathbb{C}^{2n+2})$2), the four-dimensional argument generalises fully: a TFG admits an Einstein scale if and only if there exists a closed one-form $F_2(\mathbb{C}^{2n+2})$3 satisfying

$F_2(\mathbb{C}^{2n+2})$4

Weak non-degeneracy (injectivity of $F_2(\mathbb{C}^{2n+2})$5) permits solving algebraically for a candidate $F_2(\mathbb{C}^{2n+2})$6 via a left inverse $F_2(\mathbb{C}^{2n+2})$7, yielding an explicit invariant analogous to the four-dimensional obstruction. However, the question of whether integrability conditions force an algebraic solution to be closed — the step handled in four dimensions by Calderbank's Faraday-curvature theorem — is left open, with the author conjecturing a connection to a suitable generalisation of Einstein-Weyl geometry and the underlying BGG sequences.

Limitations and open questions

Several restrictions qualify the results. All statements are local and holomorphic; global topological obstructions to the decomposition $F_2(\mathbb{C}^{2n+2})$8 are set aside by working on contractible neighbourhoods. The sufficiency of the closedness condition requires genericity ($F_2(\mathbb{C}^{2n+2})$9 in four dimensions; weak or $n=1$0-non-degeneracy in general), and the degenerate cases are handled only by falling back on higher-order criteria. The tractor-based obstructions to metrisability are necessary but demonstrably not sufficient. Most substantively, the closure question for algebraic solutions of the Grassmannian Einstein-scale equation remains unresolved for $n=1$1, and the relationship between compatible metrics and the almost-Grassmannian structures arising from path geometries (via the Crampin-Saunders/Guo construction) is unexplored.

Conclusion

The paper refines the classical conformal-to-Einstein criteria for ASD conformal structures to a minimal four-equation third-order obstruction, completes the local classification of conformally Einstein Riemannian Kähler four-manifolds via a holomorphic Killing vector criterion, and transfers the entire spinorial framework to $n=1$2-Grassmannian geometry, where it yields necessary and sufficient Einstein-scale conditions under genericity, concrete algebraic obstructions to metrisability, and a sharp submaximal count of compatible metrics attained uniquely by the Kruglikov-The model. The natural next targets identified are Grassmannian structures on monopole moduli spaces and the path-geometry construction, both of which would test the reach of the tractor methods developed here.