2015/08/06 by Dusa McDuff, McDuff, Dusa, Katrin Wehrheim +1
Mathematics · #53D35 #53D45 #54B15 #57R17 #57R95 #FOS: Mathematics #Symplectic Geometry (math.SG) #math.SG #msc:53D35 #msc:53D45 #msc:54B15 #msc:57R17 #msc:57R95
paper · pdf · doi:10.48550/arxiv.1508.01560
replaces parts of arXiv:1208.1340; v2 adjusts some cross-references
arxiv created 2015/08/11 · arxiv updated 2015/08/12
Kuranishi structures were introduced in the 1990s by Fukaya and Ono for the purpose of assigning a virtual cycle to moduli spaces of pseudoholomorphic curves that cannot be regularized by geometric methods. Their core idea was to build such a cycle by patching local finite dimensional reductions. The first sections of this paper discuss topological, algebraic and analytic challenges that arise in this program. We then develop a theory of Kuranishi atlases and cobordisms that transparently resolves these challenges, for simplicity concentrating on the case of trivial isotropy. In this case, we assign to a cobordism class of additive weak Kuranishi atlases both a virtual moduli cycle (VMC - a cobordism class of smooth manifolds) and a virtual fundamental class (VFC - a Cech homology class). We moreover show that such Kuranishi atlases exist on simple Gromov-Witten moduli spaces and develop the technical results in a manner that easily transfers to more general settings.