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.
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 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 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.
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.
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 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.
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.
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.
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.
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
Investigates admissible metrics on compact Kähler varieties and their stability.
problem Existence of admissible metrics on compact Kähler varieties and their stability.
method Analyzes admissible Hermitian metrics and Hermitian-Yang-Mills metrics on slope stable coherent sheaves.
result Existence of admissible metrics and Hermitian-Yang-Mills metrics under certain conditions.
The paper examines the stability of a specific flow on complex manifolds.
problem Stability of line bundle mean curvature flow on complex manifolds.
method Analyzes the convergence of the line bundle mean curvature flow to a deformed Hermitian-Yang-Mills metric.
result The flow converges exponentially to the deformed Hermitian-Yang-Mills metric in the C∞ sense. 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.
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.
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 $-\…
Solutions to Strominger system found for square of Kähler class.
problem Finding solutions to Strominger system with specific balanced classes.
method Deforming Calabi-Yau and Hermitian-Yang-Mills metrics.
result Classes that are squares of Kähler metrics admit solutions.
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 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.
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 …
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.
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.
We prove an analogue of the Donaldson-Uhlenbeck-Yau theorem for asymptotically cylindrical Kähler manifolds: If E is a reflexive sheaf over an ACyl Kähler manifold, which is asymptotic to a μ-stable holomorphic vector bundle, then it admits an asymptotically translation-invariant protectively Hermitian Ya…
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.
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.
A new class of Higgs bundles is introduced in a natural setting. Existence and nonexistence results for Higgs-Hermitian-Yang-Mills metrics are proved.
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.
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.
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.
This paper begins to study the limiting behavior of a family of Hermitian Yang-Mills (HYM for brevity) metrics on a class of rank two slope stable vector bundles over a product of two elliptic curves with Kähler metrics ωε when ε→0. Here ωε are flat and have areas ε and ε−1 on the two elliptic curves …
Study shows stability of tangent bundle through conifold transitions.
problem Stability of tangent bundle through conifold transitions.
method Hermitian-Yang-Mills metric and conformally balanced metrics.
result Tangent bundle T1,0Xt admits a Hermitian-Yang-Mills metric Ht near vanishing cycles of Xt. 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 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.
Study Spin(7) instantons and HYM connections on Stenzel metric.
problem Relationship between Spin(7) instantons and HYM connections.
method Reduced to ODEs, analyzed invariant solutions, constructed moduli space.
result Nonexistence of global invariant Spin(7) instantons, one parameter family of irreducible instantons.
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.
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.
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…
Let X→P1 be an elliptically fibered K3 surface, admitting a sequence ωi of Ricci-flat metrics collapsing the fibers. Let V be a holomorphic SU(n) bundle over X, stable with respect to ωi. Given the corresponding sequence Ξi of Hermitian-Yang-Mills connections on V, we prove …
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.
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.
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.
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 study quiver gauge theories on the round and squashed seven-spheres, and orbifolds thereof. They arise by imposing G-equivariance on the homogeneous space G/H=SU(4)/SU(3) endowed with its Sasaki-Einstein structure, and G/H=Sp(2)/Sp(1) as a 3-Sasakian manifold. In both cases …
We consider the Toda systems of VHS type with singular sources and provide a criterion for the existence of solutions with prescribed asymptotic behaviour near singularities. We also prove the uniqueness of solution. Our approach uses Simpson's theory of constructing Higgs-Hermitian-Yang-Mills metrics from stability.