2015/10/26 by Joyce, Dominic
#Differential Geometry (math.DG) #FOS: Mathematics #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.1510.07444
This is a survey of the author's paper arXiv:1409.6908 and in-progress book. 'Kuranishi spaces' were introduced in the work of Fukaya, Oh, Ohta and Ono in symplectic geometry (see e.g. arXiv:1503.07631), 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 \bf Kur. Thus the homotopy category Ho(\bf Kur) is an ordinary category of Kuranishi spaces. Any Fukaya-Oh-Ohta-Ono (FOOO) Kuranishi space \bf X can be made into a compact Kuranishi space \bf X' uniquely up to equivalence in \bf Kur (that is, up to isomorphism in Ho(\bf Kur)), and conversely any compact Kuranishi space \bf X' comes from some (nonunique) FOOO Kuranishi space \bf 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' in arXiv:1508.01556. Using results of Yang on polyfolds and Kuranishi spaces surveyed in arXiv:1510.06849, a compact topological space X with a 'polyfold Fredholm structure' in the sense of Hofer, Wysocki and Zehnder (see e.g. arXiv:1407.3185) can be made into a Kuranishi space \bf X uniquely up to equivalence in \bf Kur. Our Kuranishi spaces are based on the author's theory of Derived Differential Geometry (see e.g. arXiv:1206.4207), the study of classes of derived manifolds and orbifolds that we call 'd-manifolds' and 'd-orbifolds'. There is an equivalence of 2-categories \bf Kur≃\bf dOrb, where \bf 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.