We describe an extension of Morse theory to smooth functions on compact Riemannian manifolds, without any non-degeneracy assumptions except that the critical locus must have only finitely many connected components.
{"title":"Morse Theory without Non-Degeneracy","authors":"Frances Kirwan;Geoffrey Penington","doi":"10.1093/qmath/haaa064","DOIUrl":"https://doi.org/10.1093/qmath/haaa064","url":null,"abstract":"We describe an extension of Morse theory to smooth functions on compact Riemannian manifolds, without any non-degeneracy assumptions except that the critical locus must have only finitely many connected components.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 1-2","pages":"455-514"},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmath/haaa064","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49950712","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We define and study the motive of the moduli stack of vector bundles of fixed rank and degree over a smooth projective curve in Voevodsky’s category of motives. We prove that this motive can be written as a homotopy colimit of motives of smooth projective Quot schemes of torsion quotients of sums of line bundles on the curve. When working with rational coefficients, we prove that the motive of the stack of bundles lies in the localizing tensor subcategory generated by the motive of the curve, using Białynicki-Birula decompositions of these Quot schemes. We conjecture a formula for the motive of this stack, inspired by the work of Atiyah and Bott on the topology of the classifying space of the gauge group, and we prove this conjecture modulo a conjecture on the intersection theory of the Quot schemes.
{"title":"On the Voevodsky Motive of the Moduli Stack of Vector Bundles on a Curve","authors":"Victoria Hoskins;Simon Pepin Lehalleur","doi":"10.1093/qmathj/haaa023","DOIUrl":"https://doi.org/10.1093/qmathj/haaa023","url":null,"abstract":"We define and study the motive of the moduli stack of vector bundles of fixed rank and degree over a smooth projective curve in Voevodsky’s category of motives. We prove that this motive can be written as a homotopy colimit of motives of smooth projective Quot schemes of torsion quotients of sums of line bundles on the curve. When working with rational coefficients, we prove that the motive of the stack of bundles lies in the localizing tensor subcategory generated by the motive of the curve, using Białynicki-Birula decompositions of these Quot schemes. We conjecture a formula for the motive of this stack, inspired by the work of Atiyah and Bott on the topology of the classifying space of the gauge group, and we prove this conjecture modulo a conjecture on the intersection theory of the Quot schemes.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 1-2","pages":"71-114"},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmathj/haaa023","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49950827","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The construction of Atiyah, Drinfeld, Hitchin and Manin provided complete description of all instantons on Euclidean four-space. It was extended by Kronheimer and Nakajima to instantons on ALE spaces, resolutions of orbifolds $mathbb{R}^4/Gamma$