Study resolves conjectures on hypercritical deformed Hermitian-Yang-Mills equation.
problem Resolving conjectures on hypercritical deformed Hermitian-Yang-Mills equation.
method Study compact Kähler manifolds and resolves conjectures of Collins-Yau.
result Resolves two conjectures of Collins-Yau.
Study on a deformed Hermitian-Yang-Mills equation on compact Kähler manifolds.
problem Existence of solutions to the hypercritical deformed Hermitian-Yang-Mills equation.
method Introduce coerciveness and properness of the J-functional on almost calibrated (1,1)-forms.
result Equivalence of coerciveness and properness to the existence of solutions.
Paper proves solutions for deformed Hermitian-Yang-Mills equation on almost Hermitian manifolds.
problem Existence of solutions for deformed Hermitian-Yang-Mills equation on almost Hermitian manifolds.
method Derive a priori estimates under the existence of an admissible C-subsolution; prove existence of solutions under the condition of existence of a supersolution. result Proves existence of solutions for the deformed Hermitian-Yang-Mills equation.
Paper solves a complex equation for smooth domains.
problem Investigates a specific type of parabolic equation in complex domains.
method Uses J-functional to prove solution convergence.
result Proves the convergence of solutions to the equation.
Introduces new equations linking Kähler-Einstein and Hermitian-Yang-Mills theories.
problem Existence of solutions to coupled Kähler-Einstein and Hermitian-Yang-Mills equations.
method Moment map interpretation, Futaki invariant, Matsushima-Lichnerowicz theorem, deformation results.
result Nontrivial solutions produced under certain conditions.
Study deformed Hermitian-Yang-Mills equation on complex projective space blowup.
problem Solving the deformed Hermitian-Yang-Mills equation on complex projective space blowup.
method Expressed the equation as an ODE and solved it using combinatorial methods under an algebraic stability condition.
result Evidence supporting a conjecture on general compact Kahler manifolds.
Surveying progress on deformed Hermitian-Yang-Mills equation and its connections to stability.
problem Solvability of the deformed Hermitian-Yang-Mills equation and its relation to geometric stability.
method Utilizing geometric invariant theory (GIT) and Bridgeland stability theory to analyze the equation.
result On the blow-up of \(\mathbb{P}^2\), line bundles admitting a solution of the deformed Hermitian-Yang-Mills equation are Bridgeland stable, but not conversely.
Study disproves conjecture about Hermitian-Yang-Mills solutions.
problem Disproving conjecture about Hermitian-Yang-Mills solutions.
method Analyzes real (1,1)-classes on compact Kähler manifolds.
result Proves conjecture is false by showing proper subset.
Paper proves solvability condition for complex equation on special submanifolds.
problem Solvability condition for supercritical deformed Hermitian-Yang-Mills equation.
method Used integrals on subvarieties to provide necessary and sufficient condition.
result Confirms mirror version of Thomas-Yau conjecture about special Lagrangian submanifolds.
The paper proves conditions for solutions of J-equation and deformed Hermitian-Yang-Mills equation on holomorphic submersions.
problem Conditions for existence of solutions of J-equation and deformed Hermitian-Yang-Mills equation on holomorphic submersions. method Adiabatic limit technique
result Conditions for existence of solutions of J-equation and deformed Hermitian-Yang-Mills equation on holomorphic submersions. New equation approximates Kähler potentials using Hermitian-Yang-Mills metrics.
problem Approximating Kähler potentials in complex domains.
method New Wess-Zumino-Witten type equation and Berndtsson's theorem on direct image bundles.
result Approximation of Kähler potentials by Hermitian-Yang-Mills metrics.
Study on a nonlinear elliptic equation on compact Hermitian manifolds.
problem Solving a nonlinear elliptic equation on compact Hermitian manifolds.
method Investigation of the deformed Hermitian-Yang-Mills equation.
result Analysis of eigenvalues and their relation to the equation.
Proves existence and uniqueness of weak solutions for specific equations.
problem Existence and uniqueness of solutions for generalized Monge-Ampère and deformed Hermitian-Yang-Mills equations.
method Combines viscosity-theoretic and pluripotential-theoretic techniques.
result Existence and uniqueness of weak solutions in boundary cases.
Solves supercritical dHYM on projective manifolds with specific conditions.
problem Solving supercritical deformed Hermitian Yang-Mills equation on compact projective manifolds.
method Extends Gao Chen's result to non-constant twisting functions, proving solvability under certain conditions.
result Solvability of the twisted supercritical dHYM equation on compact projective manifolds.
New examples of deformed Hermitian-Yang-Mills connections found.
problem Constructing deformed Hermitian-Yang-Mills connections on manifolds.
method Constructed first higher rank, irreducible deformed Hermitian-Yang-Mills connections in both small and large radius regimes.
result Existence of solutions with any possible angle and ruling out some stability conditions.
Paper establishes estimates for nonlinear equations on compact manifolds.
problem Estimating solutions to fully nonlinear equations with gradient terms on compact almost Hermitian manifolds.
method Establishes second order estimates and proves existence of solutions for specific equations.
result Proves existence of solutions for various equations, including Monge-Ampère and Hessian equations.
Proves solvability of general inverse σ_k equations with constant coefficients.
problem Solvability of general inverse σ_k equations with constant coefficients.
method Proves existence of unique solution if a C-subsolution exists.
result Confirms analytical conjecture for deformed Hermitian--Yang--Mills equation.
Study higher rank deformed Hermitian-Yang-Mills equations for stable vector bundles.
problem Stability conditions for higher rank vector bundles over complex manifolds.
method Establish equivalence between dHYM equations and Z-stability. result Equivalence between dHYM solutions and Z-stability for vortex type bundles. Study higher rank deformed Hermitian-Yang-Mills equations for stable vector bundles.
problem Stability conditions for higher rank vector bundles over complex manifolds.
method Establish equivalence between dHYM equations and Z-stability. result Equivalence between dHYM solutions and Z-stability for vortex type bundles. The paper confirms the solvability of a complex equation for a 4D manifold.
problem Solvability of a deformed Hermitian--Yang--Mills equation on a 4D Kähler manifold.
method Used eigenvalues and topological constants to prove the existence of a C-subsolution.
result The existence of a C-subsolution implies the solvability of the deformed Hermitian--Yang--Mills equation when the complex dimension is 4 and θ is close to π.
We prove an existence result for the deformed Hermitian Yang-Mills equation for the full admissible range of the phase parameter, i.e., θ^∈(2π,23π), on compact complex three-folds conditioned on a necessary subsolution condition. Our proof hinges on a delicate analysis of a new continuity path …
We study weak geodesics in the space of potentials for the deformed Hermitian-Yang-Mills equation. The geodesic equation can be formulated as a degenerate elliptic equation, allowing us to employ nonlinear Dirichlet duality theory, as developed by Harvey-Lawson. By exploiting the convexity of the level sets of the Lagr…
Paper confirms conjecture for projective manifolds in supercritical phase.
problem Stability condition for deformed Hermitian-Yang-Mills equation.
method Establishes stability result not involving uniform constants.
result Confirms conjecture for projective manifolds in supercritical phase.
In this paper, we study the deformed Hermitian-Yang-Mills equation on compact Kähler manifold with non-negative orthogonal bisectional curvature. We prove that the curvatures of deformed Hermitian-Yang-Mills metrics are parallel with respect to the background metric if there exists a positive constant C such that $-\…
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 …
We introduce Z-critical connections for holomorphic vector bundles and prove their existence under stability conditions.
problem Existence of Z-critical connections for holomorphic vector bundles. method Associated geometric PDEs to Bridgeland stability conditions and used infinite dimensional moment maps.
result In the large volume limit, a sufficiently smooth holomorphic vector bundle admits a Z-critical connection if and only if it is asymptotically Z-stable. Paper generalizes sub-slope definition and solves complex equations on compact manifolds.
problem Solving complex equations on compact almost Hermitian manifolds.
method Generalized sub-slope definition and proved existence of solutions for a class of equations.
result Solved complex Hessian quotient and deformed Hermitian-Yang-Mills equations.
Investigates J-equation on holomorphic vector bundles over Kähler manifolds.
problem Analyzes properties and solutions of J-equation on holomorphic vector bundles. method Introduces and studies J-equation, provides algebraic and numerical criteria. result Provides an algebraic condition (asymptotic J-stability) and a numerical criterion for vortex bundles. Study of deformed Hermitian Yang-Mills equations with variable Kähler metrics.
problem Solving special Lagrangian type equations with variable metrics.
method Introducing extended gauge group to couple moment maps and scalar curvature.
result Solutions satisfy a mixture of K-stability and Bridgeland-type stability.
Introduces a new PDE involving differential forms for Kähler geometry.
problem Solving a unified PDE for various important equations in Kähler geometry.
method Introduces a fully nonlinear PDE with differential form Λ and proves solvability conditions.
result Generalizes previous works and proves a conjecture for the dHYM equation.
Introduce a flow for prescribed Hermitian-Yang-Mills tensors
problem Solve the prescribed Hermitian-Yang-Mills tensor equation
method Long-time convergence of the flow
result Uniform C0-estimate of the flow We study and construct non-abelian hermitian Yang-Mills (HYM) instantons on Calabi-Yau cones. By means of a particular isometry preserving ansatz, the HYM equations are reduced to a novel Higgs-Yang-Mills flow on the Einstein-Kahler base. For any 2d-dimensional Calabi-Yau cone, we find explicit solutions of the flow eq…
We review the notions of (weak) Hermitian-Yang-Mills structure and approximate Hermitian-Yang-Mills structure for Higgs bundles. Then, we construct the Donaldson functional for Higgs bundles over compact Kähler manifolds and we present some basic properties of it. In particular, we show that its gradient flow can be wr…
The paper studies limits of flows on Kähler surfaces, proving convergence to solutions of equations.
problem Analyzing limits of flows on Kähler surfaces and their convergence to solutions of equations.
method Using a property of limits of viscosity subsolutions.
result Proves convergence of flows to weak solutions of the Monge-Ampère equation.
Solved Demailly systems for Vortex bundles on manifolds.
problem Equivalence of Hartshorne ampleness and Griffiths positivity for vector bundles.
method Applied the continuity method to prove smooth solutions for the Vortex bundle.
result Smooth solutions exist for the proposed Demailly systems.
New exponential decay estimate for Hermitian Yang-Mills metrics near branch points.
problem Understanding the behavior of Hermitian Yang-Mills metrics near branch points.
method Local radial solutions, global gluing construction, exponential estimate near branch points.
result Exponential decay estimate for local radial solutions near branch points.
Construct Hermitian-Einstein metrics on stable holomorphic vector bundles using dynamical methods.
problem Constructing Hermitian-Einstein metrics on stable holomorphic vector bundles
method Dynamical construction
result Provided a dynamical construction of Hermitian-Einstein metrics on stable holomorphic vector bundles
We review the notions of (weak) Hermitian-Yang-Mills structure and approximate Hermitian-Yang-Mills structure for Higgs bundles. Then, we construct the Donaldson functional for Higgs bundles over compact Kähler manifolds and we present some basic properties of it. In particular, we show that its gradient flow can be wr…
The paper solves the dHYM equation on rational homogeneous varieties using Lie theory.
problem Solving the deformed Hermitian Yang-Mills equation on rational homogeneous varieties.
method Using Lie theory to describe the Lagrangian phase and characterize solutions.
result Characterization of all supercritical and hypercritical homogeneous solutions of the dHYM equation.
Characterizes solutions to Z-critical equations on surfaces using effective conditions.
problem Characterizing solutions to Z-critical equations on compact Kähler surfaces.
method Uses effective conditions and Picard number bounds to characterize solutions.
result Characterizes optimally destabilizing curves for Donaldson's J-equation and deformed Hermitian Yang-Mills equation.
The note proves a metric equivalence for stable bundles on surfaces.
problem Understanding stability conditions and metrics on complex projective surfaces.
method Analyzing stability in the large scaling limit and proving equivalence with deformed Hermitian-Yang-Mills metrics.
result Equivalence of stability and deformed Hermitian-Yang-Mills metrics for smooth projective surfaces.
New system solves curvature for ample vector bundles, proving Griffiths conjecture.
problem Proving Griffiths conjecture on vector bundle positivity.
method Proposes Hermitian-Yang-Mills elliptic system for curvature.
result Solutions provide metrics with positive curvature in Griffiths sense.
The paper finds explicit instantons on a specific 6-manifold.
problem Finding explicit solutions to the Hermitian Yang-Mills equations.
method Dimensional reduction and reformulation of the equations on quotient spaces.
result Explicit abelian instantons and deformed Hermitian Yang-Mills connections are found.
The paper introduces new equations in Kähler geometry and proves their solutions and convexity.
problem Solving equations in Kähler geometry and understanding their geometric implications.
method Using moment map pictures to motivate and prove solutions for the equations.
result The Mabuchi functional for certain equations is shown to be convex.
For any locally defined G2−structure and Hermitian-Yang-Mills connection on S6, the G2−monopole equation always admits a local solution that is asymptotic to the HYM-connection.
We prove a priori estimates for a generalised Monge-Ampère PDE with "non-constant coefficients" thus improving a result of Sun in the Kähler case. We apply this result to the deformed Hermitian Yang-Mills (dHYM) equation of Jacob-Yau to obtain an existence result and a priori estimates for some ranges of the phase angl…
The paper studies conditions for graphs connecting level sets of harmonic polynomials.
problem Conditions for graphs connecting level sets of harmonic polynomials.
method Algebraic properties and Kempf-Ness functional construction.
result Stability condition equivalent to the existence of a solution to the deformed Hermitian-Yang-Mills equation.
We address the issue why Calabi-Yau manifolds exist with a mirror pair. We observe that the irreducible spinor representation of the Lorentz group Spin(6) requires us to consider the vector spaces of two-forms and four-forms on an equal footing. The doubling of the two-form vector space due to the Hodge duality doubles…