Pub Date : 2017-01-03DOI: 10.1090/pspum/097.1/01681
A. Okounkov
This is an introduction to: (1) the enumerative geometry of rational curves in equivariant symplectic resolutions, and (2) its relation to the structures of geometric representation theory. Written for the 2015 Algebraic Geometry Summer Institute.
{"title":"Enumerative geometry and geometric\u0000 representation theory","authors":"A. Okounkov","doi":"10.1090/pspum/097.1/01681","DOIUrl":"https://doi.org/10.1090/pspum/097.1/01681","url":null,"abstract":"This is an introduction to: (1) the enumerative geometry of rational curves in equivariant symplectic resolutions, and (2) its relation to the structures of geometric representation theory. Written for the 2015 Algebraic Geometry Summer Institute.","PeriodicalId":412716,"journal":{"name":"Algebraic Geometry: Salt Lake City\n 2015","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125943048","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 : 2016-11-11DOI: 10.1090/PSPUM/097.1/01670
T. Bridgeland
This is a survey article on Hall algebras and their applications to the study of motivic invariants of moduli spaces of coherent sheaves on Calabi-Yau threefolds. It is a write-up of my talks at the 2015 Salt Lake City AMS Summer Research Institute and will appear in the Proceedings. The ideas presented here are mostly due to Joyce, Kontsevich, Reineke, Soibelman and Toda.
{"title":"Hall algebras and Donaldson-Thomas\u0000 invariants","authors":"T. Bridgeland","doi":"10.1090/PSPUM/097.1/01670","DOIUrl":"https://doi.org/10.1090/PSPUM/097.1/01670","url":null,"abstract":"This is a survey article on Hall algebras and their applications to the study of motivic invariants of moduli spaces of coherent sheaves on Calabi-Yau threefolds. It is a write-up of my talks at the 2015 Salt Lake City AMS Summer Research Institute and will appear in the Proceedings. The ideas presented here are mostly due to Joyce, Kontsevich, Reineke, Soibelman and Toda.","PeriodicalId":412716,"journal":{"name":"Algebraic Geometry: Salt Lake City\n 2015","volume":"291 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124896094","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 : 2016-10-11DOI: 10.1090/PSPUM/097.1/01683
Z. Patakfalvi
This is a survey for the 2015 AMS Summer Institute on Algebraic Geometry about the Frobenius type techniques recently used extensively in positive characteristic algebraic geometry. We first explain the basic ideas through simple versions of the fundamental definitions and statements, and then we survey most of the recent algebraic geometry results obtained using these techniques.
{"title":"Frobenius techniques in birational\u0000 geometry","authors":"Z. Patakfalvi","doi":"10.1090/PSPUM/097.1/01683","DOIUrl":"https://doi.org/10.1090/PSPUM/097.1/01683","url":null,"abstract":"This is a survey for the 2015 AMS Summer Institute on Algebraic Geometry about the Frobenius type techniques recently used extensively in positive characteristic algebraic geometry. We first explain the basic ideas through simple versions of the fundamental definitions and statements, and then we survey most of the recent algebraic geometry results obtained using these techniques.","PeriodicalId":412716,"journal":{"name":"Algebraic Geometry: Salt Lake City\n 2015","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123263114","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 : 2016-09-02DOI: 10.1090/PSPUM/097.2/01705
M. Gross, Bernd S Siebert
This contribution to the 2015 AMS Summer Institute in Algebraic Geometry (Salt Lake City) announces a general mirror construction. This construction applies to log Calabi-Yau pairs (X,D) with maximal boundary D or to maximally unipotent degenerations of Calabi-Yau manifolds. The new ingredient is a notion of "punctured Gromov-Witten invariant", currently in progress with Abramovich and Chen. The mirror to a pair (X,D) is constructed as the spectrum of a ring defined using the punctured invariants of (X,D). An analogous construction leads to mirrors of Calabi-Yau manifolds. This can be viewed as a generalization of constructions developed jointly with Hacking and Keel in the case of log CY surfaces and K3 surfaces.
{"title":"Intrinsic mirror symmetry and punctured\u0000 Gromov-Witten invariants","authors":"M. Gross, Bernd S Siebert","doi":"10.1090/PSPUM/097.2/01705","DOIUrl":"https://doi.org/10.1090/PSPUM/097.2/01705","url":null,"abstract":"This contribution to the 2015 AMS Summer Institute in Algebraic Geometry (Salt Lake City) announces a general mirror construction. This construction applies to log Calabi-Yau pairs (X,D) with maximal boundary D or to maximally unipotent degenerations of Calabi-Yau manifolds. The new ingredient is a notion of \"punctured Gromov-Witten invariant\", currently in progress with Abramovich and Chen. The mirror to a pair (X,D) is constructed as the spectrum of a ring defined using the punctured invariants of (X,D). An analogous construction leads to mirrors of Calabi-Yau manifolds. This can be viewed as a generalization of constructions developed jointly with Hacking and Keel in the case of log CY surfaces and K3 surfaces.","PeriodicalId":412716,"journal":{"name":"Algebraic Geometry: Salt Lake City\n 2015","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121093741","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 : 2016-08-23DOI: 10.1090/pspum/097.1/01676
D. Greb, Stefan Kebekus, Behrouz Taji
After a historical discussion of classical uniformisation results for Riemann surfaces, of problems appearing in higher dimensions, and of uniformisation results for projective manifolds with trivial or ample canonical bundle, we introduce the basic technical concepts and sketch the ideas of the proofs for recent uniformisation theorems for singular varieties obtained by the authors in collaboration with Thomas Peternell.
{"title":"Uniformisation of higher-dimensional minimal\u0000 varieties","authors":"D. Greb, Stefan Kebekus, Behrouz Taji","doi":"10.1090/pspum/097.1/01676","DOIUrl":"https://doi.org/10.1090/pspum/097.1/01676","url":null,"abstract":"After a historical discussion of classical uniformisation results for Riemann surfaces, of problems appearing in higher dimensions, and of uniformisation results for projective manifolds with trivial or ample canonical bundle, we introduce the basic technical concepts and sketch the ideas of the proofs for recent uniformisation theorems for singular varieties obtained by the authors in collaboration with Thomas Peternell.","PeriodicalId":412716,"journal":{"name":"Algebraic Geometry: Salt Lake City\n 2015","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125990385","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 : 2016-07-11DOI: 10.1090/pspum/097.2/01714
Takeshi Saito
We study the compatibility with proper push-forward of the characteristic cycles of a constructible complex on a smooth variety over a perfect field.
研究了光滑变化域中可构造复合体的特征环与正推的相容性。
{"title":"On the proper push-forward of the\u0000 characteristic cycle of a constructible\u0000 sheaf","authors":"Takeshi Saito","doi":"10.1090/pspum/097.2/01714","DOIUrl":"https://doi.org/10.1090/pspum/097.2/01714","url":null,"abstract":"We study the compatibility with proper push-forward of the characteristic cycles of a constructible complex on a smooth variety over a perfect field.","PeriodicalId":412716,"journal":{"name":"Algebraic Geometry: Salt Lake City\n 2015","volume":"122 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115883143","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 : 2016-06-30DOI: 10.1090/PSPUM/097.2/01706
Eric Katz, Joseph Rabinoff, David Zureick-Brown
We describe recent work connecting combinatorics and tropical/non-Archimedean geometry to Diophantine geometry, particularly the uniformity conjectures for rational points on curves and for torsion packets of curves. The method of Chabauty--Coleman lies at the heart of this connection, and we emphasize the clarification that tropical geometry affords throughout the theory of $p$-adic integration, especially to the comparison of analytic continuations of $p$-adic integrals and to the analysis of zeros of integrals on domains admitting monodromy.
{"title":"Diophantine and tropical geometry, and\u0000 uniformity of rational points on curves","authors":"Eric Katz, Joseph Rabinoff, David Zureick-Brown","doi":"10.1090/PSPUM/097.2/01706","DOIUrl":"https://doi.org/10.1090/PSPUM/097.2/01706","url":null,"abstract":"We describe recent work connecting combinatorics and tropical/non-Archimedean geometry to Diophantine geometry, particularly the uniformity conjectures for rational points on curves and for torsion packets of curves. The method of Chabauty--Coleman lies at the heart of this connection, and we emphasize the clarification that tropical geometry affords throughout the theory of $p$-adic integration, especially to the comparison of analytic continuations of $p$-adic integrals and to the analysis of zeros of integrals on domains admitting monodromy.","PeriodicalId":412716,"journal":{"name":"Algebraic Geometry: Salt Lake City\n 2015","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121333472","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 : 2016-06-28DOI: 10.1090/PSPUM/097.2/01698
David Ben-Zvi, D. Nadler
We introduce and survey a Betti form of the geometric Langlands conjecture, parallel to the de Rham form developed by Beilinson-Drinfeld and Arinkin-Gaitsgory, and the Dolbeault form of Donagi-Pantev, and inspired by the work of Kapustin-Witten in supersymmetric gauge theory. The conjecture proposes an automorphic category associated to a compact Riemann surface X and complex reductive group G is equivalent to a spectral category associated to the underlying topological surface S and Langlands dual group G^. The automorphic category consists of suitable C-sheaves on the moduli stack Bun_G(X) of G-bundles on X, while the spectral category consists of suitable O-modules on the character stack Loc_G^(S) of G^-local systems on S. The conjecture is compatible with and constrained by the natural symmetries of both sides coming from modifications of bundles and local systems. On the one hand, cuspidal Hecke eigensheaves in the de Rham and Betti sense are expected to coincide, so that one can view the Betti conjecture as offering a different "integration measure" on the same fundamental objects. On the other hand, the Betti spectral categories are more explicit than their de Rham counterparts and one might hope the conjecture is less challenging. The Betti program also enjoys symmetries coming from topological field theory: it is expected to extend to an equivalence of four-dimensional topological field theories, and in particular, the conjecture for closed surfaces is expected to reduce to the case of the thrice-punctured sphere. Finally, we also present ramified, quantum and integral variants of the conjecture, and highlight connections to other topics, including representation theory of real reductive groups and quantum groups.
{"title":"Betti Geometric Langlands","authors":"David Ben-Zvi, D. Nadler","doi":"10.1090/PSPUM/097.2/01698","DOIUrl":"https://doi.org/10.1090/PSPUM/097.2/01698","url":null,"abstract":"We introduce and survey a Betti form of the geometric Langlands conjecture, parallel to the de Rham form developed by Beilinson-Drinfeld and Arinkin-Gaitsgory, and the Dolbeault form of Donagi-Pantev, and inspired by the work of Kapustin-Witten in supersymmetric gauge theory. The conjecture proposes an automorphic category associated to a compact Riemann surface X and complex reductive group G is equivalent to a spectral category associated to the underlying topological surface S and Langlands dual group G^. The automorphic category consists of suitable C-sheaves on the moduli stack Bun_G(X) of G-bundles on X, while the spectral category consists of suitable O-modules on the character stack Loc_G^(S) of G^-local systems on S. The conjecture is compatible with and constrained by the natural symmetries of both sides coming from modifications of bundles and local systems. On the one hand, cuspidal Hecke eigensheaves in the de Rham and Betti sense are expected to coincide, so that one can view the Betti conjecture as offering a different \"integration measure\" on the same fundamental objects. On the other hand, the Betti spectral categories are more explicit than their de Rham counterparts and one might hope the conjecture is less challenging. The Betti program also enjoys symmetries coming from topological field theory: it is expected to extend to an equivalence of four-dimensional topological field theories, and in particular, the conjecture for closed surfaces is expected to reduce to the case of the thrice-punctured sphere. Finally, we also present ramified, quantum and integral variants of the conjecture, and highlight connections to other topics, including representation theory of real reductive groups and quantum groups.","PeriodicalId":412716,"journal":{"name":"Algebraic Geometry: Salt Lake City\n 2015","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129269887","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 : 2016-06-24DOI: 10.1090/pspum/097.1/01677
C. Hacon, J. McKernan, Chenyang Xu
Simons foundation [DMS-1300750, DMS-1265285]; NSF [0701101, 1200656, 1265263]; Simons foundation; Mathematische Forschungsinstitut Oberwolfach; National Science Fund for Distinguished Young Scholars grant from China [11425101]
{"title":"Boundedness of varieties of log general\u0000 type","authors":"C. Hacon, J. McKernan, Chenyang Xu","doi":"10.1090/pspum/097.1/01677","DOIUrl":"https://doi.org/10.1090/pspum/097.1/01677","url":null,"abstract":"Simons foundation [DMS-1300750, DMS-1265285]; NSF [0701101, 1200656, 1265263]; Simons foundation; Mathematische Forschungsinstitut Oberwolfach; National Science Fund for Distinguished Young Scholars grant from China [11425101]","PeriodicalId":412716,"journal":{"name":"Algebraic Geometry: Salt Lake City\n 2015","volume":"1996 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134549736","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 : 2016-06-06DOI: 10.1090/pspum/097.2/01715
Tam'as Szamuely, Gergely Z'abr'adi
A detailed presentation of Beilinson's approach to p-adic Hodge theory.
详细介绍了Beilinson的p-adic Hodge理论。
{"title":"The 𝑝-adic Hodge decomposition according to\u0000 Beilinson","authors":"Tam'as Szamuely, Gergely Z'abr'adi","doi":"10.1090/pspum/097.2/01715","DOIUrl":"https://doi.org/10.1090/pspum/097.2/01715","url":null,"abstract":"A detailed presentation of Beilinson's approach to p-adic Hodge theory.","PeriodicalId":412716,"journal":{"name":"Algebraic Geometry: Salt Lake City\n 2015","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129266087","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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