Complex-structure rigidity without the Kähler hypothesis for the two exceptional $G_2$ flag manifolds $Q$ and $Z$

The exceptional Lie group $G_2$ has exactly two complex five-dimensional flag manifolds, both of the form $G_2/U(2)$ for non-conjugate embeddings of $U(2)$: the quadric fivefold $Q$, whose underlying smooth manifold is diffeomorphic to a smooth quadric hypersurface in $\mathbb{CP}^6$, and $Z$, the twistor space of the quaternionic Kähler symmetric space $G_2/SO(4)$. Both are rigid among Kählerian complex structures: Brieskorn (1964) proved that a compact Kähler manifold diffeomorphic to an odd-dimensional quadric is biholomorphic to it, and Kotschick–Thung (2020) proved that any Kählerian complex manifold homeomorphic to $Z$ is biholomorphic to $Z$. It remains open whether these rigidity statements survive without the Kähler hypothesis. The question is sharpened by a theorem of Kotschick–Thung: if the six-sphere $S^6$ admits a complex structure, then the smooth manifold of $Q$ carries two distinct non-Kähler complex structures. Hence a positive answer for $Q$ would simultaneously prove that $S^6$ admits no complex structure. The problem asks, for each of $Q$ and $Z$: is every complex structure on the underlying smooth 10-manifold equivalent to the standard one — equivalence being generated by conjugation, diffeomorphism, and homotopies of almost complex structures, as in Kotschick–Thung — or does an inequivalent, necessarily non-Kähler, complex structure exist?

Background

Let $G_2$ denote the compact exceptional Lie group of dimension $14$, realized as the automorphism group of the octonions. Among its homogeneous spaces there are two complex five-dimensional flag manifolds (homogeneous spaces that are projective algebraic varieties), both quotients $G_2/U(2)$ for two non-conjugate embeddings of $U(2)$ into $G_2$; they are distinguished by their isotropy representations, which split into three irreducible summands for one quotient and two for the other. Following Kotschick–Thung, they are denoted $Q = G_2/U(2)_-$ and $Z = G_2/U(2)_+$. The manifold $Q$ is diffeomorphic to the smooth five-dimensional quadric hypersurface $Q^5 \subset \mathbb{CP}^6$, and its standard complex structure is that of this quadric. The manifold $Z$ is the twistor space of the Wolf space $M = G_2/SO(4)$, the compact quaternionic Kähler symmetric space: a quaternionic Kähler manifold is a Riemannian manifold of dimension $4n > 4$ whose holonomy group is contained in the group $Sp(n)\cdot Sp(1)$ of quaternionic linear isometries, as is the case for quaternionic projective space, and its twistor space — the bundle of complex structures on the tangent spaces compatible with the metric — is itself a complex manifold when the scalar curvature is positive. As such, $Z$ is a Fano variety (a projective variety with ample anticanonical bundle) carrying a holomorphic contact structure (a holomorphic subbundle of the cotangent bundle that is maximally non-integrable, the complex analogue of a contact structure). Both $Q$ and $Z$ are simply connected compact complex fivefolds with second Betti number $b_2 = 1$ and with the same additive cohomology groups as $\mathbb{CP}^5$, but with cohomology ring structures different from $\mathbb{CP}^5$ and from each other; neither admits a spin structure.

A classical rigidity theme asks how unique such structures are. A theorem of Hirzebruch and Kodaira from 1957, completed in even dimensions by Yau's proof of the Calabi conjecture (recounted in the introduction of Kotschick–Thung), shows that the standard structure is the unique Kählerian complex structure — the unique complex structure admitting some Kähler metric, i.e. a Hermitian metric whose associated $(1,1)$-form is closed — on the smooth manifold underlying complex projective space. Brieskorn extended this to the quadrics: a compact Kähler manifold diffeomorphic to the smooth quadric of odd complex dimension $n$ is biholomorphic to it (for even $n > 2$ the same holds up to the sign of the first Chern class), with no homogeneity or Fano assumption. In particular the manifold $Q$ has a unique Kählerian complex structure. Hwang proved a partial rigidity result for homogeneous contact Fano manifolds with $b_2 = 1$, which covers $Z$ within a restricted deformation class. Kotschick–Thung completed the Kählerian picture with their Theorem 1: any Kählerian complex manifold homeomorphic to the twistor space $Z$ is biholomorphic to $Z$. The proof combines the homeomorphism invariance of Pontryagin classes, Hodge-theoretic constraints on Chern numbers, the Hirzebruch–Riemann–Roch theorem, and the classification of Fano manifolds of coindex three.

All of these arguments break down outside the Kähler world: Hodge decomposition, the resulting Hodge-number constraints on Chern numbers, and the Kodaira embedding theorem (which makes a Kählerian structure with $b_2 = 1$ projective) have no known counterparts for a general compact complex manifold.

The stakes are high because of the six-sphere. The sphere $S^6 = G_2/SU(3)$ carries a $G_2$-invariant almost complex structure induced by octonion multiplication, but this structure is not integrable; an almost complex structure is integrable, i.e. comes from a genuine complex structure, exactly when its Nijenhuis tensor vanishes (Newlander–Nirenberg theorem). Whether $S^6$ admits any complex structure is a famous open problem that appears on Hirzebruch's 1954 problem list. Classically, blowing up a complex $S^6$ at a point would produce a non-Kähler complex structure on the smooth manifold $\mathbb{CP}^3$, so proving uniqueness of the complex structure of $\mathbb{CP}^3$ without the Kähler hypothesis would settle the $S^6$ problem negatively. Kotschick–Thung discovered the analogous mechanism for $Q$: their Theorem 2 states that if $S^6$ admits a complex structure, then the manifold $Q$ admits two distinct non-Kähler complex structures, arising as the projectivized tangent and cotangent bundles $\mathbb{P}(TS^6)$ and $\mathbb{P}(T^*S^6)$ of the hypothetical complex sphere, at least one of which carries a holomorphic contact structure. Here two complex structures are distinct when they are inequivalent under the equivalence relation generated by conjugation, diffeomorphism, and homotopies of almost complex structures. Consequently, as Kotschick–Thung emphasize, if Brieskorn's rigidity theorem for $Q$ could be proved without the Kähler hypothesis, it would follow from Theorem 2 that $S^6$ has no complex structure.

Problem Statement

Decide, without any Kähler hypothesis, how unique the standard complex structures of the two exceptional $G_2$ flag manifolds are. Throughout, two complex structures on a fixed smooth manifold count as equivalent when they are related by the equivalence relation generated by conjugation, diffeomorphism, and homotopies of almost complex structures (continuous one-parameter families through almost complex structures) — the same relation under which Kotschick–Thung formalize distinctness in their Theorem 2; in particular, a complex structure that is biholomorphic to the standard one after conjugation and after pullback by a diffeomorphism is equivalent to it.

(1) The quadric $Q = G_2/U(2)_-$, whose underlying smooth 10-manifold is diffeomorphic to the smooth quadric fivefold in $\mathbb{CP}^6$. Is every complex structure on this smooth manifold equivalent to the standard quadric structure — that is, connected to it by a finite chain of conjugations, pullbacks by diffeomorphisms, and homotopies of almost complex structures — or does there exist an inequivalent complex structure? By Brieskorn's theorem any structure on this manifold that admits a Kähler metric is biholomorphic to the standard quadric structure, so any counterexample is necessarily non-Kähler. A positive answer, combined with Theorem 2 of Kotschick–Thung, would prove that $S^6$ admits no complex structure: the two structures supplied by Theorem 2 would then both be equivalent to the standard one, hence to each other, contradicting the distinctness asserted by that theorem.

(2) The twistor space $Z = G_2/U(2)_+$ of $G_2/SO(4)$. The same question: is every complex structure on the smooth manifold underlying $Z$ equivalent to the standard twistor space under the same equivalence relation, or does an inequivalent complex structure exist? Kotschick–Thung's Theorem 1 settles the case where the structure admits a Kähler metric, so again any counterexample is necessarily non-Kähler.

The common objective of the two parts is to determine whether the Kähler hypothesis in these rigidity theorems is essential. A complete resolution settles both (1) and (2); a resolution of either part alone — a proof of uniqueness, or an explicit inequivalent complex structure on the corresponding smooth manifold — is a coherent partial answer, with part (1) the one that carries the direct implication for the complex-structure problem on $S^6$.

References

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