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

SIGMA 18 (2022), 093, 23 pages      arXiv:2207.12946

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

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.

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