Pub Date : 2016-03-29DOI: 10.1090/PSPUM/097.2/01711
J. Nicaise
This survey paper explains how one can attach geometric invariants to semialgebraic sets defined over non-archimedean fields, using the theory of motivic integration of Hrushovski and Kazhdan. It also discusses tropical methods to compute these invariants in concrete cases, as well as an application to refined curve counting, developed in collaboration with Sam Payne and Franziska Schroeter.
{"title":"Geometric invariants for non-archimedean\u0000 semialgebraic sets","authors":"J. Nicaise","doi":"10.1090/PSPUM/097.2/01711","DOIUrl":"https://doi.org/10.1090/PSPUM/097.2/01711","url":null,"abstract":"This survey paper explains how one can attach geometric invariants to semialgebraic sets defined over non-archimedean fields, using the theory of motivic integration of Hrushovski and Kazhdan. It also discusses tropical methods to compute these invariants in concrete cases, as well as an application to refined curve counting, developed in collaboration with Sam Payne and Franziska Schroeter.","PeriodicalId":412716,"journal":{"name":"Algebraic Geometry: Salt Lake City\n 2015","volume":"10 18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125762014","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}
Pub Date : 2016-03-16DOI: 10.1090/PSPUM/097.1/01682
R. Pandharipande
This article accompanies my lecture at the 2015 AMS summer institute in algebraic geometry in Salt Lake City. I survey the recent advances in the study of tautological classes on the moduli spaces of curves. After discussing the Faber-Zagier relations on the moduli spaces of nonsingular curves and the kappa rings of the moduli spaces of curves of compact type, I present Pixton's proposal for a complete calculus of tautological classes on the moduli spaces of stable curves. Several open questions are discussed. An effort has been made to condense a great deal of mathematics into as few pages as possible with the hope that the reader will follow through to the end.
{"title":"A calculus for the moduli space of\u0000 curves","authors":"R. Pandharipande","doi":"10.1090/PSPUM/097.1/01682","DOIUrl":"https://doi.org/10.1090/PSPUM/097.1/01682","url":null,"abstract":"This article accompanies my lecture at the 2015 AMS summer institute in algebraic geometry in Salt Lake City. I survey the recent advances in the study of tautological classes on the moduli spaces of curves. After discussing the Faber-Zagier relations on the moduli spaces of nonsingular curves and the kappa rings of the moduli spaces of curves of compact type, I present Pixton's proposal for a complete calculus of tautological classes on the moduli spaces of stable curves. Several open questions are discussed. An effort has been made to condense a great deal of mathematics into as few pages as possible with the hope that the reader will follow through to the end.","PeriodicalId":412716,"journal":{"name":"Algebraic Geometry: Salt Lake City\n 2015","volume":"94 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126233546","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}
Pub Date : 2016-03-09DOI: 10.1090/PSPUM/097.2/01712
T. Pantev, G. Vezzosi
We review recent results and ongoing investigations of the symplectic and Poisson geometry of derived moduli spaces, and describe applications to deformation quantization of such spaces.
我们回顾了最近的结果和正在进行的研究的辛和泊松几何的衍生模空间，并描述了这些空间的变形量化的应用。
{"title":"Symplectic and Poisson derived geometry and\u0000 deformation quantization","authors":"T. Pantev, G. Vezzosi","doi":"10.1090/PSPUM/097.2/01712","DOIUrl":"https://doi.org/10.1090/PSPUM/097.2/01712","url":null,"abstract":"We review recent results and ongoing investigations of the symplectic and Poisson geometry of derived moduli spaces, and describe applications to deformation quantization of such spaces.","PeriodicalId":412716,"journal":{"name":"Algebraic Geometry: Salt Lake City\n 2015","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127084547","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}
Pub Date : 2015-12-30DOI: 10.1090/PSPUM/097.2/01703
H. Esnault
Those are the notes for the 2015 Summer Research Institute on Algebraic Geometry. We report on Deligne's finiteness theorem for $ell$-adic representations on smooth varieties defined over a finite field, on its crystalline version, and on how the geometric etale fundamental group of a smooth projective variety defined over a characteristic $p>0$ field controls crystals on the infinitesimal site and should control those on the crystalline site. v2: last results added to the report, and some typos corrected.
{"title":"Some fundamental groups in arithmetic\u0000 geometry","authors":"H. Esnault","doi":"10.1090/PSPUM/097.2/01703","DOIUrl":"https://doi.org/10.1090/PSPUM/097.2/01703","url":null,"abstract":"Those are the notes for the 2015 Summer Research Institute on Algebraic Geometry. We report on Deligne's finiteness theorem for $ell$-adic representations on smooth varieties defined over a finite field, on its crystalline version, and on how the geometric etale fundamental group of a smooth projective variety defined over a characteristic $p>0$ field controls crystals on the infinitesimal site and should control those on the crystalline site. v2: last results added to the report, and some typos corrected.","PeriodicalId":412716,"journal":{"name":"Algebraic Geometry: Salt Lake City\n 2015","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115524926","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}
Pub Date : 2015-07-13DOI: 10.1090/PSPUM/097.2/01704
Laurent Fargues
We begin by reviewing our joint work with J.-M. Fontaine about the fundamental curve of p-adic Hodge theory. We then explain our results obtained in [4] about the classification of G-bundles on this curve and its link with local class field theory. We finish by formulating conjectures that would extend those results.
{"title":"From local class field to the curve and vice\u0000 versa","authors":"Laurent Fargues","doi":"10.1090/PSPUM/097.2/01704","DOIUrl":"https://doi.org/10.1090/PSPUM/097.2/01704","url":null,"abstract":"We begin by reviewing our joint work with J.-M. Fontaine about the fundamental curve of p-adic Hodge theory. We then explain our results obtained in [4] about the classification of G-bundles on this curve and its link with local class field theory. We finish by formulating conjectures that would extend those results.","PeriodicalId":412716,"journal":{"name":"Algebraic Geometry: Salt Lake City\n 2015","volume":"11697 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125097435","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}
Pub Date : 2013-07-13DOI: 10.1090/PSPUM/097.1/01669
R. Berman
In the present paper and the companion paper [8] a probabilistic (statistical mechanical) approach to the study of canonical metrics and measures on a complex algebraic variety X is introduced. On any such variety with positive Kodaira dimension a canonical (birationally invariant) random point processes is defined and shown to converge in probability towards a canonical deterministic measure on X, coinciding with the canonical measure of Song-Tian and Tsuji. The proof is based on new large deviation principle for Gibbs measures with singular Hamiltonians which relies on an asymptotic submean inequality in large dimensions, proved in a companion paper. In the case of a variety X of general type we obtain as a corollary that the (possibly singular) K"ahler-Einstein metric on X with negative Ricci curvature is the limit of a canonical sequence of quasi-explicit Bergman type metrics. In the opposite setting of a Fano variety X we relate the canonical point processes to a new notion of stability, that we call Gibbs stability, which admits a natural algebro-geometric formulation and which we conjecture is equivalent to the existence of a K"ahler-Einstein metric on X and hence to K-stability as in the Yau-Tian-Donaldson conjecture.
{"title":"Kähler–Einstein metrics, canonical random\u0000 point processes and birational geometry","authors":"R. Berman","doi":"10.1090/PSPUM/097.1/01669","DOIUrl":"https://doi.org/10.1090/PSPUM/097.1/01669","url":null,"abstract":"In the present paper and the companion paper [8] a probabilistic (statistical mechanical) approach to the study of canonical metrics and measures on a complex algebraic variety X is introduced. On any such variety with positive Kodaira dimension a canonical (birationally invariant) random point processes is defined and shown to converge in probability towards a canonical deterministic measure on X, coinciding with the canonical measure of Song-Tian and Tsuji. The proof is based on new large deviation principle for Gibbs measures with singular Hamiltonians which relies on an asymptotic submean inequality in large dimensions, proved in a companion paper. In the case of a variety X of general type we obtain as a corollary that the (possibly singular) K\"ahler-Einstein metric on X with negative Ricci curvature is the limit of a canonical sequence of quasi-explicit Bergman type metrics. In the opposite setting of a Fano variety X we relate the canonical point processes to a new notion of stability, that we call Gibbs stability, which admits a natural algebro-geometric formulation and which we conjecture is equivalent to the existence of a K\"ahler-Einstein metric on X and hence to K-stability as in the Yau-Tian-Donaldson conjecture.","PeriodicalId":412716,"journal":{"name":"Algebraic Geometry: Salt Lake City\n 2015","volume":"76 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125329107","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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