The boundary between parallelizability, almost complex structure, and integrability: complex structures on compact parallelizable manifolds above dimension four, and rigidity of complex manifolds homotopy equivalent to tori

Yau's 1976 note 'Parallelizable manifolds without complex structure' established, in real dimension four, both that parallelizability does not imply the existence of a complex structure — products $H\times S^1$ of closed orientable 3-manifolds $H$ with a circle, ruled out by a nonzero real Massey product together with the Kählerness of surfaces with even first Betti number — and, via Kodaira's classification of compact complex surfaces, which complex surfaces have parallelizable underlying 4-manifolds. Two linked cores are open as of September 2026. First, the higher-dimensional integrability boundary: Yau's conjecture, recorded as Problem 5.21 (Yau) in Kirby's 1995 problem list, asserts that every closed smooth $2k$-manifold with $k>2$ admitting an almost complex structure admits some integrable complex structure; since a parallelizable manifold of even dimension is almost complex directly from a global frame, the existence of a compact parallelizable manifold of even real dimension $2k$ with $k>2$ and no complex structure is the first open case, and no example and no proof is known. Second, non-Kähler rigidity of homotopy tori: compact complex manifolds diffeomorphic to a complex torus need not be biholomorphic to one (the Blanchard and Sommese examples, all of negative Kodaira dimension), while rigidity holds in the Kähler and Fujiki class $\mathcal C$ settings and for threefolds with trivial canonical bundle admitting a nonconstant meromorphic function (Catanese–Oguiso–Peternell); whether a compact complex manifold with trivial canonical bundle can be diffeomorphic but not biholomorphic to a complex torus — Catanese's question — is open in general.

Background

A smooth manifold $X$ of real dimension $n$ is parallelizable if its tangent bundle is isomorphic to the trivial bundle $X\times\mathbb R^n$, i.e. if $X$ admits a global frame of tangent vectors. Parallelizability trivializes all characteristic classes of the tangent bundle and is the strongest condition on $TX$ expressible in homotopy terms. A complex structure on a real $2n$-manifold is a maximal atlas with charts in $\mathbb C^n$ and holomorphic transition functions; it induces an almost complex structure, that is, a vector-bundle endomorphism $J\colon TX\to TX$ with $J^2=-\mathrm{Id}$. An almost complex structure is integrable, i.e. induced by a complex structure, precisely when the eigen-distribution $T^{1,0}\subset TX\otimes\mathbb C$ (the $+i$-eigenbundle of $J$ on the complexified tangent bundle) is closed under Lie brackets (the classical integrability criterion). Integrability is a differential condition invisible to homotopy theory, so the passage from 'topologically trivial tangent bundle' to 'holomorphic atlas' is genuinely obstructed, and locating that obstruction is the subject of this problem.

In real dimension four the boundary is completely mapped. The Dold–Whitney classification of oriented sphere bundles over a 4-complex implies that a closed orientable 4-manifold is parallelizable if and only if its second Stiefel–Whitney class $w_2$ (a characteristic class of the tangent bundle with $\mathbb Z/2$ coefficients; $w_2=0$ is the spin condition), its first Pontryagin class $p_1$ (an integral characteristic class of the tangent bundle), and its Euler characteristic $\chi$ all vanish. On the complex side, a compact complex surface with even first Betti number $b_1$ (the rank of the first integral homology group) is Kähler, i.e. carries a closed positive $(1,1)$-form, and the real Massey products — tertiary cohomology operations defined whenever the underlying pairwise cup products vanish — vanish on every compact Kähler manifold. Yau combined these facts in 1976: every closed orientable 3-manifold is parallelizable, so the product $H\times S^1$ is parallelizable, and choosing $H$ so that $b_1(H\times S^1)$ is even while a real Massey product of $H\times S^1$ is nonzero yields a compact parallelizable 4-manifold with no complex structure, since were it a complex surface its even $b_1$ would force it to be Kähler and Kähler manifolds have vanishing real Massey products. Brotherton produced further parallelizable non-complex 4-manifolds by the same technique; Fernández–Gotay–Gray and Fernández–Gray charted which combinations of parallelizability, symplecticity (a symplectic structure being a closed nondegenerate 2-form), and complex structure are realized on closed 4-manifolds, including compact parallelizable symplectic solvmanifolds — compact quotients of solvable Lie groups — with no complex structure. On the classification side, Kodaira's four-part series 'On the structure of compact complex analytic surfaces' organizes compact complex surfaces into classes, sorted by their Kodaira dimension (the growth rate, as $m$ grows, of the spaces of holomorphic pluricanonical sections $H^0(X,mK_X)$, where $K_X$ is the canonical bundle, the top exterior power of the bundle of holomorphic 1-forms; the dimension is $-\infty$ by convention when all these spaces are zero): complex tori, hyperelliptic surfaces (free quotients of products of two elliptic curves by finite groups), primary and secondary Kodaira surfaces (elliptic surfaces — surfaces admitting a holomorphic map to a curve whose generic fibre is an elliptic curve — with trivial canonical bundle, and their finite quotients), Hopf surfaces (compact quotients of $\mathbb C^2\setminus\{0\}$), Inoue surfaces (compact quotients of $\mathbb C\times\mathbb H$, with $\mathbb H$ the upper half-plane), class VII surfaces (first Betti number 1 and Kodaira dimension $-\infty$), ruled and elliptic surfaces, and surfaces of general type. Yau's note determined, through this classification, which compact complex surfaces have a parallelizable underlying smooth manifold; Deschamps made this precise and completed it: the list comprises the Hopf surfaces, the Inoue surfaces, complex tori, hyperelliptic surfaces, primary Kodaira surfaces, the spin secondary Kodaira surfaces, genus-one ruled surfaces (bundles of projective lines over an elliptic curve) with even intersection form (the cup-product form on second cohomology having even diagonal, equivalently the spin condition), and certain spin minimal elliptic surfaces — and no others (for instance, K3 surfaces, the simply connected compact complex surfaces with trivial canonical bundle, are spin with topologically trivial twistor bundle but not parallelizable, having Euler characteristic 24). Deschamps also proved, by twistor methods (the twistor space of an oriented Riemannian 4-manifold being the $S^2$-bundle of orthogonal almost complex structures on its tangent spaces, which carries a natural complex structure for suitable metrics), that for every closed orientable hyperbolic 3-manifold $H$ — one carrying a metric of constant sectional curvature $-1$ — the 6-manifold $H\times S^1\times S^2$ is complex even though $H\times S^1$ is not.

Above real dimension four almost nothing is known. Yau's conjecture, recorded as Problem 5.21 (Yau) in Kirby's 1995 problem list 'Problems in Low-Dimensional Topology', states: all smooth $2k$-manifolds, $k>2$, with a given almost complex structure, are actually complex manifolds (the complex structure may not deform to the given almost complex structure). The low-dimensional cases behave oppositely — all orientable 2-manifolds are complex, while in real dimension four many almost complex manifolds, such as the connected sum of three copies of $\mathbb CP^2$ (the complex projective plane), carry no complex structure because of characteristic-class restrictions on complex surfaces. A compact parallelizable manifold of even dimension $2k$ is almost complex directly from any global frame (pair consecutive frame vectors into complex lines), so a single compact parallelizable non-complex $2k$-manifold with $k>2$ would refute the conjecture; conversely a proof of the conjecture would show that every compact parallelizable $2k$-manifold with $k>2$ is complex. No example and no proof is known.

The second core concerns how far a complex torus $T=\mathbb C^n/\Lambda$ — the quotient of $\mathbb C^n$ by a lattice $\Lambda\cong\mathbb Z^{2n}$ acting by translations — is determined by its topology. A compact Kähler manifold homotopy equivalent to a complex torus is biholomorphic to a torus; Catanese–Oguiso–Peternell proved the same under the much weaker Fujiki class $\mathcal C$ hypothesis (bimeromorphic to a compact Kähler manifold, for which equality of the integral cohomology rings suffices), and that a Moishezon manifold — a compact complex manifold whose algebraic dimension, i.e. the maximal number of algebraically independent meromorphic functions, equals its dimension — homotopy equivalent to a torus is an abelian variety. For complex surfaces the statement holds by Kodaira's classification of surfaces. But in complex dimension at least three it fails without such hypotheses: the examples of Blanchard and of Sommese are compact complex manifolds diffeomorphic to complex tori yet not biholomorphic to them, all of negative Kodaira dimension and fibred over elliptic curves. Catanese–Oguiso–Peternell prove rigidity for threefolds with trivial canonical bundle $K_X\cong\mathcal O_X$ that admit a dominant meromorphic map to a lower-dimensional analytic space, and pose, following Catanese's 2004 question, the general problem: is there a compact complex manifold with trivial canonical bundle which is diffeomorphic, but not biholomorphic, to a complex torus? This is open, as is the analogous problem in complex dimension at least four without hypotheses on meromorphic functions or on the Albanese map (the canonical holomorphic map to the dual torus of the space of holomorphic 1-forms).

The two cores meet in a single objective inherited from Yau's note: characterize, topologically, the compact manifolds on which complex structures can exist, and how tightly topology pins them down when they do. In real dimension four the answer is complete enough to exhibit the obstruction phenomenon and to preclude the rigidity phenomenon's failure; above dimension four, both the obstruction side (does topology ever forbid integrability when an almost complex structure exists?) and the rigidity side (when does topology force integrable structures to be biholomorphically standard?) are open.

Problem Statement

Determine how far topological conditions force the existence — or the rigidity — of complex structures on compact smooth manifolds, following the questions left open by Yau's 1976 construction of compact parallelizable 4-manifolds with no complex structure. The problem has two linked parts.

(1) The integrability boundary in even real dimension at least six. Does every closed smooth $2k$-manifold with $k>2$ that admits an almost complex structure (a vector-bundle endomorphism $J\colon TX\to TX$ with $J^2=-\mathrm{Id}$) admit a complex structure, i.e. a holomorphic atlas (not necessarily one deforming to the given almost complex structure)? This is the conjecture recorded as Problem 5.21 (Yau) in Kirby's 1995 problem list, whose proof or refutation is the target of this part. A compact parallelizable manifold (one whose tangent bundle is trivial, i.e. which admits a global frame of tangent vectors) of even dimension carries an almost complex structure directly from any global frame (pair consecutive frame vectors into complex lines), so a compact parallelizable non-complex $2k$-manifold with $k>2$ is a counterexample, and dimension 6 is the first open dimension; conversely, a proof of the conjecture would imply that every compact parallelizable $2k$-manifold with $k>2$ is complex. A counterexample is a closed almost complex non-complex $2k$-manifold with $k>2$, parallelizable or not; a proof of the conjecture would establish that no such counterexample exists.

(2) Rigidity of complex tori beyond the Kähler and Fujiki class $\mathcal C$ worlds. Which compact complex manifolds that are homotopy equivalent (or diffeomorphic) to a complex torus $T=\mathbb C^n/\Lambda$ — the quotient of $\mathbb C^n$ by a lattice $\Lambda\cong\mathbb Z^{2n}$ acting by translations — are biholomorphic to a complex torus? The answer is affirmative for $n\le2$ (by Kodaira's classification of surfaces), for Kähler manifolds (those carrying a closed positive $(1,1)$-form) and more generally for manifolds in the Fujiki class $\mathcal C$ (bimeromorphic to a compact Kähler manifold) with the torus cohomology ring, and for Moishezon manifolds (whose algebraic dimension, i.e. the maximal number of algebraically independent meromorphic functions, equals their dimension), which are then abelian varieties — Catanese–Oguiso–Peternell; it is negative in general in dimension at least 3, the known examples (Blanchard, Sommese) having negative Kodaira dimension (all pluricanonical sections vanish) and being fibred over elliptic curves. The open core, posed by Catanese: does there exist a compact complex manifold $X$ with trivial canonical bundle $K_X\cong\mathcal O_X$ (the top exterior power of the bundle of holomorphic 1-forms being holomorphically trivial) that is diffeomorphic to a complex torus but not biholomorphic to one? Catanese–Oguiso–Peternell settle affirmatively-for-rigidity the case of threefolds admitting a dominant meromorphic map to a lower-dimensional analytic space (or a nonconstant meromorphic function, or a nontrivial Albanese map); the question is open for threefolds with no such map and in every complex dimension at least 4.

The common objective of (1) and (2) is the boundary between topology and integrable complex structure: (1) asks whether topology can forbid integrability when an almost complex structure exists; (2) asks whether topology can force integrable structures to be biholomorphically standard. Answers are required unconditional in the stated generality — no restrictions on fundamental group, algebraic dimension, or method of construction beyond those explicitly stated.

References

1. Shing-Tung Yau, Parallelizable manifolds without complex structure, Topology 15 (1976), no. 1, 51–53. doi:10.1016/0040-9383(76)90049-5. https://doi.org/10.1016/0040-9383(76)90049-5
2. Rob Kirby (ed.), Problems in Low-Dimensional Topology, manuscript problem list, University of California, Berkeley, 22 December 1995; Problem 5.21 (Yau). https://math.berkeley.edu/~kirby/problems.ps.gz (compressed PostScript file; title page reads 'Problems in Low-Dimensional Topology, Edited by Rob Kirby, Berkeley – 22 Dec 95')
3. Neil Brotherton, Some parallelizable four manifolds not admitting a complex structure, Bulletin of the London Mathematical Society 10 (1978), no. 3, 303–304. doi:10.1112/blms/10.3.303. https://doi.org/10.1112/blms/10.3.303
4. Marisa Fernández, Mark J. Gotay, Alfred Gray, Compact parallelizable four-dimensional symplectic and complex manifolds, Proceedings of the American Mathematical Society 103 (1988), no. 4, 1209–1212. doi:10.1090/S0002-9939-1988-0955011-9. https://doi.org/10.1090/S0002-9939-1988-0955011-9
5. Marisa Fernández, Alfred Gray, Compact symplectic solvmanifolds not admitting complex structures, Geometriae Dedicata 34 (1990), no. 3, 295–299. doi:10.1007/BF00181691. https://doi.org/10.1007/BF00181691
6. Kunihiko Kodaira, On the structure of compact complex analytic surfaces, I–IV: part I, American Journal of Mathematics 86 (1964), no. 4, 751–798, doi:10.2307/2373157; part II, American Journal of Mathematics 88 (1966), no. 3, 682–721, doi:10.2307/2373150; part III, American Journal of Mathematics 90 (1968), no. 1, 55–83, doi:10.2307/2373426; part IV, published under the title 'On the structure of complex analytic surfaces, IV', American Journal of Mathematics 90 (1968), no. 4, 1048–1066, doi:10.2307/2373289. https://doi.org/10.2307/2373157
7. Guillaume Deschamps, 4-variétés parallélisables sans structure complexe dont l'espace twistoriel est complexe, Comptes Rendus Mathématique 341 (2005), no. 1, 35–38. doi:10.1016/j.crma.2005.05.027. https://doi.org/10.1016/j.crma.2005.05.027
8. Guillaume Deschamps, Espaces twistoriels et structures complexes non standards, Publicacions Matemàtiques 52 (2008), no. 2, 435–457. doi:10.5565/PUBLMAT_52208_09. https://doi.org/10.5565/PUBLMAT_52208_09
9. Mustafa Kalafat, Hülya Argüz, Complex and symplectic structures on panelled web 4-manifolds, Topology and its Applications 159 (2012), no. 8, 2168–2173. doi:10.1016/j.topol.2012.02.009. https://doi.org/10.1016/j.topol.2012.02.009
10. Fabrizio Catanese, Keiji Oguiso, Thomas Peternell, On volume-preserving complex structures on real tori, Kyoto Journal of Mathematics 50 (2010), no. 4, 753–775. doi:10.1215/0023608X-2010-013. https://doi.org/10.1215/0023608X-2010-013
11. Andrew John Sommese, Quaternionic manifolds, Mathematische Annalen 212 (1975), no. 3, 191–214. doi:10.1007/BF01357140. https://doi.org/10.1007/BF01357140
12. André Blanchard, Sur les variétés analytiques complexes, Annales scientifiques de l'École Normale Supérieure (3) 73 (1956), no. 2, 157–202. doi:10.24033/asens.1045. https://doi.org/10.24033/asens.1045
13. Fabrizio Catanese, Deformation in the large of some complex manifolds, I, Annali di Matematica Pura ed Applicata (4) 183 (2004), no. 3, 261–289. doi:10.1007/s10231-003-0096-y. https://doi.org/10.1007/s10231-003-0096-y
14. F. E. A. Johnson, J. P. Walton, Parallelizable manifolds and the fundamental group, Mathematika 47 (2000), no. 1–2, 165–172. doi:10.1112/S0025579300015795. https://doi.org/10.1112/S0025579300015795
15. A. Dold, H. Whitney, Classification of oriented sphere bundles over a 4-complex, Annals of Mathematics (2) 69 (1959), no. 3, 667–677. doi:10.2307/1970030. https://doi.org/10.2307/1970030
16. Valentina Bais, On Dold–Whitney's parallelizability of 4-manifolds, arXiv:2410.22117 (2024). https://arxiv.org/abs/2410.22117
17. Indranil Biswas, Junyan Cao, Sorin Dumitrescu, Henri Guenancia, Geometry of K-trivial Moishezon manifolds: decomposition theorem and holomorphic geometric structures, Mathematische Annalen 391 (2024), no. 2, 3181–3220. doi:10.1007/s00208-024-02997-4. Preprint arXiv:2306.16729. https://doi.org/10.1007/s00208-024-02997-4
18. A. van de Ven, On the Chern numbers of certain complex and almost complex manifolds, Proceedings of the National Academy of Sciences 55 (1966), no. 6, 1624–1627. doi:10.1073/pnas.55.6.1624. https://doi.org/10.1073/pnas.55.6.1624