Pub Date : 2016-06-05DOI: 10.1090/PSPUM/097.2/01699
B. Bhatt
We present a semicontinuity result, proven in recent joint work with Morrow and Scholze, relating the mod $p$ singular cohomology of a smooth projective complex algebraic variety X to the de Rham cohomology of a smooth characteristic $p$ specialization of X: the rank of the former is bounded above by that of the latter. The path to this result passes through $p$-adic Hodge theory and perfectoid geometry, so we survey the relevant aspects of those subjects as well.
{"title":"Specializing varieties and their cohomology\u0000 from characteristic 0 to characteristic 𝑝","authors":"B. Bhatt","doi":"10.1090/PSPUM/097.2/01699","DOIUrl":"https://doi.org/10.1090/PSPUM/097.2/01699","url":null,"abstract":"We present a semicontinuity result, proven in recent joint work with Morrow and Scholze, relating the mod $p$ singular cohomology of a smooth projective complex algebraic variety X to the de Rham cohomology of a smooth characteristic $p$ specialization of X: the rank of the former is bounded above by that of the latter. The path to this result passes through $p$-adic Hodge theory and perfectoid geometry, so we survey the relevant aspects of those subjects as well.","PeriodicalId":412716,"journal":{"name":"Algebraic Geometry: Salt Lake City\n 2015","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128066233","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 : 2016-06-04DOI: 10.1090/PSPUM/097.1/01675
E. Gonz'alez, P. Solis, C. Woodward
We give an introduction to moduli stacks of gauged maps satisfying a stability conditition introduced by Mundet and Schmitt, and the associated integrals giving rise to gauged Gromov-Witten invariants. We survey various applications to cohomological and K-theoretic Gromov-Witten invariants.
{"title":"Stable gauged maps","authors":"E. Gonz'alez, P. Solis, C. Woodward","doi":"10.1090/PSPUM/097.1/01675","DOIUrl":"https://doi.org/10.1090/PSPUM/097.1/01675","url":null,"abstract":"We give an introduction to moduli stacks of gauged maps satisfying a stability conditition introduced by Mundet and Schmitt, and the associated integrals giving rise to gauged Gromov-Witten invariants. We survey various applications to cohomological and K-theoretic Gromov-Witten invariants.","PeriodicalId":412716,"journal":{"name":"Algebraic Geometry: Salt Lake City\n 2015","volume":"9 5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133834304","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 : 2016-06-01DOI: 10.1090/PSPUM/097.2/01701
L. Caporaso
In recent years a series of remarkable advances in tropical geometry and in non-archimedean geometry have brought new insights to the moduli theory of algebraic curves and their Jacobians. The goal of this survey, an expanded version of my talks at the 2015 AMS symposium in algebraic geometry and at AGNES 2016, is to present some of the results in this area.
{"title":"Tropical methods in the moduli theory of\u0000 algebraic curves","authors":"L. Caporaso","doi":"10.1090/PSPUM/097.2/01701","DOIUrl":"https://doi.org/10.1090/PSPUM/097.2/01701","url":null,"abstract":"In recent years a series of remarkable advances in tropical geometry and in non-archimedean geometry have brought new insights to the moduli theory of algebraic curves and their Jacobians. The goal of this survey, an expanded version of my talks at the 2015 AMS symposium in algebraic geometry and at AGNES 2016, is to present some of the results in this area.","PeriodicalId":412716,"journal":{"name":"Algebraic Geometry: Salt Lake City\n 2015","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124367388","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 an expository article. In the first part we recall the definition and a few results concerning singular Hermitian metrics on torsion-free coherent sheaves. They offer the perfect platform for the study of properties of direct images of twisted pluricanonical bundles which we will survey in the second part.
{"title":"Singualar Hermitian metrics and positivity of\u0000 direct images of pluricanonical bundles","authors":"Mihai Păun","doi":"10.1090/pspum/097.1/18","DOIUrl":"https://doi.org/10.1090/pspum/097.1/18","url":null,"abstract":"This is an expository article. In the first part we recall the definition and a few results concerning singular Hermitian metrics on torsion-free coherent sheaves. They offer the perfect platform for the study of properties of direct images of twisted pluricanonical bundles which we will survey in the second part.","PeriodicalId":412716,"journal":{"name":"Algebraic Geometry: Salt Lake City\n 2015","volume":"91 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121680523","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 : 2016-05-25DOI: 10.1090/PSPUM/097.1/01685
M. Popa
This is a survey of vanishing and positivity theorems for Hodge modules, and their recent applications to birational and complex geometry, expanding on my lecture at the 2015 AMS Summer Institute.
{"title":"Positivity for Hodge modules and geometric\u0000 applications","authors":"M. Popa","doi":"10.1090/PSPUM/097.1/01685","DOIUrl":"https://doi.org/10.1090/PSPUM/097.1/01685","url":null,"abstract":"This is a survey of vanishing and positivity theorems for Hodge modules, and their recent applications to birational and complex geometry, expanding on my lecture at the 2015 AMS Summer Institute.","PeriodicalId":412716,"journal":{"name":"Algebraic Geometry: Salt Lake City\n 2015","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129395009","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 : 2016-05-24DOI: 10.1090/PSPUM/097.1/01674
L. Ein, R. Lazarsfeld
This paper is a survey of recent work on the asymptotic behavior of the syzygies of a smooth complex projective variety as the positivity of the embedding line bundle grows. After a quick overview of results from the 1980s and 1990s concerning the linearity of the first few terms of a resolution, we discuss a non-vanishing theorem to the effect that from an asymptotic viewpoint, essentially all of the syzygy modules that could be non-zero are in fact non-zero. We explain the quick new proof of this result in the case of Veronese varieties due to Erman and authors, and we explore some results and conjectures about the asymptotics of Betti numbers. Finally we discuss the case of syzygies of weight one, and the gonality conjecture on the syzygies of curves of large degree. The exposition also discusses numerous open questions and conjectures.
{"title":"Syzygies of projective varieties of large\u0000 degree: Recent progress and open problems","authors":"L. Ein, R. Lazarsfeld","doi":"10.1090/PSPUM/097.1/01674","DOIUrl":"https://doi.org/10.1090/PSPUM/097.1/01674","url":null,"abstract":"This paper is a survey of recent work on the asymptotic behavior of the syzygies of a smooth complex projective variety as the positivity of the embedding line bundle grows. \u0000After a quick overview of results from the 1980s and 1990s concerning the linearity of the first few terms of a resolution, we discuss a non-vanishing theorem to the effect that from an asymptotic viewpoint, essentially all of the syzygy modules that could be non-zero are in fact non-zero. We explain the quick new proof of this result in the case of Veronese varieties due to Erman and authors, and we explore some results and conjectures about the asymptotics of Betti numbers. Finally we discuss the case of syzygies of weight one, and the gonality conjecture on the syzygies of curves of large degree. \u0000The exposition also discusses numerous open questions and conjectures.","PeriodicalId":412716,"journal":{"name":"Algebraic Geometry: Salt Lake City\n 2015","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122306778","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}
We report on progress in the qualitative study of rational points on rationally connected varieties over number fields, also examining integral points, zero-cycles, and non-rationally connected varieties. One of the main objectives is to highlight and explain the many recent interactions with analytic number theory.
{"title":"Rational points and zero-cycles on rationally\u0000 connected varieties over number fields","authors":"Olivier Wittenberg","doi":"10.1090/pspum/097.2/20","DOIUrl":"https://doi.org/10.1090/pspum/097.2/20","url":null,"abstract":"We report on progress in the qualitative study of rational points on rationally connected varieties over number fields, also examining integral points, zero-cycles, and non-rationally connected varieties. One of the main objectives is to highlight and explain the many recent interactions with analytic number theory.","PeriodicalId":412716,"journal":{"name":"Algebraic Geometry: Salt Lake City\n 2015","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133999174","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 : 2016-04-27DOI: 10.1090/PSPUM/097.1/01668
Arend Bayer
Over the last few years, wall-crossing for Bridgeland stability conditions has led to a large number of results in algebraic geometry, particular on birational geometry of moduli spaces. We illustrate some of the methods behind these result by reproving Lazarsfeld's Brill-Noether theorem for curves on K3 surfaces via wall-crossing. We conclude with a survey of recent applications of stability conditions on surfaces. The intended reader is an algebraic geometer with a limited working knowledge of derived categories. This article is based on the author's talk at the AMS Summer Institute on Algebraic Geometry in Utah, July 2015.
{"title":"Wall-crossing implies Brill-Noether\u0000 applications of stability conditions on\u0000 surfaces","authors":"Arend Bayer","doi":"10.1090/PSPUM/097.1/01668","DOIUrl":"https://doi.org/10.1090/PSPUM/097.1/01668","url":null,"abstract":"Over the last few years, wall-crossing for Bridgeland stability conditions has led to a large number of results in algebraic geometry, particular on birational geometry of moduli spaces. \u0000We illustrate some of the methods behind these result by reproving Lazarsfeld's Brill-Noether theorem for curves on K3 surfaces via wall-crossing. We conclude with a survey of recent applications of stability conditions on surfaces. \u0000The intended reader is an algebraic geometer with a limited working knowledge of derived categories. This article is based on the author's talk at the AMS Summer Institute on Algebraic Geometry in Utah, July 2015.","PeriodicalId":412716,"journal":{"name":"Algebraic Geometry: Salt Lake City\n 2015","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134209010","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 : 2016-04-25DOI: 10.1090/PSPUM/097.2/01702
R. Cavalieri, P. Johnson, H. Markwig, Dhruv Ranganathan
We explore the explicit relationship between the descendant Gromov--Witten theory of target curves, operators on Fock spaces, and tropical curve counting. We prove a classical/tropical correspondence theorem for descendant invariants and give an algorithm that establishes a tropical Gromov--Witten/Hurwitz equivalence. Tropical curve counting is related to an algebra of operators on the Fock space by means of bosonification. In this manner, tropical geometry provides a convenient "graphical user interface" for Okounkov and Pandharipande's celebrated GW/H correspondence. An important goal of this paper is to spell out the connections between these various perspectives for target dimension 1, as a first step in studying the analogous relationship between logarithmic descendant theory, tropical curve counting, and Fock space formalisms in higher dimensions.
{"title":"A graphical interface for the Gromov-witten\u0000 theory of curves","authors":"R. Cavalieri, P. Johnson, H. Markwig, Dhruv Ranganathan","doi":"10.1090/PSPUM/097.2/01702","DOIUrl":"https://doi.org/10.1090/PSPUM/097.2/01702","url":null,"abstract":"We explore the explicit relationship between the descendant Gromov--Witten theory of target curves, operators on Fock spaces, and tropical curve counting. We prove a classical/tropical correspondence theorem for descendant invariants and give an algorithm that establishes a tropical Gromov--Witten/Hurwitz equivalence. Tropical curve counting is related to an algebra of operators on the Fock space by means of bosonification. In this manner, tropical geometry provides a convenient \"graphical user interface\" for Okounkov and Pandharipande's celebrated GW/H correspondence. An important goal of this paper is to spell out the connections between these various perspectives for target dimension 1, as a first step in studying the analogous relationship between logarithmic descendant theory, tropical curve counting, and Fock space formalisms in higher dimensions.","PeriodicalId":412716,"journal":{"name":"Algebraic Geometry: Salt Lake City\n 2015","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127216426","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 : 2016-03-30DOI: 10.1090/PSPUM/097.2/01713
Alena Pirutka
This is a survey of recent examples of varieties that are not stably rational. We review the specialization method based on properties of the Chow group of zero-cycles used in these examples and explain the point of view of unramified cohomology for the construction of nontrivial stable invariants of the special fiber. In particular, we find an explicit formula for the Brauer group of fourfolds fibered in quadrics of dimension two over a rational surface.
{"title":"Varieties that are not stably rational,\u0000 zero-cycles and unramified cohomology","authors":"Alena Pirutka","doi":"10.1090/PSPUM/097.2/01713","DOIUrl":"https://doi.org/10.1090/PSPUM/097.2/01713","url":null,"abstract":"This is a survey of recent examples of varieties that are not stably rational. We review the specialization method based on properties of the Chow group of zero-cycles used in these examples and explain the point of view of unramified cohomology for the construction of nontrivial stable invariants of the special fiber. In particular, we find an explicit formula for the Brauer group of fourfolds fibered in quadrics of dimension two over a rational surface.","PeriodicalId":412716,"journal":{"name":"Algebraic Geometry: Salt Lake City\n 2015","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115235897","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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