2022/01/05 by Chao-Ming Lin, Lin, Chao-Ming · 2 citations
Mathematics · #32Q1 #32W50 #53C55 #Advanced Topics in Algebra #Algebraic Geometry and Number Theory #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #math.AP #math.DG #msc:32Q1 #msc:32W50 #msc:53C55
paper · pdf · doi:10.48550/arxiv.2201.01438
56 pages, 7 figures; in version 2, we fully resolve the conjecture when the complex dimension equals four
openalex publication_date 2022/01/05 · arxiv created 2022/01/09 · arxiv updated 2022/01/11 · openalex created_date 2022/04/03 · openalex updated_date 2026/07/28
Let (M, ω) be a compact connected Kähler manifold of complex dimension four and let [χ] ∈ H1,1(M; ℝ). We confirmed the conjecture by Collins--Jacob--Yau [arXiv:1508.01934] of the solvability of the deformed Hermitian--Yang--Mills equation, which is given by the following nonlinear elliptic equation ∑i \arctan (λi) = θ, where λi are the eigenvalues of χ with respect to ω and θ is a topological constant. This conjecture was stated in [arXiv:1508.01934], wherein they proved that the existence of a supercritical C-subsolution or the existence of a C-suboslution when θ ∈ [ ( (n-2) + 2/n ) π/2, nπ/2 ) will give the solvability of the deformed Hermitian--Yang--Mills equation. Collins--Jacob--Yau conjectured that their existence theorem can be improved when θ ∈ ( (n-2 ) π/2, ( (n-2) + 2/n ) π/2 ), where n is the complex dimension of the manifold. In this paper, we confirmed their conjecture that when the complex dimension equals four and θ is close to the supercritical phase π from the right, then the existence of a C-subsolution implies the solvability of the deformed Hermitian--Yang--Mills equation.