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The deformed Hermitian-Yang-Mills equation in geometry and physics
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The Hermitian-Yang-Mills equation (HMM) is a fully nonlinear geometric PDE on Kahler manifolds which plays an important role in mirror symmetry.Abstract:
We provide an introduction to the mathematics and physics of the deformed Hermitian-Yang-Mills equation, a fully nonlinear geometric PDE on Kahler manifolds which plays an important role in mirror symmetry. We discuss the physical origin of the equation, and some recent progress towards its solution. In dimension 3 we prove a new Chern number inequality and discuss the relationship with algebraic stability conditions.read more
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Moment maps, nonlinear PDE, and stability in mirror symmetry
TL;DR: In this article, the deformed Hermitian-Yang-Mills (dHYM) equation is studied from the variational point of view via an infinite dimensional GIT problem mirror to Thomas' GIT picture for special Lagrangians.
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A numerical criterion for generalised Monge-Ampere equations on projective manifolds
Ved Datar,Vamsi Pritham Pingali +1 more
TL;DR: In this article, it was shown that generalised Monge-Ampere equations (a family of equations which includes the inverse Hessian equations like the J-equation, as well as the monge-ampere equation) on projective manifolds have smooth solutions if certain intersection numbers are positive.
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On J-equation
TL;DR: In this paper, it was shown that for any constant scalar curvature Kahler metric, there exists a constant curvature metric that satisfies the J-equation if and only if it is uniformly J-stable.
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The deformed Hermitian Yang-Mills equation on three-folds
TL;DR: In this article, the authors proved an existence result for the deformed Hermitian Yang-Mills equation for the full admissible range of the phase parameter on compact complex three-folds conditioned on a necessary subsolution condition.
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On The Ricci Curvature of a Compact Kahler Manifold and the Complex Monge-Ampere Equation, I*
TL;DR: In this paper, the Ricci form of some Kahler metric is shown to be closed and its cohomology class must represent the first Chern class of M. This conjecture of Calabi can be reduced to a problem in non-linear partial differential equation.
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A Pair of Calabi-Yau manifolds as an exactly soluble superconformal theory
TL;DR: In this paper, the prepotentials and geometry of the moduli spaces for a Calabi-Yau manifold and its mirror were derived and all the sigma model corrections to the Yukawa couplings and moduli space metric were obtained.
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Scalar Curvature and Stability of Toric Varieties
TL;DR: In this paper, a stability condition for a polarised algebraic variety is defined and a conjecture relating this to the existence of a Kahler metric of constant scalar curvature.