Kuranishi spaces as a 2-category

Book section

‘Kuranishi spaces’ were introduced in the work of Fukaya, Oh, Ohta and Ono [10–19] in symplectic geometry, as the geometric structure on moduli spaces of J-holomorphic curves. We propose a new definition of Kuranishi space, which has the nice property that they form a 2-category Kur. Any Fukaya–Oh–Ohta–Ono (FOOO) Kuranishi space X can be made into a compact Kuranishi space X′ uniquely up to equivalence in Kur, and conversely any compact Kuranishi space X′ comes from some (nonunique) FOOO Kuranishi space X. So FOOO Kuranishi spaces are equivalent to ours at one level, but our definition has better categorical properties. The same holds for McDuff and Wehrheim’s ‘Kuranishi atlases’ [39–42]. A compact topological space X with a ‘polyfold Fredholm structure’ in the sense of Hofer, Wysocki and Zehnder [23–29] can be made into a Kuranishi space X uniquely up to equivalence in Kur. Our Kuranishi spaces are based on the author’s theory of Derived Differential Geometry [31–33], the study of classes of derived manifolds and orbifolds that we call ‘d-manifolds’ and ‘d-orbifolds’. There is an equivalence of 2-categories Kur ≃ dOrb, where dOrb is the 2-category of d-orbifolds. So Kuranishi spaces are really a form of derived orbifold. We discuss the differential geometry of Kuranishi spaces, and the author’s programme for applying these ideas in symplectic geometry.

Authors: Joyce, D

• Publication status: Published
• Peer review status: Not peer reviewed
• Publisher: American Mathematical Society
• Host title: Virtual Fundamental Cycles in Symplectic Topology
• Volume: 237
• Pages: 253-300
• Series: Mathematical Surveys and Monographs
• Publication date: 2019-05-30
• ISBN: 9781470450144