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.
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.
Construct a Hermitian metric on non-Hermitian Yang--Mills moduli spaces near the Hermitian locus.
problem Moduli space of non-Hermitian Yang--Mills connections over a compact Kähler manifold
method Using normalized harmonic metrics
result Near the Hermitian locus, the unobstructed locus carries an almost hypercomplex structure compatible with the associated Riemannian metric.
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.
Study on hermitian Yang-Mills connections on blown-up manifolds.
problem Analyzing hermitian Yang-Mills connections on blown-up Kähler manifolds.
method Investigates connections for pullback vector bundles under specific conditions.
result Provides numerical criterion for convergence of hermitian Yang-Mills connections.
Study the long-time behavior of Hermitian-Yang-Mills flow on non-Kähler manifolds.
problem Understanding the long-time behavior of Hermitian-Yang-Mills flow on non-Kähler manifolds.
method Monotonicity of eigenvalues of mean curvature, convergence to geometric invariants.
result Eigenvalues of mean curvature converge to geometric invariants in the Gauduchon case.
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 Compactifies moduli spaces of Hermitian-Yang-Mills connections on balanced manifolds.
problem Analyzing Ω-Yang-Mills connections on Riemannian manifolds. method Extending known results on Yang-Mills connections to Ω-Yang-Mills connections, proving weak compactness and removable singularity theorems. result Compactification of moduli spaces of smooth Hermitian-Yang-Mills connections on unitary bundles over balanced manifolds.
Paper solves tensor problem for holomorphic vector bundles.
problem Existence of Hermitian metrics with prescribed Hermitian-Yang-Mills tensors.
method New comparison theorem for Hermitian-Yang-Mills tensors.
result Existence of unique smooth Hermitian metric for any positive-definite tensor.
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.
Characterizes tangent cones for specific connections on reflexive sheaves.
problem Analyzing tangent cones of admissible Hermitian-Yang-Mills connections over reflexive sheaves.
method Algebro-geometric characterization of analytic tangent cones.
result Complete characterization of tangent cones for admissible Hermitian-Yang-Mills connections over reflexive sheaves.
The paper solves a problem related to Higgs bundles and Hermitian metrics.
problem Existence of Hermitian metrics with prescribed Hermitian-Yang-Mills tensors for Higgs bundles.
method Solving the prescribed Hermitian-Yang-Mills tensor problem for Higgs bundles over compact complex manifolds.
result For any Hermitian positive definite tensor, there exists a unique smooth Hermitian metric on the Higgs bundle.
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.
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.
Study on existence of harmonic metrics for non-Hermitian Yang-Mills bundles.
problem Existence of harmonic metrics in non-Hermitian Yang-Mills bundles.
method Examined compact Kähler manifolds and equivalence to semisimplicity of NHYM bundles.
result Existence of harmonic metrics is equivalent to semisimplicity of NHYM 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…
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 hermitian Yang-Mills connections on pullback bundles for holomorphic submersions.
problem Investigate hermitian Yang-Mills connections on pullback bundles for holomorphic submersions.
method Obtain a criterion for the existence of hermitian Yang-Mills connections on pullback bundles, using intersection numbers on the base.
result Determine conditions under which pullback bundles of stable or unstable bundles remain stable or unstable for adiabatic classes.
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.
New criterion found for Hermitian-Yang-Mills metrics on non-compact Kähler manifolds.
problem Existence of Hermitian-Yang-Mills metrics on non-compact Kähler manifolds.
method Algebraic criterion and stability condition introduced.
result New stability condition is both sufficient and necessary for the existence of Hermitian-Yang-Mills metrics.
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.
Study on HYM connections on Kähler manifolds, calculating moduli space virtual dimension.
problem Calculating the moduli space of Hermitian-Yang-Mills connections.
method Analytic proof of stability of Higgs bundles on compact Kähler manifolds.
result Virtual dimension of the moduli space of HYM connections calculated.
The paper examines inequalities for Chern numbers on specific 4D Kähler manifolds.
problem Investigating Chern number inequalities on 4D Kähler manifolds with deformed Hermitian-Yang-Mills metrics.
method Analyzing 4D Kähler manifolds with deformed Hermitian-Yang-Mills metrics under the condition θ^∈(π,2π). result Established Chern number inequalities for the specified 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.
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.
Vortices and coupled vortices arise from Yang-Mills-Higgs theories and can be viewed as generalizations or analogues to Yang-Mills connections and, in particular, Hermitian-Yang-Mills connections. We proved an analytic compactification of the moduli spaces of vortices and coupled vortices on hermitian vector bundles ov…
We generalize the Hitchin-Kobayashi correspondence between semistability and the existence of approximate Hermitian-Yang-Mills structures to the case of principal Higgs bundles. We prove that a principal Higgs bundle on a compact Kaehler manifold, with structure group a connected linear algebraic reductive group, is se…
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…
In this paper, we analyze the asymptotic behaviour of the Hermitian-Yang-Mills flow over a compact non-Kähler manifold (X,g) with the Hermitian metric g satisfying the Gauduchon and Astheno-Kähler condition.
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.
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
Consider a vector bundle over a Kähler manifold which admits a Hermitian Yang-Mills connection. We show that the pullback bundle on the blowup of the Kähler manifold at a collection of points also admits a Hermitian Yang-Mills connection, for Kähler classes on the blowup which make the exceptional divisors small. Our p…
We construct local examples of singular Hermitian Yang-Mills connections over B1⊂C3 with uniformly bounded L2-energy, but the number of essential singular points can be arbitrarily large.
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. 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.
In this paper, we study the curvature estimate of the Hermitian-Yang-Mills flow on holomorphic vector bundles. In one simple case, we show that the curvature of the evolved Hermitian metric is uniformly bounded away from the analytic subvariety determined by the Harder-Narasimhan-Seshadri filtration of the holomorphic …
We give a simple direct proof of uniqueness of tangent cones for singular projectively Hermitian Yang-Mills connections on reflexive sheaves at isolated singularities modelled on μ-polystable holomorphic bundles over Pn−1.
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.
We first study the degeneration of a sequence of Hermitian-Yang-Mills metrics with respect to a sequence of balanced metrics on a Calabi-Yau threefold X^ that degenerates to the balanced metric constructed by Fu, Li, and Yau on the complement of finitely many (-1,-1)-curves in X^. Then under some assumpti…
Paper proves Hermitian-Yang-Mills metrics for stable Kähler bundles.
problem Existence of Hermitian-Yang-Mills metrics for Kähler vector bundles.
method Analyzes slope polystability and uses Kähler currents with singularities.
result Validates existence of Hermitian-Yang-Mills metrics for stable bundles and nef-big currents.
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.
We give an example of a homogeneous reflexive sheaf over C3 which admits a non-conical Hermitian Yang-Mills connection. This is expected to model bubbling phenomenon along complex codimension 2 submanifolds when the Fueter section takes zero value.
The flow proves a theorem for Fano manifolds.
problem Proving a theorem for Fano manifolds using the prescribed Hermitian-Yang-Mills flow.
method Using the prescribed Hermitian-Yang-Mills flow to prove the Donaldson-Uhlenbeck-Yau theorem.
result The flow converges to a Hermitian metric satisfying the prescribed tensor condition.
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.
We study Yang-Mills connections on holomorphic bundles over complex Kähler manifolds of arbitrary dimension, in the spirit of Hitchin's and Simpson's study of flat connections. The space of non-Hermitian Yang-Mills (NHYM) connections has dimension twice the space of Hermitian Yang-Mills connections, and is locally isom…
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.
In this paper we study the analytic tangent cones of admissible Hermitian-Yang-Mills connections near a homogeneous singularity of a reflexive sheaf, and relate it to the Harder-Narasimhan-Seshadri filtration. We also give an algebro-geometric characterization of the bubbling set. This strengthens our previous result.