{"title":"$(1,1)$ forms with specified Lagrangian phase: a priori estimates and algebraic obstructions","authors":"Tristan C. Collins, Adam Jacob, S. Yau","doi":"10.4310/cjm.2020.v8.n2.a4","DOIUrl":null,"url":null,"abstract":"Let $(X,\\alpha)$ be a K\\\"ahler manifold of dimension n, and let $[\\omega] \\in H^{1,1}(X,\\mathbb{R})$. We study the problem of specifying the Lagrangian phase of $\\omega$ with respect to $\\alpha$, which is described by the nonlinear elliptic equation \\[ \\sum_{i=1}^{n} \\arctan(\\lambda_i)= h(x) \\] where $\\lambda_i$ are the eigenvalues of $\\omega$ with respect to $\\alpha$. When $h(x)$ is a topological constant, this equation corresponds to the deformed Hermitian-Yang-Mills (dHYM) equation, and is related by Mirror Symmetry to the existence of special Lagrangian submanifolds of the mirror. We introduce a notion of subsolution for this equation, and prove a priori $C^{2,\\beta}$ estimates when $|h|>(n-2)\\frac{\\pi}{2}$ and a subsolution exists. Using the method of continuity we show that the dHYM equation admits a smooth solution in the supercritical phase case, whenever a subsolution exists. Finally, we discover some stability-type cohomological obstructions to the existence of solutions to the dHYM equation and we conjecture that when these obstructions vanish the dHYM equation admits a solution. We confirm this conjecture for complex surfaces.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.8000,"publicationDate":"2015-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"76","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Cambridge Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4310/cjm.2020.v8.n2.a4","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 76
Abstract
Let $(X,\alpha)$ be a K\"ahler manifold of dimension n, and let $[\omega] \in H^{1,1}(X,\mathbb{R})$. We study the problem of specifying the Lagrangian phase of $\omega$ with respect to $\alpha$, which is described by the nonlinear elliptic equation \[ \sum_{i=1}^{n} \arctan(\lambda_i)= h(x) \] where $\lambda_i$ are the eigenvalues of $\omega$ with respect to $\alpha$. When $h(x)$ is a topological constant, this equation corresponds to the deformed Hermitian-Yang-Mills (dHYM) equation, and is related by Mirror Symmetry to the existence of special Lagrangian submanifolds of the mirror. We introduce a notion of subsolution for this equation, and prove a priori $C^{2,\beta}$ estimates when $|h|>(n-2)\frac{\pi}{2}$ and a subsolution exists. Using the method of continuity we show that the dHYM equation admits a smooth solution in the supercritical phase case, whenever a subsolution exists. Finally, we discover some stability-type cohomological obstructions to the existence of solutions to the dHYM equation and we conjecture that when these obstructions vanish the dHYM equation admits a solution. We confirm this conjecture for complex surfaces.