Virtual Fundamental Cycles in Symplectic Topology最新文献

筛选

英文中文

Gromov-Witten theory via Kuranishi structures 通过Kuranishi结构的Gromov-Witten理论

Virtual Fundamental Cycles in Symplectic Topology Pub Date : 2017-01-26 DOI: 10.1090/surv/237/02

M. Tehrani, K. Fukaya

引用次数: 15

Kuranishi spaces as a 2-category Kuranishi空间作为2范畴

Virtual Fundamental Cycles in Symplectic Topology Pub Date : 2015-10-26 DOI: 10.1090/surv/237/03

D. joyce

{"title":"Kuranishi spaces as a 2-category","authors":"D. joyce","doi":"10.1090/surv/237/03","DOIUrl":"https://doi.org/10.1090/surv/237/03","url":null,"abstract":"This is a survey of the author's in-progress book arXiv:1409.6908. '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. \u0000Any 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. \u0000Using 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 $bf Kur$. \u0000Our 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 ${bf Kur}simeq{bf dOrb}$, where $bf dOrb$ is the 2-category of d-orbifolds. So Kuranishi spaces are really a form of derived orbifold. \u0000We discuss the differential geometry of Kuranishi spaces, and the author's programme for applying these ideas in symplectic geometry.","PeriodicalId":422349,"journal":{"name":"Virtual Fundamental Cycles in Symplectic\n Topology","volume":"199 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128681204","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 13

Notes on Kuranishi atlases 仓西地图集注释

Virtual Fundamental Cycles in Symplectic Topology Pub Date : 2014-11-16 DOI: 10.1090/surv/237/01

D. Mcduff

引用次数: 10