In this expository manuscript, we review the construction of Gromov-Witten virtual fundamental class via FOOO's theory of Kuranishi structures for moduli spaces of pseudo-holomorphic maps defined on closed Riemann surfaces. We consider constraints coming from the ambient space and Deligne-Mumford moduli, called primary insertions, as well as intrinsic classes such as $psi$-classes and Hodge classes.
{"title":"Gromov-Witten theory via Kuranishi\u0000 structures","authors":"M. Tehrani, K. Fukaya","doi":"10.1090/surv/237/02","DOIUrl":"https://doi.org/10.1090/surv/237/02","url":null,"abstract":"In this expository manuscript, we review the construction of Gromov-Witten virtual fundamental class via FOOO's theory of Kuranishi structures for moduli spaces of pseudo-holomorphic maps defined on closed Riemann surfaces. We consider constraints coming from the ambient space and Deligne-Mumford moduli, called primary insertions, as well as intrinsic classes such as $psi$-classes and Hodge classes.","PeriodicalId":422349,"journal":{"name":"Virtual Fundamental Cycles in Symplectic\n Topology","volume":"63 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124581730","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}
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. 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 ${bf Kur}simeq{bf dOrb}$, 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, and the author's programme for applying these ideas in symplectic geometry.
{"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}
These notes aim to explain a joint project with Katrin Wehrheim that uses finite dimensional reductions to construct a virtual fundamental class for the Gromov--Witten moduli space of closed genus zero curves. Our method is based on work by Fukaya and Ono as well as more recent work by Fukaya, Oh, Ohta, and Ono. We reformulated their ideas in order to clarify the formal structures underlying the construction and make explicit all important choices (of tamings, shrinkings and reductions), thus creating tools with which to give an explicit proof that the virtual fundamental class is independent of these choices. After summarizing the main ideas and proofs in the arXiv preprint 1208.1340, these notes explain the modifications needed to deal with isotropy. Further sections outline the construction of a Kuranishi atlas in the genus zero case, and give some examples of their use. We also show that every finite dimensional orbifold has a Kuranishi atlas.
{"title":"Notes on Kuranishi atlases","authors":"D. Mcduff","doi":"10.1090/surv/237/01","DOIUrl":"https://doi.org/10.1090/surv/237/01","url":null,"abstract":"These notes aim to explain a joint project with Katrin Wehrheim that uses finite dimensional reductions to construct a virtual fundamental class for the Gromov--Witten moduli space of closed genus zero curves. Our method is based on work by Fukaya and Ono as well as more recent work by Fukaya, Oh, Ohta, and Ono. We reformulated their ideas in order to clarify the formal structures underlying the construction and make explicit all important choices (of tamings, shrinkings and reductions), thus creating tools with which to give an explicit proof that the virtual fundamental class is independent of these choices. After summarizing the main ideas and proofs in the arXiv preprint 1208.1340, these notes explain the modifications needed to deal with isotropy. Further sections outline the construction of a Kuranishi atlas in the genus zero case, and give some examples of their use. We also show that every finite dimensional orbifold has a Kuranishi atlas.","PeriodicalId":422349,"journal":{"name":"Virtual Fundamental Cycles in Symplectic\n Topology","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134015219","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}
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