### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 093, 23 pages      arXiv:2207.12946      https://doi.org/10.3842/SIGMA.2022.093

### Topology of Almost Complex Structures on Six-Manifolds

Gustavo Granja a and Aleksandar Milivojević b
a) Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
b) Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany

Received August 08, 2022, in final form November 20, 2022; Published online December 02, 2022

Abstract
We study the space of (orthogonal) almost complex structures on closed six-dimensional manifolds as the space of sections of the twistor space for a given metric. For a connected six-manifold with vanishing first Betti number, we express the space of almost complex structures as a quotient of the space of sections of a seven-sphere bundle over the manifold by a circle action, and then use this description to compute the rational homotopy theoretic minimal model of the components that satisfy a certain Chern number condition. We further obtain a formula for the homological intersection number of two sections of the twistor space in terms of the Chern classes of the corresponding almost complex structures.

Key words: almost complex structure; twistor space; space of almost complex structures.

pdf (538 kb)   tex (31 kb)

References

1. Armstrong J., On four-dimensional almost Kähler manifolds, Quart. J. Math. Oxford Ser. (2) 48 (1997), 405-415.
2. Atiyah M.F., Hitchin N.J., Singer I.M., Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A 362 (1978), 425-461.
3. Blanchard A., Recherche de structures analytiques complexes sur certaines variétés, C. R. Acad. Sci. Paris 236 (1953), 657-659.
4. Borel A., Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts, Ann. of Math. 57 (1953), 115-207.
5. Bredon G.E., Topology and geometry, Grad. Texts in Math., Vol. 139, Springer, New York, 1993.
6. Calabi E., Gluck H., What are the best almost-complex structures on the $6$-sphere?, in Differential Geometry: Geometry in Mathematical Physics and Related Topics (Los Angeles, CA, 1990), Proc. Sympos. Pure Math., Vol. 54, Amer. Math. Soc., Providence, RI, 1993, 99-106.
7. Crabb M.C., Sutherland W.A., Function spaces and Hurwitz-Radon numbers, Math. Scand. 55 (1984), 67-90.
8. Demailly J.-P., Complex analytic and differential geometry, available at https://www-fourier.ujf-grenoble.fr/ demailly/manuscripts/agbook.pdf.
9. Duan H., The characteristic classes and Weyl invariants of spinor groups, arXiv:1810.03799.
10. Eastwood M.G., Singer M.A., The Fröhlicher spectral sequence on a twistor space, J. Differential Geom. 38 (1993), 653-669.
11. Eells J., Salamon S., Twistorial construction of harmonic maps of surfaces into four-manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 12 (1985), 589-640.
12. Evans J.D., Quantum cohomology of twistor spaces and their Lagrangian submanifolds, J. Differential Geom. 96 (2014), 353-397, arXiv:1106.3959.
13. Ferlengez B., Granja G., Milivojevic A., On the topology of the space of almost complex structures on the six sphere, New York J. Math. 27 (2021), 1258-1273, arXiv:2108.00750.
14. Godinho L., Sabatini S., New tools for classifying Hamiltonian circle actions with isolated fixed points, Found. Comput. Math. 14 (2014), 791-860, arXiv:1206.3195.
15. Haefliger A., Rational homotopy of the space of sections of a nilpotent bundle, Trans. Amer. Math. Soc. 273 (1982), 609-620.
16. Hitchin N.J., Kählerian twistor spaces, Proc. London Math. Soc. 43 (1981), 133-150.
17. Lawson H.B., Michelsohn M.-L., Spin geometry, Princeton Math. Ser., Vol. 38, Princeton University Press, Princeton, NJ, 1989.
18. LeBrun C., Orthogonal complex structures on $S^6$, Proc. Amer. Math. Soc. 101 (1987), 136-138.
19. LeBrun C., Topology versus Chern numbers for complex $3$-folds, Pacific J. Math. 191 (1999), 123-131, arXiv:math.AG/9801133.
20. Mimura M., Toda H., Topology of Lie groups. I, II, Transl. Math. Monogr., Vol. 91, Amer. Math. Soc., Providence, RI, 1991.
21. Møller J.M., Nilpotent spaces of sections, Trans. Amer. Math. Soc. 303 (1987), 733-741.
22. Møller J.M., Raussen M., Rational homotopy of spaces of maps into spheres and complex projective spaces, Trans. Amer. Math. Soc. 292 (1985), 721-732.
23. Salamon S., Harmonic and holomorphic maps, in Geometry Seminar ''Luigi Bianchi'' II - 1984, Lecture Notes in Math., Vol. 1164, Springer, Berlin, 1985, 161-224.
24. Salamon S.M., Orthogonal complex structures, in Differential Geometry and Applications (Brno, 1995), Masaryk University, Brno, 1996, 103-117.
25. Sullivan D., Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269-331.
26. Taubes C.H., The existence of anti-self-dual conformal structures, J. Differential Geom. 36 (1992), 163-253.
27. Vigué-Poirrier M., Sullivan D., The homology theory of the closed geodesic problem, J. Differential Geometry 11 (1976), 633-644.