{"title":"Geometric pluripotential theory on Kähler\u0000 manifolds","authors":"Tam'as Darvas","doi":"10.1090/conm/735/14822","DOIUrl":"https://doi.org/10.1090/conm/735/14822","url":null,"abstract":"Finite energy pluripotential theory accommodates the variational theory of equations of complex Monge-Amp`ere type arising in K\"ahler geometry. Recently it has been discovered that many of the potential spaces involved have a rich metric geometry, effectively turning the variational problems in question into problems of infinite dimensional convex optimization, yielding existence results for solutions of the underlying complex Monge-Amp`ere equations. The purpose of this survey is to describe these developments from basic principles.","PeriodicalId":139005,"journal":{"name":"Advances in Complex Geometry","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126250829","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}
{"title":"Pluriharmonics in general potential\u0000 theories","authors":"F. R. Harvey, H. Lawson, Jr.","doi":"10.1090/conm/735/14824","DOIUrl":"https://doi.org/10.1090/conm/735/14824","url":null,"abstract":"The general purpose of this paper is to investigate the notion of \"pluriharmonics\" for the general potential theory associated to a convex cone $Fsubset {rm Sym}^2({bf R}^n)$. For such $F$ there exists a maximal linear subspace $Esubset F$, called the edge, and $F$ decomposes as $F=E oplus F_0$. The pluriharmonics or edge functions are $u$'s with $D^2u in E$. Many subequations $F$ have the same edge $E$, but there is a unique smallest such subequation. These are the focus of this investigation. Structural results are given. Many examples are described, and a classification of highly symmetric cases is given. Finally, the relevance of edge functions to the solutions of the Dirichlet problem is established.","PeriodicalId":139005,"journal":{"name":"Advances in Complex Geometry","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125629832","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}
{"title":"The Anomaly flow on unimodular Lie\u0000 groups","authors":"D. Phong, Sebastien Picard, Xiangwen Zhang","doi":"10.1090/conm/735/14828","DOIUrl":"https://doi.org/10.1090/conm/735/14828","url":null,"abstract":"The Hull-Strominger system for supersymmetric vacua of the heterotic string allows general unitary Hermitian connections with torsion and not just the Chern unitary connection. Solutions on unimodular Lie groups exploiting this flexibility were found by T. Fei and S.T. Yau. The Anomaly flow is a flow whose stationary points are precisely the solutions of the Hull-Strominger system. Here we examine its long-time behavior on unimodular Lie groups with general unitary Hermitian connections. We find a diverse and intricate behavior, which depends very much on the Lie group and the initial data.","PeriodicalId":139005,"journal":{"name":"Advances in Complex Geometry","volume":"486 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123557370","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}
{"title":"Orbifold regularity of weak Kähler-Einstein\u0000 metrics","authors":"Chi Li, G. Tian","doi":"10.1090/conm/735/14825","DOIUrl":"https://doi.org/10.1090/conm/735/14825","url":null,"abstract":"In this note, we prove that any non-collapsing and compact Gromov-Hausdorff limit of Kahler-Einstein manifolds is either smooth or is orbifold outside a subvariety of complex codimension at least 3.","PeriodicalId":139005,"journal":{"name":"Advances in Complex Geometry","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124981894","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}
{"title":"On orthogonal Ricci curvature","authors":"Lei Ni, F. Zheng","doi":"10.1090/conm/735/14827","DOIUrl":"https://doi.org/10.1090/conm/735/14827","url":null,"abstract":"In this paper we discuss some recent progresses in the study of compact Kähler manifolds with positive orthogonal Ricci curvature, a curvature condition defined as the difference between Ricci curvature and holomorphic sectional curvature. In the recent works by authors and the joint work of authors with Q. Wang the comparison theorems, vanishing theorems, and structural theorems for such manifolds have been proved. We also constructed examples of this type of manifolds, and give some classification results in low dimensions. 1. Orthogonal Ricci curvature Let (M, g) be a Kähler manifold of complex dimension n. Its orthogonal Ricci curvature Ric⊥ is defined by (cf. [21]): Ric⊥ XX = Ric(X,X)−R(X,X,X,X)/|X|, where X is a non-zero type (1, 0) tangent vector at a point x ∈ M. This curvature arises in the study of the comparison theorem for Kähler manifolds and the previous study of manifolds with so-called nonnegative quadratic orthogonal bisectional curvature (cf. [4], [26], [16], [5]). We refer the readers to [21] for a more detailed account on this topic. Clearly this curvature is closely related to Ricci curvature Ric and holomorphic sectional curvature H. It is natural to ask, what is the relationship between Ric⊥ and Ric or H (other than the obvious one that Ric⊥ + H = Ric for unit length tangent vectors), and what kind of compact complex manifolds M can admit Kähler metrics with Ric⊥ > 0 (or ≥ 0, or ≤ 0, or < 0, or ≡ 0) everywhere? In this paper, we will focus on the curvature condition Ric⊥ and pay particular attention to the class of compact Kähler manifolds with Ric⊥ > 0 everywhere, except in Section 2 where complete noncompact Kähler manifolds are also considered. Throughout this paper, we will assume that the complex dimension n ≥ 2 unless stated otherwise, since Ric⊥ ≡ 0 when n = 1. We start with the following observation. At a point x ∈ M, let us denote by S2n−1 x the unit sphere of all type (1, 0) tangent vector at x of unit length. By a classic result of Berger, The research of LN is partially supported by NSF grant DMS-1401500 and the “Capacity Building for Sci-Tech Innovation-Fundamental Research Funds”. The research of FZ is partially supported by a Simons Collaboration Grant 355557.","PeriodicalId":139005,"journal":{"name":"Advances in Complex Geometry","volume":"90 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117325985","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}
{"title":"Analysis of the Laplacian on the moduli space\u0000 of polarized Calabi-Yau manifolds","authors":"Zhiqin Lu, Hangjun Xu","doi":"10.1090/conm/735/14826","DOIUrl":"https://doi.org/10.1090/conm/735/14826","url":null,"abstract":"","PeriodicalId":139005,"journal":{"name":"Advances in Complex Geometry","volume":"43 4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131588941","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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