The paper studies the modified J-equation on Kähler manifolds.
problem Solvability of the modified J-equation on compact Kähler manifolds.
method Characterization of solvability via coercivity of the modified J-functional; Nakai-Moishezon criterion formulation and verification.
result Extension of existence criteria for extremal Kähler metrics.
The paper proves conditions for solutions of J J J -equation and deformed Hermitian-Yang-Mills equation on holomorphic submersions.
problem Conditions for existence of solutions of J J J -equation and deformed Hermitian-Yang-Mills equation on holomorphic submersions. method Adiabatic limit technique
result Conditions for existence of solutions of J J J -equation and deformed Hermitian-Yang-Mills equation on holomorphic submersions. Characterizes solvability of J-equation on Kähler surfaces with singularities.
problem Solvability of J-equation on Kähler surfaces with Poincaré type singularities.
method Two-parameter continuity path for J-equation, Kähler metrics with Poincaré type singularities.
result Existence of Poincaré type solutions implies boundedness of K-energy.
Investigates J J J -equation on holomorphic vector bundles over Kähler manifolds.
problem Analyzes properties and solutions of J J J -equation on holomorphic vector bundles. method Introduces and studies J J J -equation, provides algebraic and numerical criteria. result Provides an algebraic condition (asymptotic J J J -stability) and a numerical criterion for vortex bundles. New criterion for solving inverse Hessian equations, including J-equation.
problem Existence of solutions to inverse Hessian equations, including J-equation.
method Stability of pairs in the sense of Paul, formulated in terms of GIT criterion.
result New numerical criterion for existence of solutions to inverse Hessian equations.
Proof that certain complex equations have only simple solutions.
problem Characterizing solutions to complex equations.
method Analyzing fully nonlinear elliptic operators.
result Entire smooth solutions are quadratic polynomials.
This paper proves a Nakai-Moishezon criterion for complex Hessian equations.
problem The solvability of complex Hessian equations on Kähler manifolds.
method Establishing a Nakai-Moishezon criterion for Kähler classes on analytic Kähler varieties.
result Proves Lejmi-Szekelyhidi's conjecture for the J J J -equation. In this paper, we prove that for any Kähler metrics ω 0 ω_0 ω 0 and χ χ χ on M M M , there exists ω φ = ω 0 + − 1 ∂ ∂ ˉ φ > 0 ω_\varphi=ω_0+\sqrt{-1}\partial\bar\partial\varphi>0 ω φ = ω 0 + − 1 ∂ ∂ ˉ φ > 0 satisfying the J-equation t r ω φ χ = c \mathrm{tr}_{ω_\varphi}χ=c tr ω φ χ = c if and only if ( M , [ ω 0 ] , [ χ ] ) (M,[ω_0],[χ]) ( M , [ ω 0 ] , [ χ ]) is uniformly J-stable. As a corollary, we can find many constant scalar curvature Kähler met…
Solves open problems for fully nonlinear elliptic equations on manifolds.
problem Solving fully nonlinear elliptic equations on manifolds.
method Analytic slope invariant and Nakai-Moishezon criterion.
result Solves open problems including hessian and hessian quotient equations.
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.
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.
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.
Proves smooth solutions for generalised Monge-Ampère equations on projective manifolds.
problem Existence of smooth solutions for generalised Monge-Ampère equations on projective manifolds.
method Intersection numbers and degenerate concentration of mass result.
result Proves existence of smooth solutions for generalised Monge-Ampère equations on projective manifolds.
In this paper we provide an explicit formula for the optimal lower bound of Donaldson's J-functional, in the sense of finding explicitly the optimal constant in the definition of coercivity, which always exists and takes negative values in general. This constant is positive precisely if the J-equation admits a solution…
Solves complex Hessian equations in unstable cases, proving unique canonical solutions with singularities.
problem Existence of smooth solutions to complex Hessian equations in unstable cases.
method Parabolic flows and moment-map energy functionals, focusing on J-equation and deformed Hermitian Yang-Mills equation.
result Proves existence of unique canonical solutions with singularities on Kahler surfaces.
Study of J-flow on Kähler manifolds confirms energy properness.
problem Properness of Mabuchi K-energy on Kähler manifolds.
method Degenerate twisted J-flow on compact Kähler manifolds.
result Flow converges to weak solution of degenerate twisted J-equation.
Study proves estimate for Hessian quotient equations on 2D Riemannian manifolds.
problem Problems posed by Delanoë and Urbas related to Hessian quotient equations.
method Maximum principle argument and new test function introduced to prove estimate.
result Unobstructed second order a priori estimate for real Hessian quotient equation in 2D.
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 C C such that $-\…
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.
Study of singularity formation in dHYM flow on a blown-up CP^3.
problem Analyzing singularity formation in the dHYM flow on a blown-up CP^3.
method Using Calabi ansatz, the paper studies the singularity formation of the dHYM cotangent flow on the one-point blow up of CP^3.
result An explicit example of singularity formation along the exceptional divisor, with a limit satisfying the corresponding singular dHYM equation.
The paper characterizes when numerical criteria for PDE solvability fail and provides effective criteria for existence.
problem Characterizing when numerical criteria for PDE solvability fail.
method Finite number of subvarieties violating Nakai type criterion, and their rigidity.
result Finite number of subvarieties violating the Nakai type criterion, and these subvarieties are rigid.
Proves convexity of level sets of general inverse σ_k equations.
problem Convexity of level sets of general inverse σ_k equations.
method Analyzes level sets of degree n general inverse σ_k equations and uses numerical conditions to verify convexity.
result Proves convexity of level sets of general inverse σ_k equations.
Develops methods to solve complex and real Hessian equations.
problem Solving complex and real Hessian equations on various domains.
method Introduces an ansatz to reduce PDEs to systems of ODEs, integrating via abelian integrals.
result Constructs entire solutions of arbitrary subcritical phase for dHYM/LYZ and special Lagrangian equations.